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+# Special Functions
+
+_Author’s note—This RFC is a work-in-progress._
+
+## Authors
+
+* Allen Goodman (@0x00b1)
+
+## Summary
+
+This proposal concerns adding new operators to PyTorch's special functions module (i.e., `torch.special`). The proposed operators have a wide range of use in scientific computing and numerical methods.
+
+This RFC proposes:
+
+* a coherent philosophy for PyTorch’s special functions module ([torch.special](https://pytorch.org/docs/stable/special.html)) that clearly distinguishes PyTorch’s elementary from special functions; and
+* a set of new [torch](https://pytorch.org/docs/stable/torch.html) and [torch.special](https://pytorch.org/docs/stable/special.html) operators that provide a robust numerical foundation for PyTorch and adhere to the aforementioned philosophy.
+
+## Motivation
+
+PyTorch users:
+
+* solves the variety of common scientific and engineering problems that special functions address.
+
+PyTorch maintainers:
+
+* provides much needed standardization to committing future operators to PyTorch.
+* provides an extremely useful set of operators that can and should be used for tricky numerical problems (e.g., implementing challenging distribution functions and gradients) and useful decomposition targets.
+
+## Special Functions
+
+There’s no formal definition of a *special function*. Colloquially, and for the purpose of this RFC, a special function is a non-elementary function that has an established name and notation due to its importance and ubiquity.
+
+### Elementary Functions
+
+Unlike special functions, *elementary functions* have a rigorous definition but, for simplicity, this RFC uses a simplified definition, categorizing the following functions as elementary:
+
+* Power Functions
+ * torch.pow
+ * torch.sqrt
+* Exponential Functions
+ * torch.exp
+ * torch.exp2
+ * torch.expm1
+* Logarithmic Functions
+ * torch.log
+ * torch.log10
+ * torch.log1p
+ * torch.log2
+ * torch.xlog1py
+ * torch.xlogy
+* Trigonometric Functions
+ * torch.sin
+ * torch.cos
+ * torch.tan
+ * torch.cot
+ * torch.csc
+ * torch.sec
+* Inverse Trigonometric Functions
+ * torch.asin
+ * torch.acos
+ * torch.atan
+ * torch.atan
+ * torch.acot
+ * torch.acsc
+ * torch.asec
+* Hyperbolic Functions
+ * torch.sinh
+ * torch.cosh
+ * torch.tanh
+ * torch.coth
+ * torch.csch
+ * torch.sech
+* Inverse Hyperbolic Functions
+ * torch.asinh
+ * torch.acosh
+ * torch.atanh
+ * torch.atanh
+ * torch.acoth
+ * torch.acsch
+ * torch.asech
+* Product Logarithms
+* Roots
+ * `torch.root` *See the “Polynomial Root” section for the proposed operator*
+* Maximum and Minimum
+ * torch.max
+ * torch.min
+* Cardinal Functions
+ * torch.sinc
+* Greatest Common Divisor (GCD) and Least Common Multiple (LCM)
+ * torch.gcd
+ * torch.lcm
+* Rounding and Congruence Functions
+ * torch.ceil
+ * torch.floor
+ * torch.floor_divide
+ * torch.remainder
+ * torch.round
+* Discontinuous Functions
+* Functions with Singular Support
+
+### Special Functions
+
+PyTorch’s mathematical operators are categorized as either “elementary” or “special.”
+
+An elementary function is a mathematical function whose corresponding operator is available from the `torch` module. A special function is a mathematical function whose corresponding operator is available from the `torch.special` module. Regardless of whether an operator implements an elementary or special function, users should expect that each operator share the following properties:
+
+* A name that adheres to the naming policy.
+* CPU and CUDA implementations
+* C++ and Python function documentation that state a function’s:
+ * common mathematical definition;
+ * real and complex domains;
+ * real and complex graphs; and, if relevant,
+ * mathematical and scientific applications.
+* If differentiable, derivtatives for each variable.
+
+#### Naming Policy
+
+## Implementation
+
+### c10
+
+#### Constants
+
+To ease implementation, c10 will provide the following lookup tables (LUTs):
+
+* factorial, $n!$, where $n = 1, 2, \cdots, 170;$
+* natural logarithm of factorial, $\ln{\left(n!\right)}$, where $n = 1, 2, \cdots, 300;$
+* double factorial, $n!!$, where $n = -1,000, -999, \cdots, 300;$
+* natural logarithm of double factorial, $\ln{\left(n!!\right)}$, where $n = -1,000, -999, \cdots, 300;$ and
+* prime number, $p\left(n\right)$, where $n = 1, 2, \cdots, 10,000.$
+
+#### Functions
+
+To ease implementation, c10 will provide the following numerical approximations:
+
+* Lanczos approximation;
+* Spouge’s approximation; and
+* Stirling’s approximation.
+
+##### Lanczos Approximation
+
+Numerical method for computing the gamma function. It exhibits faster convergence than Spouge’s approximation.
+
+##### Spouge’s approximation
+
+Numerical method for computing the gamma function. Its coefficients are faster to compute than Lanczos approximation coefficients.
+
+##### Stirling’s Approximation
+
+Numerical method for computing factorials.
+
+### ATen
+
+Each special function has a corresponding ATen operator.
+
+#### CPU
+
+#### CUDA
+
+### torch
+
+## C++ API
+
+## Python API
+
+
+Factorials
+
+### Factorials
+
+#### Factorial
+
+```Python
+factorial(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+$n^{\text{th}}$ factorial, if $n \in \mathbb{N}$:
+
+$$n! = \prod_{k = 1}^{n}k.$$
+
+Otherwise:
+
+$$n! = \Gamma(n + 1),$$
+
+where $\Gamma$ is the gamma function.
+
+$n!$ is defined for $\left\\{n \in \mathbb{R} \mid n \geq 0 \vee n \notin \mathbb{Z} \right\\}$ and $\left\\{n \in \mathbb{C} \mid \operatorname{Re}(n) \geq 0 \vee n \notin \mathbb{Z} \right\\}$.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Natural Logarithm of Factorial
+
+```Python
+ln_factorial(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Natural logarithm of $n^{\text{th}}$ factorial, $\ln{(n!)}$.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Double Factorial
+
+```Python
+double_factorial(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+$n^{\text{th}}$ double factorial, if $n \in \mathbb{N}$:
+
+$$n!!=\prod_{k = 0}^{\left\lceil\tfrac{n}{2}\right\rceil-1}(n-2k).$$
+
+Otherwise:
+
+$$n!!=\left(\frac{2}{\pi}\right)^{\frac{1}{4}(1-\cos(\pi n))}2^{\tfrac{2}{n}}\Gamma\left(\frac{n}{2}+1\right),$$
+
+where $\Gamma$ is the gamma function.
+
+$n!!$ is defined for $\left\\{n \in \mathbb{R} \mid n \geq 0 \vee \tfrac{n}{2} \notin \mathbb{Z} \right\\}$ and $\left\\{n \in \mathbb{C} \mid n \geq 0 \vee \tfrac{n}{2} \notin \mathbb{Z} \right\\}$.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Natural Logarithm of Double Factorial
+
+```Python
+ln_double_factorial(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Natural logarithm of $n^{\text{th}}$ double factorial, $\ln{(n!!)}$.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Rising Factorial
+
+```Python
+rising_factorial(
+ z: Tensor,
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Rising factorial:
+
+$$z^{n}=\frac{\Gamma(z + n)}{\Gamma(z)},$$
+
+where $\Gamma$ is the gamma function.
+
+$z^{n}$ is defined for all real and complex $z$.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $n$ is a number, $z$ must be a tensor.
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – exponent. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Natural Logarithm of Rising Factorial
+
+```Python
+ln_rising_factorial(
+ z: Tensor,
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Natural logarithm of rising factorial, $\operatorname{ln}{(z^{n})}$.
+
+$\operatorname{ln}{(z^{n})}$ is defined for all real and complex $z$.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $n$ is a number, $z$ must be a tensor.
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – exponent. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Falling Factorial
+
+```Python
+falling_factorial(
+ z: Tensor,
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Falling factorial:
+
+$$z_{n}=\frac{\Gamma(z + 1)}{\Gamma(z - n + 1)},$$
+
+where $\Gamma$ is the gamma function.
+
+$z_{n}$ is defined for all real and complex $z$.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $n$ is a number, $z$ must be a tensor.
