kvp#
- scipy.special.kvp(v, z, n=1)[source]#
- Compute derivatives of real-order modified Bessel function Kv(z) - Kv(z) is the modified Bessel function of the second kind. Derivative is calculated with respect to z. - Parameters:
- varray_like of float
- Order of Bessel function 
- zarray_like of complex
- Argument at which to evaluate the derivative 
- nint, default 1
- Order of derivative. For 0 returns the Bessel function - kvitself.
 
- Returns:
- outndarray
- The results 
 
 - See also - Notes - The derivative is computed using the relation DLFM 10.29.5 [2]. - References [1]- Zhang, Shanjie and Jin, Jianming. “Computation of Special Functions”, John Wiley and Sons, 1996, chapter 6. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html [2]- NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.29.E5 - Examples - Compute the modified bessel function of the second kind of order 0 and its first two derivatives at 1. - >>> from scipy.special import kvp >>> kvp(0, 1, 0), kvp(0, 1, 1), kvp(0, 1, 2) (0.42102443824070834, -0.6019072301972346, 1.0229316684379428) - Compute the first derivative of the modified Bessel function of the second kind for several orders at 1 by providing an array for v. - >>> kvp([0, 1, 2], 1, 1) array([-0.60190723, -1.02293167, -3.85158503]) - Compute the first derivative of the modified Bessel function of the second 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.]) >>> kvp(0, points, 1) array([-1.65644112, -0.2773878 , -0.04015643]) - Plot the modified bessel function of the second kind and its first three derivatives. - >>> import matplotlib.pyplot as plt >>> x = np.linspace(0, 5, 1000) >>> fig, ax = plt.subplots() >>> ax.plot(x, kvp(1, x, 0), label=r"$K_1$") >>> ax.plot(x, kvp(1, x, 1), label=r"$K_1'$") >>> ax.plot(x, kvp(1, x, 2), label=r"$K_1''$") >>> ax.plot(x, kvp(1, x, 3), label=r"$K_1'''$") >>> ax.set_ylim(-2.5, 2.5) >>> plt.legend() >>> plt.show() 