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

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

Confluent hypergeometric function U

It is defined as the solution to the equation

\[x \frac{d^2w}{dx^2} + (b - x) \frac{dw}{dx} - aw = 0\]

which satisfies the property

\[U(a, b, x) \sim x^{-a}\]

as \(x \to \infty\). See [dlmf] for more details.

Parameters:

a, barray_like

Real-valued parameters

xarray_like

Real-valued argument

outndarray, optional

Optional output array for the function values

Returns:

scalar or ndarray

Values of U

References

Examples

import numpy as np import scipy.special as sc

It has a branch cut along the negative x axis.

x = np.linspace(-0.1, -10, 5) sc.hyperu(1, 1, x) array([nan, nan, nan, nan, nan])

It approaches zero as x goes to infinity.

x = np.array([1, 10, 100]) sc.hyperu(1, 1, x) array([0.59634736, 0.09156333, 0.00990194])

It satisfies Kummer’s transformation.

a, b, x = 2, 1, 1 sc.hyperu(a, b, x) 0.1926947246463881 x**(1 - b) * sc.hyperu(a - b + 1, 2 - b, x) 0.1926947246463881