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – exponent. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Natural Logarithm of Falling Factorial
+
+```Python
+ln_falling_factorial(
+ z: Tensor,
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Natural logarithm of falling factorial, $\operatorname{ln}{(z_{n})}$.
+
+$\operatorname{ln}{(z_{n})}$ is defined for all real and complex $z$.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $n$ is a number, $z$ must be a tensor.
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – exponent. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Combinatorial Numbers and Functions
+
+### Combinatorial Numbers and Functions
+
+#### Binomial Coefficient
+
+```Python
+binomial_coefficient(
+ n: Tensor,
+ k: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Binomial coefficient:
+
+$${\binom{n}{k}} = {\frac{n!}{k!(n - k)!}}.$$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $k$ is a number, $n$ must be a tensor.
+
+**k** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – exponent. If $n$ is a number, $k$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Natural Logarithm of Binomial Coefficient
+
+```Python
+ln_binomial_coefficient(
+ n: Tensor,
+ k: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Natural logarithm of binomial coefficient, $\operatorname{ln}{{\binom{n}{k}}}$.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $k$ is a number, $n$ must be a tensor.
+
+**k** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – exponent. If $n$ is a number, $k$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Catalan Number
+
+```Python
+catalan_number_c(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+$n^{\text{th}}$ Catalan number:
+
+$$C_{z} = \frac{2^{2z}\Gamma(z + \tfrac{1}{2})}{\sqrt{\pi} \Gamma(z + 2)}.$$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Stirling Number of the First Kind
+
+```Python
+stirling_number_s_1(
+ n: Tensor,
+ m: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Stirling number of the first kind:
+
+$$s(n, k) = $$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Stirling Number of the Second Kind
+
+```Python
+stirling_number_s_2(
+ n: Tensor,
+ m: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Stirling number of the second kind:
+
+$$S(n, k) = {\frac{1}{k!}}\sum_{i = 0}^{k}(-1)^{i}{\binom{k}{i}}(k - i)^{n}.$$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Bell Number
+
+```Python
+bell_number_b(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+$n^{\text{th}}$ Bell number:
+
+$$B_{n}=\sum_{k = 0}^{n}S(n, k).$$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Delannoy Number
+
+```Python
+delannoy_number_d(
+ m: Tensor,
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Delannoy number:
+
+$$D(m, n) = \sum_{k = 0}^{\min(m, n)}\binom{m + n - k}{m}\binom{m}{k}.$$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Motzkin Number
+
+```Python
+motzkin_number_m(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+$n^{\text{th}}$ Motzkin number:
+
+$$M_{n} = \sum_{k = 0}^{\lfloor \frac{n}{2} \rfloor}{\binom{n}{2k}}C_{k}.$$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Narayana Number
+
+```Python
+narayana_number_n(
+ n: Tensor,
+ k: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Narayana number:
+
+$$N(n, k) = {\frac{1}{n}}\binom{n}{k}\binom{n}{k - 1}.$$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Schröder Number
+
+```Python
+schroder_number_r(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+$n^{\text{th}}$ Schröder number:
+
+$$r_{n} = D(n, n) - D(n + 1, n - 1).$$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Gamma and Related Functions
+
+### Gamma and Related Functions
+
+#### Gamma Function
+
+```Python
+gamma(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Gamma function:
+
+$$\Gamma(z)=\int_{0}^{\infty}t^{z-1}e^{-t}dt.$$
+
+$\Gamma(z)$ is defined for $\left\\{n \in \mathbb{R} \mid n > 0 \vee n \notin \mathbb{Z} \right\\}$ and $\left\\{n \in \mathbb{C} \mid \operatorname{Re}(n) > 0 \vee n \notin \mathbb{Z} \right\\}$.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Reciprocal Gamma Function
+
+```Python
+reciprocal_gamma(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Polygamma Function
+
+```Python
+polygamma(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Polygamma function, if, and only if, $n \in \mathbb{Z} \land n > 0$:
+
+$$\psi^{n}(z)=(-1)^{n+1} n! \sum _{k=0}^{\infty }\frac{1}{(k+z)^{n+1}}.$$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – derivative.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Digamma Function
+
+```Python
+digamma(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Digamma function:
+
+$$\psi(z)=\sum _{k=1}^{\infty } \left(\frac{1}{k}-\frac{1}{k+z-1}\right)-\gamma.$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Trigamma Function
+
+```Python
+trigamma(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Logarithm of the Gamma Function
+
+```Python
+log_gamma(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Logarithm of the gamma function:
+
+$$\log{\Gamma}(z)=\sum _{k=1}^{\infty } \left(\frac{z}{k}-\log \left(1+\frac{z}{k}\right)\right)-\gamma z-\log (z).$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Sign of the Gamma Function
+
+```Python
+sign_gamma(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Beta Function
+
+```Python
+beta(
+ a: Tensor,
+ b: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Beta function:
+
+$$\operatorname{B}(a, b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a + b)}$$
+
+where $\Gamma$ is the gamma function.
+
+##### Parameters
+
+**a** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**b** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Logarithm of the Beta Function
+
+```Python
+log_beta(
+ a: Tensor,
+ b: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**a** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**b** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Exponential and Logarithmic Integrals
+
+### Exponential and Logarithmic Integrals
+
+#### Exponential Integral, $\operatorname{Ein}$
+
+```Python
+exponential_integral_ein(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Exponential integral:
+
+$$\operatorname{Ein}(z)=\int_{0}^{z}(1-e^{-t}){\frac{dt}{t}}.$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Exponential Integral, $\operatorname{Ei}$
+
+```Python
+exponential_integral_ei(
+ x: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Exponential integral:
+
+$$\operatorname{Ei}(z) = \sum_{k = 1}^{\infty} \frac{z^{k}}{k k!} + \gamma + \frac{1}{2}(\ln{(z)} - \ln{(\tfrac{1}{z})}).$$
+
+##### Parameters
+
+**x** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Exponential Integral, $E_{1}$
+
+```Python
+exponential_integral_e_1(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Exponential integral:
+
+$$E_{1}(z)=\int _{z}^{\infty}{\frac{e^{-t}}{t}}.$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Exponential Integral, $E_{n}$
+
+```Python
+exponential_integral_e(
+ n: Tensor,
+ x: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Exponential integral:
+
+$$E_{n}(x)=\int_{1}^{\infty}{\frac{e^{-xt}}{t^{n}}}dt.$$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**x** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Logarithmic Integral
+
+```Python
+logarithmic_integral_li(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Logarithmic integral:
+
+$$\operatorname{li}(z)=\int_{0}^{z}{\frac{1}{\ln{(t)}}}dt.$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Error and Related Functions
+
+### Error and Related Functions
+
+#### Error Function
+
+```Python
+error_erf(z: Tensor, *, out: Optional[Tensor] = None) -> Tensor
+```
+
+Error function:
+
+$$\operatorname{erf}(z) = \frac{2}{\sqrt{\pi}} \sum_{k = 0}^{\infty } \frac{-1^{k} z^{2k + 1}}{k!(2k + 1)}.$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Complementary Error Function
+
+```Python
+error_erfc(z: Tensor, *, out: Optional[Tensor] = None) -> Tensor
+```
+
+Complementary error function:
+
+$$\operatorname{erfc}(z) = 1 - \frac{2}{\sqrt{\pi}} \sum_{k = 0}^{\infty} \frac{-1^{k} z^{2k + 1}}{k!(2k + 1)}.$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Imaginary Error Function
+
+```Python
+error_erfi(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Imaginary error function:
+
+$$\operatorname{erfc}(z) = \frac{2}{\sqrt{\pi}} \sum_{k = 0}^{\infty} \frac{z^{2k + 1}}{k!(2k + 1)}.$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Inverse Error Function
+
+```Python
+error_inverse_erf(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Inverse error function:
+
+$$\operatorname{erf}^{-1}(z)=\sum_{k=0}^{\infty}{\frac{c_{k}}{2k+1}}({\frac{\sqrt{\pi}}{2}}z)^{2k+1}$$
+
+where $c_{0}=1$ and:
+
+$$c_{k}=\sum_{m=0}^{k-1}{\frac{c_{m}c_{k-1-m}}{(m+1)(2m+1)}}.$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Inverse Complementary Error Function
+
+```Python
+error_inverse_erfc(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Dawson and Fresnel Integrals
+
