scipy.special.hyperu

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

dlmf

NIST Digital Library of Mathematics Functions https://dlmf.nist.gov/13.2#E6

Examples

>>> 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