Module « scipy.special »

Fonction hyp0f1 - module scipy.special

Signature de la fonction hyp0f1

Description

hyp0f1.__doc__

hyp0f1(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

hyp0f1(v, z, out=None)

Confluent hypergeometric limit function 0F1.

Parameters
----------
v : array_like
    Real-valued parameter
z : array_like
    Real- or complex-valued argument
out : ndarray, optional
    Optional output array for the function results

Returns
-------
scalar or ndarray
    The confluent hypergeometric limit function

Notes
-----
This function is defined as:

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

It's also the limit as :math:`q \to \infty` of :math:`_1F_1(q; v; z/q)`,
and satisfies the differential equation :math:`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])