+### Dawson and Fresnel Integrals
+
+#### Dawson’s Integral
+
+```Python
+dawson_integral_f(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Dawson’s integral:
+
+$$\operatorname{F}(z)=e^{-z^{2}}\int_{0}^{z}e^{t^{2}}dt.$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Sine Fresnel Integral
+
+```Python
+fresnel_integral_s(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Fresnel integral:
+
+$$\operatorname{S}(z)=\int_{0}^{x}\sin{t^{2}}dt.$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Cosine Fresnel Integral
+
+```Python
+fresnel_integral_c(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Fresnel integral:
+
+$$\operatorname{C}(z)=\int_{0}^{x}\cos{t^{2}}dt.$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Trigonometric and Hyperbolic Integrals
+
+### Trigonometric and Hyperbolic Integrals
+
+#### Sine Integral ($operatorname{Sin}$)
+
+```Python
+sine_integral_sin(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Sine integral:
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Sine Integral ($\operatorname{Si}$)
+
+$$\operatorname{Sin}(z)=\int_{0}^{z}{\frac{\sin{t}}{t}}dt.$$
+
+```Python
+sine_integral_si(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Sine integral:
+
+$$\operatorname{Si}(z)=-\int_{z}^{\infty }{\frac{\sin{t}}{t}}dt.$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Cosine Integral ($\operatorname{Cin}$)
+
+```Python
+cosine_integral_cin(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Cosine integral:
+
+$$\operatorname{Cin}(z)=\int_{0}^{z}{\frac{1-\cos{t}}{t}}dt.$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Cosine Integral ($\operatorname{Ci}$)
+
+```Python
+cosine_integral_ci(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Cosine integral:
+
+$$\operatorname{Ci}(z)=\gamma+\ln{z}-\int_{0}^{z}{\frac{1-\cos{t}}{t}}$$
+
+where $\gamma$ is the Euler–Mascheroni constant.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Hyperbolic Sine Integral
+
+```Python
+hyperbolic_sine_integral_shi(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Hyperbolic sine integral:
+
+$$\operatorname{Shi}(z)=\int_{0}^{z}{\frac{\sinh{t}}{t}}dt.$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Hyperbolic Cosine Integral
+
+```Python
+hyperbolic_cosine_integral_chi(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Hyperbolic cosine integral:
+
+$$\operatorname{Chi}(z)=\gamma+\ln{z}+\int_{0}^{z}{\frac{\cosh{t-1}}{t}}dt.$$
+
+where $\gamma$ is the Euler–Mascheroni constant.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Incomplete Gamma and Related Functions
+
+### Incomplete Gamma and Related Functions
+
+#### Incomplete Gamma Function ($\gamma$)
+
+```Python
+lower_incomplete_gamma(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Lower incomplete gamma function:
+
+$$\Gamma(s,z)=\int_{z}^{\infty}t^{s-1}e^{-t}dt.$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Incomplete Gamma Function ($\Gamma$)
+
+```Python
+upper_incomplete_gamma(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Upper incomplete gamma function:
+
+$$\gamma(s,z)=\int_{0}^{z}t^{s-1}e^{-t}dt.$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Incomplete Beta Function
+
+```Python
+incomplete_beta(
+ z: Tensor,
+ a: Tensor,
+ b: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+**a** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**b** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Airy Functions
+
+### Airy Functions
+
+Functions defined as the two, linearly independent solutions to:
+
+$$y'' - yz = 0.$$
+
+#### Airy Function of the First Kind
+
+```Python
+airy_ai(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Airy function of the first kind:
+
+$$\operatorname{Ai}(z)={\frac{1}{3^{\tfrac{2}{3}}\Gamma(\tfrac{2}{3})}} {_0F_1(; \tfrac{2}{3}; \tfrac{1}{9}; z^{3})} - \frac{z}{3^{\tfrac{1}{3}}\Gamma(\tfrac{1}{3})} {_0F_1(; \tfrac{4}{3}; \tfrac{1}{9}; z^{3})}$$
+
+where $\Gamma$ is the gamma function and $_0F_1(; a; z)$ is the confluent hypergeometric limit function.
+
+$\operatorname{Ai}(z)$ is defined for all real and complex values.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Airy Function of the Second Kind
+
+```Python
+airy_bi(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Airy function of the second kind:
+
+$$\operatorname{Bi}(z)=\frac{_0F_1\left(;\frac{2}{3};\frac{z^3}{9}\right)}{\sqrt[6]{3} \Gamma \left(\frac{2}{3}\right)}+\frac{\sqrt[6]{3} z _0F_1\left(;\frac{4}{3};\frac{z^3}{9}\right)}{\Gamma \left(\frac{1}{3}\right)}$$
+
+where $\Gamma$ is the gamma function and $_0F_1(; a; z)$ is the confluent hypergeometric limit function.
+
+$\operatorname{Bi}(z)$ is defined for all real and complex values.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Derivative of the Airy Function of the First Kind
+
+```Python
+airy_ai_derivative(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Derivative of the Airy function of the first kind:
+
+$$\operatorname{Ai}'(z)=\frac{z^2 \\,_0F_1\left(;\frac{5}{3};\frac{z^3}{9}\right)}{2\ 3^{\tfrac{2}{3}} \Gamma \left(\frac{2}{3}\right)}-\frac{\\, _0F_1\left(;\frac{1}{3};\frac{z^3}{9}\right)}{\sqrt[3]{3} \Gamma \left(\frac{1}{3}\right)},$$
+
+where $\Gamma$ is the gamma function and $_0F_1(; a; z)$ is the confluent hypergeometric limit function.
+
+$\operatorname{Ai}'(z)$ is defined for all real and complex values.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Derivative of the Airy Function of the Second Kind
+
+```Python
+airy_bi_derivative(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Derivative of the Airy function of the second kind:
+
+$$\operatorname{Bi}'(z)=\frac{z^2 \\, _0F_1\left(;\frac{5}{3};\frac{z^3}{9}\right)}{2 \sqrt[6]{3} \\, \Gamma \left(\frac{2}{3}\right)}+\frac{\sqrt[6]{3} \\, _0F_1\left(;\frac{1}{3};\frac{z^3}{9}\right)}{\Gamma \left(\frac{1}{3}\right)},$$
+
+where $\Gamma$ is the gamma function and $_0F_1(; a; z)$ is the confluent hypergeometric limit function.
+
+$\operatorname{Bi}'(z)$ is defined for all real and complex values.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Exponentially Scaled Airy Function of the First Kind
+
+```Python
+exp_airy_ai(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Exponentially scaled Airy function of the first kind, $\operatorname{exp}(\operatorname{Ai}(z))$.
+
+$\exp{(\operatorname{Ai}(z))}$ is defined for all real and complex values.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Exponentially Scaled Airy Function of the Second Kind
+
+```Python
+exp_airy_bi(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Exponentially scaled Airy function of the second kind, $\operatorname{exp}(\operatorname{Bi}(z))$.
+
+$\exp{(\operatorname{Bi}(z))}$ is defined for all real and complex values.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Exponentially Scaled Derivative of the Airy Function of the First Kind
+
+```Python
+exp_airy_ai_derivative(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Exponentially scaled Derivative of the Airy function of the first kind, $\operatorname{exp}(\operatorname{Ai'}(z))$.
+
+$\exp{(\operatorname{Ai}'(z))}$ is defined for all real and complex values.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Exponentially Scaled Derivative of the Airy Function of the Second Kind
+
+```Python
+exp_airy_bi_derivative(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Exponentially scaled Derivative of the Airy function of the second kind, $\operatorname{exp}(\operatorname{Bi'}(z))$.
+
+$\exp{(\operatorname{Bi}'(z))}$ is defined for all real and complex values.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Scorer’s Functions
+
+### Scorer’s Functions
+
+#### Scorer’s Function $\left(\operatorname{Gi}\right)$
+
+```Python
+scorer_gi(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Scorer’s function:
+
+$$\mathrm{Gi}(z)=\frac{1}{\pi}\int_0^{\infty}\sin\left(\frac{t^3}{3}+zt\right)\text{d}t.$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Scorer’s Function $\left(\operatorname{Hi}\right)$
+
+```Python
+scorer_hi(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Scorer’s function:
+
+$$\mathrm{Hi}(z)=\frac{1}{\pi}\int_0^{\infty}\exp\left(-\frac{t^3}{3}+zt\right)\text{d}t.$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Bessel Functions
+
+### Bessel Functions
+
+#### Bessel Function of the First Kind
+
+```Python
+bessel_j(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Bessel function of the first kind:
+
+$$J_{n}(z)=\sum _{k=0}^{\infty } \frac{(-1)^k \left(\frac{z}{2}\right)^{2k+n}}{\Gamma (k+\nu +1) k!},$$
+
+where $\Gamma$ is the gamma function.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – order. If $z$ is a number, $n$ must be a tensor.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $n$ is a number, $z$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Bessel Function of the First Kind of Order 0
+
+```Python
+bessel_j_0(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Bessel function of the first kind of order $0$, $J_{0}(z)$
+
+$J_{0}(z)$ is defined for all real and complex $z$.
