scipy.special.hyp1f1#

scipy.special.hyp1f1(a, b, x, out=None) = <ufunc 'hyp1f1'>#

Confluent hypergeometric function \({}_1F_1(a; b; x)\).

The confluent hypergeometric function is defined by the series

\[{}_1F_1(a; b; x) = \sum_{k = 0}^\infty \frac{(a)_k}{(b)_k k!} x^k.\]

See [DLMF] for more details. Here \((\cdot)_k\) is the Pochhammer symbol; see poch.

Parameters:

a, barray_like: Real parameters
xarray_like: Real or complex argument
outndarray, optional: Optional output array for the function results

Returns:

scalar or ndarray: Values of the confluent hypergeometric function

See also

hyperu: another confluent hypergeometric function
hyp0f1: confluent hypergeometric limit function
hyp2f1: Gaussian hypergeometric function

Notes

For real values, this function uses the hyp1f1 routine from the C++ Boost library [2], for complex values a C translation of the specfun Fortran library [3].

Array API Standard Support

hyp1f1 has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	⛔
PyTorch	✅	⛔
JAX	✅	✅
Dask	✅	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable. hyp1f1 does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

References

[DLMF]

NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/13.2#E2

[2]

The Boost Developers. “Boost C++ Libraries”. https://www.boost.org/.

[3]

Zhang and J.M. Jin, “Computation of Special Functions”, Wiley 1996.

Examples

>>> import numpy as np
>>> import scipy.special as sc

It is one when x is zero:

>>> sc.hyp1f1(0.5, 0.5, 0)
1.0

It is singular when b is a nonpositive integer.

>>> sc.hyp1f1(0.5, -1, 0)
inf

It is a polynomial when a is a nonpositive integer.

>>> a, b, x = -1, 0.5, np.array([1.0, 2.0, 3.0, 4.0])
>>> sc.hyp1f1(a, b, x)
array([-1., -3., -5., -7.])
>>> 1 + (a / b) * x
array([-1., -3., -5., -7.])

It reduces to the exponential function when a = b.

>>> sc.hyp1f1(2, 2, [1, 2, 3, 4])
array([ 2.71828183,  7.3890561 , 20.08553692, 54.59815003])
>>> np.exp([1, 2, 3, 4])
array([ 2.71828183,  7.3890561 , 20.08553692, 54.59815003])