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 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])