+
+#### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Bessel Function of the First Kind of Order 1
+
+```Python
+bessel_j_1(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Bessel function of the first kind of order $1$, $J_{1}(z)$
+
+$J_{1}(z)$ is defined for all real and complex $z$.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Bessel Function of the Second Kind
+
+```Python
+bessel_y(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Bessel function of the second kind, if, and only if $\nu \notin \mathbb{Z}$:
+
+$$Y_{\nu }(z)=\csc (\pi \nu ) (\cos (\nu \pi ) J_{\nu }(z)-J_{-\nu }(z)),$$
+
+where $J_{n}(z)$ is the Bessel function of the first kind.
+
+If $z \in \mathbb{R}$, $Y_{n}(z)$ is defined for $z > 0$.
+
+If $z \in \mathbb{C}$, $Y_{n}(z)$ is defined for $z \neq 0$.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – order. If $z$ is a number, $n$ must be a tensor.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $n$ is a number, $z$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Bessel Function of the Second Kind of Order 0
+
+```Python
+bessel_y_0(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Bessel function of the second kind of order $0$, $Y_{0}(z)$.
+
+$Y_{0}(z)$ is defined for $\\{z \in \mathbb{R}\\}$ and $\\{z \in \mathbb{C} \mid z \neq 0\\}$.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Bessel Function of the Second Kind of Order 1
+
+```Python
+bessel_y_1(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Bessel function of the second kind of order $1$, $Y_{1}(z)$.
+
+$Y_{1}(z)$ is defined for $\\{z \in \mathbb{R} \mid z > 0\\}$ and $\\{z \in \mathbb{C} \mid z \neq 0\\}$.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Hankel Functions
+
+### Hankel Functions
+
+#### Hankel Function of the First Kind
+
+```Python
+hankel_h_1(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Hankel function of the first kind:
+
+$$H_{n}^{1}(z) = J_{n}(z) + i Y_{n}(z),$$
+
+where $J_{n}(z)$ is the Bessel function of the first kind, $i$ is the imaginary unit, and $Y_{n}(z)$ is the Bessel function of the second kind.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – order. If $z$ is a number, $n$ must be a tensor.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Hankel Function of the Second Kind
+
+```Python
+hankel_h_2(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Hankel function of the second kind:
+
+$$H_{n}^{2}(z) = J_{n}(z) - i Y_{n}(z),$$
+
+where $J_{n}(z)$ is the Bessel function of the first kind, $i$ is the imaginary unit, and $Y_{n}(z)$ is the Bessel function of the second kind.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – order. If $z$ is a number, $n$ must be a tensor.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Modified Bessel Functions
+
+### Modified Bessel Functions
+
+#### Modified Bessel Function of the First Kind
+
+```Python
+modified_bessel_i(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Modified Bessel function of the first kind:
+
+$$I_{n}(z)=\sum _{k=0}^{\infty} \frac{\left(\frac{z}{2}\right)^{2k+n}}{\Gamma(k+n+1) k!},$$
+
+where $n!$ is the $n^{\text{th}}$ factorial and $\Gamma{\left(z\right)}$ is the gamma function.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – order. If $z$ is a number, $n$ must be a tensor.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Modified Bessel Function of the First Kind of Order 0
+
+```Python
+modified_bessel_i_0(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Modified Bessel function of the first kind of order $0$, $I_{0}(z)$.
+
+$I_{0}(z)$ is defined for all real and complex $z$.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Modified Bessel Function of the First Kind of Order 1
+
+```Python
+modified_bessel_i_1(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Modified Bessel function of the first kind of order $1$, $I_{1}(z)$.
+
+$I_{1}(z)$ is defined for all real and complex $z$.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Modified Bessel Function of the Second Kind
+
+```Python
+modified_bessel_k(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Modified Bessel function of the second kind:
+
+$$K_{n}(z) = \frac{1}{2} \pi i^{n + 1} H_n^{1}(i z),$$
+
+where $H_{n}^{1}(z)$ is the Hankel function of the first kind.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – order. If $z$ is a number, $n$ must be a tensor.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Modified Bessel Function of the Second Kind of Order 0
+
+```Python
+modified_bessel_k_0(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Modified Bessel function of the first kind of order $0$, $K_{0}(z)$.
+
+$K_{0}(z)$ is defined for $\\{z \in \mathbb{R} \mid z > 0\\}$ and $\\{z \in \mathbb{C} \mid z \neq 0\\}$.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Modified Bessel Function of the Second Kind of Order 1
+
+```Python
+modified_bessel_k_1(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Modified Bessel function of the first kind of order $1$, $K_{1}(z)$.
+
+$K_{1}(z)$ is defined for $\\{z \in \mathbb{R} \mid z > 0\\}$ and $\\{z \in \mathbb{C} \mid z \neq 0\\}$.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Spherical Bessel Functions
+
+### Spherical Bessel Functions
+
+#### Spherical Bessel Function of the First Kind
+
+```Python
+spherical_bessel_j(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Spherical Bessel function of the first kind:
+
+$$j_{n}(x)={\sqrt{\frac{\pi}{2x}}}J_{n + {\frac{1}{2}}}(x).$$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – order. If $z$ is a number, $n$ must be a tensor.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Spherical Bessel Function of the First Kind of Order 0
+
+```Python
+spherical_bessel_j_0(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Spherical Bessel function of the first kind:
+
+$$j_{n}(x)={\sqrt{\frac{\pi}{2x}}}J_{n + {\frac{1}{2}}}(x)$$
+
+where $n = 0$.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Spherical Bessel Function of the First Kind of Order 1
+
+```Python
+spherical_bessel_j_1(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Spherical Bessel function of the first kind:
+
+$$j_{n}(x)={\sqrt{\frac{\pi}{2x}}}J_{n + {\frac{1}{2}}}(x)$$
+
+where $n = 1$.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Spherical Bessel Function of the Second Kind
+
+```Python
+spherical_bessel_y(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Spherical Bessel function of the second kind:
+
+$$y_{n}(x)={\sqrt{\frac{\pi}{2x}}}Y_{n+{\frac{1}{2}}}(x).$$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – order. If $z$ is a number, $n$ must be a tensor.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Spherical Bessel Function of the Second Kind of Order 0
+
+```Python
+spherical_bessel_y_0(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Spherical Bessel function of the second kind:
+
+$$y_{n}(x)={\sqrt{\frac{\pi}{2x}}}Y_{n+{\frac{1}{2}}}(x)$$
+
+where $n = 0$.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Spherical Bessel Function of the Second Kind of Order 1
+
+```Python
+spherical_bessel_y_1(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Spherical Bessel function of the second kind:
+
+$$y_{n}(x)={\sqrt{\frac{\pi}{2x}}}Y_{n+{\frac{1}{2}}}(x)$$
+
+where $n = 1$.
