scipy.special.h1vp#

scipy.special.h1vp(v, z, n=1)[source]#

Compute derivatives of Hankel function H1v(z) with respect to z.

Parameters:

varray_like: Order of Hankel function
zarray_like: Argument at which to evaluate the derivative. Can be real or complex.
nint, default 1: Order of derivative. For 0 returns the Hankel function h1v itself.

Returns:

scalar or ndarray: Values of the derivative of the Hankel function.

See also

hankel1

Notes

The derivative is computed using the relation DLFM 10.6.7 [2].

References

[1]

Zhang, Shanjie and Jin, Jianming. “Computation of Special Functions”, John Wiley and Sons, 1996, chapter 5. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

[2]

NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.6.E7

Examples

Compute the Hankel function of the first kind of order 0 and its first two derivatives at 1.

>>> from scipy.special import h1vp
>>> h1vp(0, 1, 0), h1vp(0, 1, 1), h1vp(0, 1, 2)
((0.7651976865579664+0.088256964215677j),
 (-0.44005058574493355+0.7812128213002889j),
 (-0.3251471008130329-0.8694697855159659j))

Compute the first derivative of the Hankel function of the first kind for several orders at 1 by providing an array for v.

>>> h1vp([0, 1, 2], 1, 1)
array([-0.44005059+0.78121282j,  0.3251471 +0.86946979j,
       0.21024362+2.52015239j])

Compute the first derivative of the Hankel function of the first kind of order 0 at several points by providing an array for z.

>>> import numpy as np
>>> points = np.array([0.5, 1.5, 3.])
>>> h1vp(0, points, 1)
array([-0.24226846+1.47147239j, -0.55793651+0.41230863j,
       -0.33905896-0.32467442j])