# scipy.special.it2i0k0

scipy.special.it2i0k0(x, out=None) = <ufunc 'it2i0k0'>

  Integrals related to modified Bessel functions of order 0.

  Computes the integrals

  \[\begin{split}\int_0^x \frac{I_0(t) - 1}{t} dt \\ \int_x^\infty \frac{K_0(t)}{t} dt.\end{split}\]
  Parameters:
    x array_like

      Values at which to evaluate the integrals.

    out tuple of ndarrays, optional

      Optional output arrays for the function results.

  Returns:
    ii0 scalar or ndarray

      The integral for `i0`

    ik0 scalar or ndarray

      The integral for `k0`

  References

  [1]

  S. Zhang and J.M. Jin, “Computation of Special Functions”, Wiley 1996

  Examples

  Evaluate the functions at one point.

      >>> from scipy.special import it2i0k0
      >>> int_i, int_k = it2i0k0(1.)
      >>> int_i, int_k
      (0.12897944249456852, 0.2085182909001295)

  Evaluate the functions at several points.

      >>> import numpy as np
      >>> points = np.array([0.5, 1.5, 3.])
      >>> int_i, int_k = it2i0k0(points)
      >>> int_i, int_k
      (array([0.03149527, 0.30187149, 1.50012461]),
       array([0.66575102, 0.0823715 , 0.00823631]))

  Plot the functions from 0 to 5.

      >>> import matplotlib.pyplot as plt
      >>> fig, ax = plt.subplots()
      >>> x = np.linspace(0., 5., 1000)
      >>> int_i, int_k = it2i0k0(x)
      >>> ax.plot(x, int_i, label=r"$\int_0^x \frac{I_0(t)-1}{t}\,dt$")
      >>> ax.plot(x, int_k, label=r"$\int_x^{\infty} \frac{K_0(t)}{t}\,dt$")
      >>> ax.legend()
      >>> ax.set_ylim(0, 10)
      >>> plt.show()