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Spherical Hankel Functions
+
+### Spherical Hankel Functions
+
+#### Spherical Hankel Function of the First Kind
+
+```Python
+spherical_hankel_h_1(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Spherical Hankel function of the first kind:
+
+$$h_{n}^{1}(z)=\frac{\sqrt{\frac{\pi}{2}}H_{n +\frac{1}{2}}^{1}(z)}{\sqrt{z}},$$
+
+where $H_{n}^{1}$ is the Hankel function of the first kind.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – order. If $z$ is a number, $n$ must be a tensor.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Spherical Hankel Function of the Second Kind
+
+```Python
+spherical_hankel_h_2(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Spherical Hankel function of the second kind:
+
+$$h_{n}^{2}(z)=\frac{\sqrt{\frac{\pi}{2}}H_{n+\frac{1}{2}}^{2}(z)}{\sqrt{z}},$$
+
+where $H_{n}^{2}$ is the Hankel function of the second kind.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – order. If $z$ is a number, $n$ must be a tensor.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Modified Spherical Bessel Functions
+
+### Modified Spherical Bessel Functions
+
+#### Modified Spherical Bessel Function of the First Kind
+
+```Python
+modified_spherical_bessel_i(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Modified spherical Bessel function of the first kind:
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – order. If $z$ is a number, $n$ must be a tensor.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Modified Spherical Bessel Function of the First Kind of Order 0
+
+```Python
+modified_spherical_bessel_i_0(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Modified Spherical Bessel Function of the First Kind of Order 1
+
+```Python
+modified_spherical_bessel_i_1(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Modified Spherical Bessel Function of the Second Kind
+
+```Python
+modified_spherical_bessel_k(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – order. If $z$ is a number, $n$ must be a tensor.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Modified Spherical Bessel Function of the Second Kind of Order 0
+
+```Python
+modified_spherical_bessel_k_0(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Modified Spherical Bessel Function of the Second Kind of Order 1
+
+```Python
+modified_spherical_bessel_k_1(
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Kelvin Functions
+
+### Kelvin Functions
+
+#### Kelvin Function of the First Kind ($\operatorname{ber}$)
+
+```Python
+kelvin_ber(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+$$\mathrm {ber} _{n}(x)=({\frac {x}{2}})^{n}\sum _{k\geq 0}{\frac {\cos [({\frac {3n}{4}}+{\frac {k}{2}})\pi ]}{k!\Gamma (n+k+1)}}({\frac {x^{2}}{4}})^{k}.$$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – order. If $z$ is a number, $n$ must be a tensor.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Kelvin Function of the First Kind ($\operatorname{bei}$)
+
+```Python
+kelvin_bei(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – order. If $z$ is a number, $n$ must be a tensor.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Kelvin Function of the Second Kind ($\operatorname{kei}$)
+
+```Python
+kelvin_kei(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – order. If $z$ is a number, $n$ must be a tensor.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Kelvin Function of the Second Kind ($\operatorname{ker}$)
+
+```Python
+kelvin_ker(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – order. If $z$ is a number, $n$ must be a tensor.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Struve and Modified Struve Functions
+
+### Struve and Modified Struve Functions
+
+#### Struve Function
+
+```Python
+struve_h(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Struve function:
+
+$$\mathbf{H}_{n}(z)=\left(\frac{z}{2}\right)^{n+1} \sum _{k=0}^{\infty } \frac{(-1)^k \left(\frac{z}{2}\right)^{2 k}}{\Gamma \left(k+\frac{3}{2}\right) \Gamma \left(k+n+\frac{3}{2}\right)},$$
+
+where $\Gamma(z)$ is the gamma function.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – order. If $z$ is a number, $n$ must be a tensor.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Modified Struve Function
+
+```Python
+modified_struve_l(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Modified Struve function:
+
+$$\mathbf{L}_{n}(z)=\left(\frac{z}{2}\right)^{n+1} \sum _{k=0}^{\infty } \frac{\left(\frac{z}{2}\right)^{2 k}}{\Gamma \left(k+\frac{3}{2}\right) \Gamma \left(k+n+\frac{3}{2}\right)},$$
+
+where $\Gamma(z)$ is the gamma function.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – order. If $z$ is a number, $n$ must be a tensor.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Lommel Functions
+
+### Lommel Functions
+
+Functions defined as the solutions to the Lommel differential equation:
+
+$$z^{2}{\frac{d^{2}y}{dz^{2}}}+z{\frac{dy}{dz}}+(z^{2}-n^{2})y=z^{m+1}.$$
+
+#### Lommel Function of the First Kind
+
+#### Lommel Function of the Second Kind
+
+
+
+Anger and Weber Functions
+
+### Anger and Weber Functions
+
+#### Anger Function
+
+```Python
+anger_j(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Anger function:
+
+$$\mathbf{J}_{n}(z)={\frac{1}{\pi}}\int_{0}^{\pi}\cos(n\theta-z\sin\theta)d\theta.$$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Weber Function
+
+```Python
+weber_e(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Weber function:
+
+$$\mathbf{E}_{n}(z)={\frac{1}{\pi}}\int_{0}^{\pi}\sin(n\theta-z\sin \theta )d\theta.$$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Parabolic Cylinder Functions
+
+### Parabolic Cylinder Functions
+
+#### Parabolic Cylinder Function
+
+```Python
+parabolic_cylinder_d(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None
+) -> Tensor
+```
+
+Parabolic cylinder function:
+
+$$D_{n}(z) = \sqrt{\pi} 2^{\tfrac{n}{2}} e^{-\frac{z^2}{4}} \left(\frac{ _1F_1(-\frac{n }{2}; \frac{1}{2}; \frac{z^2}{2})}{\Gamma (\frac{1-n }{2})}-\frac{\sqrt{2} z _1F_1(\frac{1-n }{2};\frac{3}{2};\frac{z^2}{2})}{\Gamma (-\frac{n }{2})} \right),$$
+
+where $_1F_1$ is the confluent hypergeometric function of the first kind and $\Gamma$ is the gamma function.
+
+
+
+Confluent Hypergeometric Functions
+
+### Confluent Hypergeometric Functions
+
+#### Confluent Hypergeometric Function of the First Kind
+
+```Python
+confluent_hypergeometric_1_f_1(
+ a: Tensor,
+ b: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Confluent hypergeometric function of the first kind:
+
+$$_{1}{F}_{1}(a; b; z) = \sum_{k = 0}^{\infty} \frac{a_{k}}{b_{k}} \frac{z^{k}}{k!},$$
+
+where $a_{k}$ and $b_{k}$ are rising factorials.
+
+#### Confluent Hypergeometric Function of the Second Kind
+
+
+
+Whittaker Functions
+
+### Whittaker Functions
+
+Whittaker functions are defined as a special solution of Whittaker’s equation:
+
+$${\frac {d^{2}w}{dz^{2}}}+(-{\frac {1}{4}}+{\frac {\kappa }{z}}+{\frac {1/4-\mu ^{2}}{z^{2}}})w=0.$$
+
+#### Whittaker Function ($M_{\kappa, \mu}$)
+
+```Python
+whittaker_m(
+ k: Tensor,
+ m: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Whittaker function:
+
+$${\displaystyle M_{\kappa ,\mu }(z)=\exp (-z/2)z^{\mu +{\tfrac {1}{2}}}M(\mu -\kappa +{\tfrac {1}{2}},1+2\mu ,z)}.$$
+
+##### Parameters
+
+**k** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**m** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Whittaker Function ($W_{\kappa, \mu}$)
+
+```Python
+whittaker_w(
+ k: Tensor,
+ m: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Whittaker function:
+
+$${\displaystyle W_{\kappa ,\mu }(z)=\exp (-z/2)z^{\mu +{\tfrac {1}{2}}}U(\mu -\kappa +{\tfrac {1}{2}},1+2\mu ,z).}$$
+
+##### Parameters
+
+**k** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**m** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Legendre Functions
+
+### Legendre Functions
+
+#### Legendre Function of the First Kind
+
+```Python
+legendre_p(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Legendre function of the first kind:
+
+$$P_{n}(z)=\operatorname{_2F_2}\left(-n,n+1;1;\frac{1-z}{2}\right),$$
+
+where $\operatorname{_2F_2}$ is the generalized hypergeometric function.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – order. If $z$ is a number, $n$ must be a tensor.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Legendre Function of the Second Kind
+
+```Python
+legendre_q(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Legendre function of the second kind:
+
+$$Q_{n}(z)=Q_{n}^0(z),$$
+
+where $Q_{n}^m$ is the associated Legendre function of the second kind.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – order. If $z$ is a number, $n$ must be a tensor.
