scipy.special.hyp1f1#

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

Confluent hypergeometric function 1F1.

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 experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. 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

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, Jin, “Computation of Special Functions”, John Wiley and Sons, Inc, 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])