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scipy.special.

# jvp

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

  Compute derivatives of Bessel functions of the first kind.

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

  Parameters:
    varray_like or float

      Order of Bessel function

    zcomplex

      Argument at which to evaluate the derivative; can be real or complex.

    nint, default 1

      Order of derivative. For 0 returns the Bessel function `jv` itself.

  Returns:
    scalar or ndarray

      Values of the derivative of the Bessel function.

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

      >>> from scipy.special import jvp
      >>> jvp(0, 1, 0), jvp(0, 1, 1), jvp(0, 1, 2)
      (0.7651976865579666, -0.44005058574493355, -0.3251471008130331)

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

      >>> jvp([0, 1, 2], 1, 1)
      array([-0.44005059,  0.3251471 ,  0.21024362])

  Compute the first derivative of the 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.])
      >>> jvp(0, points, 1)
      array([-0.        , -0.55793651, -0.33905896])

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

      >>> import matplotlib.pyplot as plt
      >>> x = np.linspace(-10, 10, 1000)
      >>> fig, ax = plt.subplots()
      >>> ax.plot(x, jvp(1, x, 0), label=r"$J_1$")
      >>> ax.plot(x, jvp(1, x, 1), label=r"$J_1'$")
      >>> ax.plot(x, jvp(1, x, 2), label=r"$J_1''$")
      >>> ax.plot(x, jvp(1, x, 3), label=r"$J_1'''$")
      >>> plt.legend()
      >>> plt.show()