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) – input. If $z$ is a number, $n$ must be a tensor.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Associated Legendre Functions
+
+### Associated Legendre Functions
+
+#### Associated Legendre Function of the First Kind
+
+```Python
+associated_legendre_p(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Associated Legendre function of the first kind:
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**m** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Associated Legendre Function of the Second Kind
+
+```Python
+associated_legendre_q(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Associated Legendre function of the second kind:
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**m** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Ferrers Functions
+
+### Ferrers Functions
+
+#### Ferrers Function of the First Kind
+
+#### Ferrers Function of the Second Kind
+
+
+
+Appell Functions
+
+### Appell Functions
+
+#### Appell Function $\left(F_{1}\right)$
+
+```Python
+appell_f_1(
+ a: Tensor,
+ b: Tensor,
+ c: Tensor,
+ d: Tensor,
+ x: Tensor,
+ y: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Appell function:
+
+$$F_{1}(a; b, c; d; x, y) = \sum_{m, n=0}^{\infty}{\frac{a_{m + n}b_{m}c_{n}}{d_{m + n}m!n!}}x^{m}y^{n}.$$
+
+##### Parameters
+
+**a** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**b** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**c** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**d** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**x** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**y** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Appell Function $\left(F_{2}\right)$
+
+```Python
+appell_f_2(
+ a: Tensor,
+ b: Tensor,
+ c: Tensor,
+ d: Tensor,
+ e: Tensor,
+ x: Tensor,
+ y: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**a** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**b** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**c** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**d** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**e** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**x** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**y** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Appell Function $\left(F_{3}\right)$
+
+```Python
+appell_f_3(
+ a: Tensor,
+ b: Tensor,
+ c: Tensor,
+ d: Tensor,
+ e: Tensor,
+ x: Tensor,
+ y: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**a** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**b** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**c** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**d** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**e** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**x** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**y** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Appell Function $\left(F_{4}\right)$
+
+```Python
+appell_f_4(
+ a: Tensor,
+ b: Tensor,
+ c: Tensor,
+ d: Tensor,
+ x: Tensor,
+ y: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**a** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**b** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**c** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**d** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**x** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**y** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+$q$-Hypergeometric and Related Functions
+
+### $q$-Hypergeometric and Related Functions
+
+#### $q$-Factorial
+
+$q$-factorial:
+
+$$[n]_q! = \prod^{n - 1} _{i = 0} \left(\textstyle\sum^{i} _{j = 0} q^j\right).$$
+
+#### $q$-Binomial Coefficient
+
+#### $q$-Gamma Function
+
+#### $q$-Digamma Function
+
+#### $q$-Polygamma Function
+
+
+
+Orthogonal Polynomials
+
+### Orthogonal Polynomials
+
+#### Legendre Polynomial
+
+#### Laguerre Polynomial
+
+
+
+Chebyshev Polynomials
+
+### Chebyshev Polynomials
+
+#### Chebyshev Polynomial of the First Kind
+
+```Python
+chebyshev_polynomial_t(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Chebyshev polynomial of the first kind, if, and only if, $n \in \mathbb{N} \wedge n \geq 0$:
+
+$$T_n(z)=\frac{\delta _{n,0}}{2}+\frac{1}{2} n \sum _{k=1}^{\left\lfloor\frac{n}{2}\right\rfloor } \frac{(-1)^k (n-k-1)! (2 z)^{n-2 k}}{k! (n-2 k)!}+2^{n-1}z^n,$$
+
+where $\delta$ is the Kronecker delta function.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Chebyshev Polynomial of the Second Kind
+
+```Python
+chebyshev_polynomial_u(
+ n: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Chebyshev polynomial of the second kind, if, and only if, $n \in \mathbb{N} \wedge n \geq 0$:
+
+$$U_n(z)=\sum _{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor } \frac{(-1)^k (n-k)! (2 z)^{n-2 k}}{k! (n-2 k)!}.$$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Chebyshev Polynomial of the Third Kind
+
+```Python
+chebyshev_polynomial_v(
+ n: Tensor,
+ x: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Chebyshev polynomial of the third kind, $V_{n}(x).$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**x** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Chebyshev Polynomial of the Fourth Kind
+
+```Python
+chebyshev_polynomial_w(
+ n: Tensor,
+ x: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Chebyshev polynomial of the fourth kind, $W_{n}(x).$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**x** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Shifted Chebyshev Polynomials
+
+### Shifted Chebyshev Polynomials
+
+#### Shifted Chebyshev Polynomial of the First Kind
+
+```Python
+shifted_chebyshev_polynomial_t(
+ n: Tensor,
+ x: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Shifted Chebyshev polynomial of the first kind, $T_{n}^{\ast}(x).$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**x** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Shifted Chebyshev Polynomial of the Second Kind
+
+```Python
+shifted_chebyshev_polynomial_u(
+ n: Tensor,
+ x: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Shifted Chebyshev polynomial of the second kind, $U_{n}^{\ast}(x).$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**x** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Shifted Chebyshev Polynomial of the Third Kind
+
+```Python
+shifted_chebyshev_polynomial_v(
+ n: Tensor,
+ x: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Shifted Chebyshev polynomial of the third kind, $V_{n}^{\ast}(x).$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**x** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Shifted Chebyshev Polynomial of the Fourth Kind
+
+```Python
+shifted_chebyshev_polynomial_w(
+ n: Tensor,
+ x: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Shifted Chebyshev polynomial of the fourth kind, $W_{n}^{\ast}(x).$
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**x** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Hermite Polynomials
+
+### Hermite Polynomials
+
+#### Probabilist’s Hermite Polynomial
+
+```Python
+hermite_polynomial_he(
+ n: Tensor,
+ x: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Probabilist’s Hermite polynomial:
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**x** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Physicist’s Hermite Polynomial
+
+```Python
+hermite_polynomial_h(
+ n: Tensor,
+ x: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Physicist’s Hermite polynomial:
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**x** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Legendre Forms of Elliptic Integrals
+
+### Legendre Forms of Elliptic Integrals
+
+#### Elliptic Integral of the First Kind
+
+```Python
+legendre_elliptic_integral_f(
+ m: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Legendre form of the incomplete elliptic integral of the first kind:
+
+The incomplete elliptic integral is defined in terms of the parameter $m$ instead of the elliptic modulus $k$. $m$ is defined as $m = k^{2}$.
+
+##### Parameters
+
+**m** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Elliptic Integral of the Second Kind
+
+```Python
+legendre_elliptic_integral_e(
+ m: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Legendre form of the incomplete elliptic integral of the second kind:
+
+The incomplete elliptic integral is defined in terms of the parameter $m$ instead of the elliptic modulus $k$. $m$ is defined as $m = k^{2}$.
+
+##### Parameters
+
+**m** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Elliptic Integral of the Third Kind
+
+```Python
+legendre_elliptic_integral_pi(
+ n: Tensor,
+ m: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Legendre form of the incomplete elliptic integral of the third kind:
+
+The incomplete elliptic integral is defined in terms of the parameter $m$ instead of the elliptic modulus $k$. $m$ is defined as $m = k^{2}$.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**m** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Legendre Forms of Complete Elliptic Integrals
+
+### Legendre Forms of Complete Elliptic Integrals
+
+#### Complete Elliptic Integral of the First Kind
+
+```Python
+complete_legendre_elliptic_integral_k(
+ m: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Legendre form of the complete elliptic integral of the first kind:
+
+$$K(m) = F({\tfrac{\pi}{2}}, m).$$
+
+The complete elliptic integral is defined in terms of the parameter $m$ instead of the elliptic modulus $k$. $m$ is defined as $m = k^{2}$.
+
+##### Parameters
+
+**m** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Complete Elliptic Integral of the Second Kind
+
+```Python
+complete_legendre_elliptic_integral_e(
+ m: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Legendre form of the complete elliptic integral of the second kind:
+
+$$E(m) = \int_{0}^{\tfrac{\pi}{2}}{\sqrt{1 - m \sin^{2} \theta}}d\theta.$$
+
+The complete elliptic integral is defined in terms of the parameter $m$ instead of the elliptic modulus $k$. $m$ is defined as $m = k^{2}$.
