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scipy.special.

# ivp

scipy.special.ivp(v, z, n=1)

  Compute derivatives of modified Bessel functions of the first kind.

  Compute the nth derivative of the modified Bessel function Iv with respect to z.

  Parameters:
    varray_like or float

      Order of Bessel 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 Bessel function `iv` itself.

  Returns:
    scalar or ndarray

      nth derivative of the modified Bessel function.

  See also

  `iv`

  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 first kind of order 0 and its first two derivatives at 1.

      >>> from scipy.special import ivp
      >>> ivp(0, 1, 0), ivp(0, 1, 1), ivp(0, 1, 2)
      (1.2660658777520084, 0.565159103992485, 0.7009067737595233)

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

      >>> ivp([0, 1, 2], 1, 1)
      array([0.5651591 , 0.70090677, 0.29366376])

  Compute the first derivative of the modified Bessel 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., 1.5, 3.])
      >>> ivp(0, points, 1)
      array([0.        , 0.98166643, 3.95337022])

  Plot the modified Bessel function of the first kind of order 1 and its first three derivatives.

      >>> import matplotlib.pyplot as plt
      >>> x = np.linspace(-5, 5, 1000)
      >>> fig, ax = plt.subplots()
      >>> ax.plot(x, ivp(1, x, 0), label=r"$I_1$")
      >>> ax.plot(x, ivp(1, x, 1), label=r"$I_1'$")
      >>> ax.plot(x, ivp(1, x, 2), label=r"$I_1''$")
      >>> ax.plot(x, ivp(1, x, 3), label=r"$I_1'''$")
      >>> plt.legend()
      >>> plt.show()