scipy.special.hyp0f1

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

1

Wolfram MathWorld, “Confluent Hypergeometric Limit Function”, http://mathworld.wolfram.com/ConfluentHypergeometricLimitFunction.html

Examples

>>> import scipy.special as sc

It is one when z is zero.

>>> sc.hyp0f1(1, 0)
1.0

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 * x**2)
array([0.        , 0.12402598, 0.24226846, 0.3492436 , 0.44005059])