# 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

References

[dlmf]

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

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])