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 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