+
+##### Parameters
+
+**m** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Complete Elliptic Integral of the Third Kind
+
+```Python
+complete_legendre_elliptic_integral_pi(
+ n: Tensor,
+ m: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Legendre form of the complete elliptic integral of the third kind:
+
+$$\Pi(n, m) = \int_{0}^{\frac{\pi}{2}}{\frac{d\theta}{(1 - n\sin^{2}\theta){\sqrt{1 - m\sin ^{2}\theta }}}}.$$
+
+The complete elliptic integral is defined in terms of the parameter $m$ instead of the elliptic modulus $k$. $m$ is defined as $m = k^{2}$.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**m** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Carlson Symmetric Forms of Elliptic Integrals
+
+### Carlson Symmetric Forms of Elliptic Integrals
+
+#### Carlson Elliptic Integral $\left(R_{C}\right)$
+
+```Python
+carlson_elliptic_integral_r_c(
+ x: Tensor,
+ y: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**x** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**y** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Carlson Elliptic Integral $\left(R_{D}\right)$
+
+```Python
+carlson_elliptic_integral_r_d(
+ x: Tensor,
+ y: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**x** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**y** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Carlson Elliptic Integral $\left(R_{E}\right)$
+
+```Python
+carlson_elliptic_integral_r_e(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+#### Carlson Elliptic Integral $\left(R_{F}\right)$
+
+```Python
+carlson_elliptic_integral_r_f(
+ x: Tensor,
+ y: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**x** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**y** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Carlson Elliptic Integral $\left(R_{G}\right)$
+
+```Python
+carlson_elliptic_integral_r_g(
+ x: Tensor,
+ y: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**x** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**y** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Carlson Elliptic Integral $\left(R_{J}\right)$
+
+```Python
+carlson_elliptic_integral_r_j(
+ x: Tensor,
+ y: Tensor,
+ z: Tensor,
+ p: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**x** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**y** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Carlson Elliptic Integral $\left(R_{K}\right)$
+
+```Python
+carlson_elliptic_integral_r_k(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+#### Carlson Elliptic Integral $\left(R_{M}\right)$
+
+```Python
+carlson_elliptic_integral_r_m(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+
+
+Theta Functions
+
+### Theta Functions
+
+#### Theta Function $\left(\vartheta_1\right)$
+
+```Python
+theta_1(
+ z: Tensor,
+ q: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Theta function:
+
+$$\vartheta _1(z,q)=2 \sqrt[4]{q} \sum _{k=0}^{\infty } (-1)^k q^{k (k+1)} \sin ((2 k+1) z).$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**q** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Derivative of the Theta Function $\left(\vartheta_1'\right)$
+
+```Python
+theta_1_prime(
+ z: Tensor,
+ q: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Derivative of the theta function:
+
+$$\vartheta_1^{\prime}(z,q)=2 \sqrt[4]{q} \sum _{k=0}^{\infty } (-1)^k q^{k (k+1)} (2 k+1) \cos ((2 k+1) z).$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**q** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Theta Function $\left(\vartheta_2\right)$
+
+```Python
+theta_2(
+ z: Tensor,
+ q: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Theta function:
+
+$$\vartheta _2(z,q)=2 \sqrt[4]{q} \sum _{k=0}^{\infty } q^{k (k+1)} \cos ((2 k+1) z).$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**q** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Derivative of the Theta Function $\left(\vartheta_2'\right)$
+
+```Python
+theta_2_prime(
+ z: Tensor,
+ q: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Derivative of the theta function:
+
+$$\vartheta _2^{\prime }(z,q)=-2 \sqrt[4]{q} \sum _{k=0}^{\infty } q^{k (k+1)} (2 k+1) \sin ((2 k+1) z).$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**q** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Theta Function $\left(\vartheta_3\right)$
+
+```Python
+theta_3(
+ z: Tensor,
+ q: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Theta function:
+
+$$\vartheta _3(z,q)=1+2 \sum _{k=1}^{\infty } q^{k^2} \cos (2 k z).$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**q** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Derivative of the Theta Function $\left(\vartheta_3'\right)$
+
+```Python
+theta_3_prime(
+ z: Tensor,
+ q: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Derivative of the theta function:
+
+$$\vartheta _3^{\prime }(z,q)=-4 \sum _{k=1}^{\infty } q^{k^2} k \sin (2 k z).$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**q** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Theta Function $\left(\vartheta_4\right)$
+
+```Python
+theta_4(
+ z: Tensor,
+ q: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Theta function:
+
+$$\vartheta _4(z,q)=1+2 \sum _{k=1}^{\infty } (-1)^k q^{k^2} \cos (2 k z).$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**q** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Derivative of the Theta Function $\left(\vartheta_4'\right)$
+
+```Python
+theta_4_prime(
+ z: Tensor,
+ q: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Derivative of the theta function:
+
+$$\vartheta _4^{\prime }(z,q)=-4 \sum _{k=1}^{\infty } (-1)^k k q^{k^2} \sin (2 k z).$$
+
+##### Parameters
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**q** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Jacobi Elliptic and Related Functions
+
+### Jacobi Elliptic and Related Functions
+
+#### Jacobi Amplitude Function
+
+```Python
+jacobi_amplitude_am(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+#### Jacobi Elliptic Function $\left(\operatorname{sn}\right)$
+
+```Python
+jacobi_elliptic_sn(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+$$\operatorname{sn}(z \mid m) = \sin(\operatorname{am}(z \mid m))$$
+
+#### Jacobi Elliptic Function $\left(\operatorname{cn}\right)$
+
+```Python
+jacobi_elliptic_cn(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+$$\operatorname{cn}(z \mid m) = \cos(\operatorname{am}(z \mid m))$$
+
+#### Jacobi Elliptic Function $\left(\operatorname{dn}\right)$
+
+```Python
+jacobi_elliptic_dn(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+#### Jacobi Elliptic Function $\left(\operatorname{sd}\right)$
+
+```Python
+jacobi_elliptic_sd(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+#### Jacobi Elliptic Function $\left(\operatorname{cd}\right)$
+
+```Python
+jacobi_elliptic_cd(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+$$\operatorname{cd}(z \mid m) = \frac{\operatorname{cn}(z \mid m)}{\operatorname{dn}(z \mid m)}$$
+
+#### Jacobi Elliptic Function $\left(\operatorname{sc}\right)$
+
+```Python
+jacobi_elliptic_sc(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+
+
+Inverse Jacobi Elliptic Functions
+
+### Inverse Jacobi Elliptic Functions
+
+#### Inverse Jacobi Elliptic Function $\left(\operatorname{sn}\right)$
+
+```Python
+inverse_jacobi_elliptic_sn(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+#### Inverse Jacobi Elliptic Function $\left(\operatorname{cn}\right)$
+
+```Python
+inverse_jacobi_elliptic_cn(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+#### Inverse Jacobi Elliptic Function $\left(\operatorname{dn}\right)$
+
+```Python
+inverse_jacobi_elliptic_dn(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+#### Inverse Jacobi Elliptic Function $\left(\operatorname{sd}\right)$
+
+```Python
+inverse_jacobi_elliptic_sd(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+#### Inverse Jacobi Elliptic Function $\left(\operatorname{cd}\right)$
+
+```Python
+inverse_jacobi_elliptic_cd(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+#### Inverse Jacobi Elliptic Function $\left(\operatorname{sc}\right)$
+
+```Python
+inverse_jacobi_elliptic_sc(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+
+
+Weierstrass Elliptic Functions
+
+### Weierstrass Elliptic Functions
+
+#### Weierstrass Elliptic Function (p)
+
+#### Weierstrass Elliptic Function (\zeta)
+
+#### Weierstrass Elliptic Function (\sigma)
+
+
+
+Inverse Weierstrass Elliptic Functions
+
+### Inverse Weierstrass Elliptic Functions
+
+#### Inverse Weierstrass Elliptic Function (p)
+
+#### Inverse Weierstrass Elliptic Function (q)
+
+
+
+Modular Functions
+
+### Modular Functions
+
+#### Elliptic Function (\lambda)
+
+#### Klein’s Complete Invariant Function
+
+
+
+Bernoulli Number and Polynomial
+
+### Bernoulli Number and Polynomial
+
+#### Bernoulli Number
+
+```Python
+bernoulli_number_b(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+$n^{\text{th}}$ Bernoulli number, $B_{n}$.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Bernoulli Polynomial
+
+```Python
+bernoulli_polynomial_b(
+ n: Tensor,
+ x: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Bernoulli polynomial, $B_{n}(x)$.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**x** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Euler Number and Polynomial
+
+### Euler Number and Polynomial
+
+#### Euler Number
+
+```Python
+euler_number_e(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+$n^{\text{th}}$ Euler number, $E_{n}$.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Euler Polynomial
+
+```Python
+euler_polynomial_e(
+ n: Tensor,
+ x: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Euler polynomial, $E_{n}(x)$.
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Zeta and Related Functions
+
+### Zeta and Related Functions
+
+#### Riemann Zeta Function
+
+```Python
+riemann_zeta(
+ s: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Riemann zeta function:
+
+$$\zeta(s)=\sum _{n=1}^{\infty}{\frac{1}{n^{s}}}.$$
+
+##### Parameters
+
+**s** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) – input.
