scipy.special.hyp0f1 — SciPy v1.15.2 Manual (original) (raw)

scipy.special.hyp0f1(v, z, out=None) = <ufunc 'hyp0f1'>#

Confluent hypergeometric limit function 0F1.

Parameters:

varray_like

Real-valued parameter

zarray_like

Real- or complex-valued argument

outndarray, optional

Optional output array for the function results

Returns:

scalar or ndarray

The confluent hypergeometric limit function

Notes

This function is defined as:

\[_0F_1(v, z) = \sum_{k=0}^{\infty}\frac{z^k}{(v)_k k!}.\]

It’s also the limit as \(q \to \infty\) of \(_1F_1(q; v; z/q)\), and satisfies the differential equation \(f''(z) + vf'(z) = f(z)\). See [1] for more information.

References

Examples

import numpy as np import scipy.special as sc

It is one when z is zero.

It is the limit of the confluent hypergeometric function as _q_goes to infinity.

q = np.array([1, 10, 100, 1000]) v = 1 z = 1 sc.hyp1f1(q, v, z / q) array([2.71828183, 2.31481985, 2.28303778, 2.27992985]) sc.hyp0f1(v, z) 2.2795853023360673

It is related to Bessel functions.

n = 1 x = np.linspace(0, 1, 5) sc.jv(n, x) array([0. , 0.12402598, 0.24226846, 0.3492436 , 0.44005059]) (0.5 * x)n / sc.factorial(n) * sc.hyp0f1(n + 1, -0.25 * x2) array([0. , 0.12402598, 0.24226846, 0.3492436 , 0.44005059])