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Hurwitz Zeta Function
+
+```Python
+hurwitz_zeta(
+ s: Tensor,
+ a: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Hurwitz zeta function:
+
+$$\zeta(s, a) = \sum _{n = 0}^{\infty}{\frac{1}{(n + a)^{s}}}.$$
+
+##### Parameters
+
+**s** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**a** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Polylogarithm
+
+```Python
+polylogarithm_li(
+ s: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Polylogarithm:
+
+$$\operatorname{Li}_{s + 1}(z ) = \int_{0}^{z}{\frac{\operatorname{Li}_{s}(t)}{t}}dt.$$
+
+##### Parameters
+
+**s** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Lerch Zeta Function
+
+```Python
+lerch_zeta_l(
+ l: Tensor,
+ z: Tensor,
+ a: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**l** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**a** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Lerch Transcendent
+
+#### Dirichlet L-Function
+
+
+
+Multiplicative Number Theoretic Functions
+
+### Multiplicative Number Theoretic Functions
+
+#### Prime Number
+
+```Python
+prime_number_p(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+$n^{\text{th}}$ prime number, $p(n)$.
+
+#### Euler’s Totient Function
+
+```Python
+euler_totient_phi(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+#### Divisor Function
+
+#### Jordan’s Totient Function
+
+#### Möbius Function
+
+```Python
+mobius_mu(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+#### Liouville Function
+
+```Python
+liouville_lambda(
+ n: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+
+
+Matthieu Characteristic Values
+
+### Mathieu Characteristic Values
+
+#### Matthieu Characteristic Value $\left(a\right)$
+
+```Python
+matthieu_characteristic_a(
+ r: Tensor,
+ q: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+#### Matthieu Characteristic Value $\left(b\right)$
+
+```Python
+matthieu_characteristic_b(
+ r: Tensor,
+ q: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+
+
+Angular Matthieu Functions
+
+### Angular Mathieu Functions
+
+#### Angular Mathieu Function ($\operatorname{ce}$)
+
+Even Mathieu function:
+
+#### Angular Mathieu Function ($\operatorname{se}$)
+
+
+
+Radial Mathieu Functions
+
+### Radial Mathieu Functions
+
+#### Radial Matthieu Function ($\operatorname{M}c$)
+
+#### Radial Matthieu Function ($\operatorname{M}s$)
+
+
+
+Lamé Functions
+
+### Lamé Functions
+
+#### Lamé Function $\left(Ec_{n}^{j}\right)$
+
+```Python
+lame_ec(
+ n: Tensor,
+ j: Tensor,
+ z: Tensor,
+ m: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**j** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**m** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Derivative of the Lamé Function $\left(Ec_{n}^{j}\right)$
+
+```Python
+lame_ec_prime(
+ n: Tensor,
+ j: Tensor,
+ z: Tensor,
+ m: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**j** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**m** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Lamé Function $\left(Es_{n}^{j}\right)$
+
+```Python
+lame_es(
+ n: Tensor,
+ j: Tensor,
+ z: Tensor,
+ m: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**j** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**m** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Derivative of the Lamé Function $\left(Es_{n}^{j}\right)$
+
+```Python
+lame_es_prime(
+ n: Tensor,
+ j: Tensor,
+ z: Tensor,
+ m: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**n** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**j** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+**m** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor)) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Heun Functions
+
+### Heun Functions
+
+#### Heun Function
+
+```Python
+heun(
+ p: Tensor,
+ q: Tensor,
+ a: Tensor,
+ b: Tensor,
+ g: Tensor,
+ d: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Heun function:
+
+##### Parameters
+
+**p** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**q** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**a** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**b** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**g** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**d** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Derivative of the Heun Function
+
+```Python
+heun_prime(
+ p: Tensor,
+ q: Tensor,
+ a: Tensor,
+ b: Tensor,
+ g: Tensor,
+ d: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Derivative of the Heun function:
+
+##### Parameters
+
+**p** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**q** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**a** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**b** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**g** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**d** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Confluent Heun Function
+
+```Python
+confluent_heun(
+ q: Tensor,
+ a: Tensor,
+ g: Tensor,
+ d: Tensor,
+ e: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Confluent Heun function:
+
+##### Parameters
+
+**q** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**a** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**g** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**d** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**e** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Derivative of the Confluent Heun Function
+
+```Python
+confluent_heun_prime(
+ q: Tensor,
+ a: Tensor,
+ g: Tensor,
+ d: Tensor,
+ e: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Derivative of the confluent Heun function:
+
+##### Parameters
+
+**q** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**a** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**g** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**d** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**e** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Doubly-Confluent Heun Function
+
+```Python
+confluent_heun(
+ q: Tensor,
+ a: Tensor,
+ g: Tensor,
+ d: Tensor,
+ e: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Doubly-confluent Heun function:
+
+##### Parameters
+
+**q** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**a** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**g** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**d** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**e** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Derivative of the Doubly-Confluent Heun Function
+
+```Python
+confluent_heun_prime(
+ q: Tensor,
+ a: Tensor,
+ g: Tensor,
+ d: Tensor,
+ e: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Derivative of the doubly-confluent Heun function:
+
+##### Parameters
+
+**q** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**a** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**g** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**d** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**e** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Bi-Confluent Heun Function
+
+```Python
+biconfluent_heun(
+ q: Tensor,
+ a: Tensor,
+ g: Tensor,
+ d: Tensor,
+ e: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Bi-confluent Heun function:
+
+##### Parameters
+
+**q** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**a** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**g** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**d** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**e** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Derivative of the Bi-Confluent Heun Function
+
+```Python
+biconfluent_heun_prime(
+ q: Tensor,
+ a: Tensor,
+ g: Tensor,
+ d: Tensor,
+ e: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Derivative of the bi-confluent Heun function:
+
+##### Parameters
+
+**q** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**a** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**g** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**d** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**e** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Tri-Confluent Heun Function
+
+```Python
+triconfluent_heun(
+ q: Tensor,
+ a: Tensor,
+ g: Tensor,
+ d: Tensor,
+ e: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Tri-confluent Heun function:
+
+##### Parameters
+
+**q** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**a** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**g** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**d** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**e** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Derivative of the Tri-Confluent Heun Function
+
+```Python
+triconfluent_heun_prime(
+ q: Tensor,
+ a: Tensor,
+ g: Tensor,
+ d: Tensor,
+ e: Tensor,
+ z: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+Derivative of the tri-confluent Heun function:
+
+##### Parameters
+
+**q** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**a** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**g** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**d** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**e** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**z** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+
+Coulomb Wave Functions
+
+#### Coulomb Wave Function
+
+```Python
+coulomb_f(
+ l: Tensor,
+ e: Tensor,
+ r: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**l** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**e** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**r** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Irregular Coulomb Wave Function
+
+```Python
+coulomb_g(
+ l: Tensor,
+ e: Tensor,
+ r: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**l** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**e** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**r** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Outgoing Irregular Coulomb Wave Function
+
+```Python
+coulomb_h_positive(
+ l: Tensor,
+ e: Tensor,
+ r: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**l** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**e** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**r** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+#### Incoming Irregular Coulomb Wave Function
+
+```Python
+coulomb_h_negative(
+ l: Tensor,
+ e: Tensor,
+ r: Tensor,
+ *,
+ out: Optional[Tensor] = None,
+) -> Tensor
+```
+
+##### Parameters
+
+**l** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**e** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+**r** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor) *or Number*) –
+
+##### Keyword Arguments
+
+**out** ([Tensor](https://pytorch.org/docs/stable/tensors.html#torch.Tensor), *optional*) – output.
+
+
+## Prior Art
+
+### Cephes Mathematical Library
+
+Suite of elementary and special functions written by Stephen L. Moshier. It was initially created as a supplement to his numerical analysis textbook, “Methods and Programs for Mathematical Functions.” Unsure of the original publication date, but I estimate somewhere between 1984 and 1986. It was most recently updated, by Moshier, in 2018. Its use is ubiquitous (it’s even presently used in PyTorch), and DeepMind recently maintained a library wrapper for PyTorch.
+
+### specfun
+
+Suite of elementary and special functions authored by W. J. Cody, a numerical analysis pioneer, written in Fortran. It’s abandonware.
+
+### Wolfram Language
+
+General-purpose, commercial, multi-paradigm programming language written and maintained by Wolfram Research. The Wolfram Language touts one of the best suites of elementary and special functions. Most of these functions support either symbolic, or relevant to PyTorch, numerical evaluation and differentiation.
+
+### MATLAB
+### International Mathematics and Statistics Library (IMSL)
+### NAG Numerical Library
+### GNU Octave
+### GNU Scientific Library (GSL)
+
+### SciPy
+
+SciPy is a free and open-source Python package for scientific computing. It contains modules for linear algebra, integration, optimization, etc. SciPy also contains a robust suite of special functions. Most of the special function implementations rely on third-party packages, many featured elsewhere in this subsection (e.g., Cephes and specfun).