Skip to main content
`Ctrl`+`K`

# scipy.special.iti0k0

scipy.special.iti0k0(x, out=None) = <ufunc 'iti0k0'>

  Integrals of modified Bessel functions of order 0.

  Computes the integrals

  \[\begin{split}\int_0^x I_0(t) dt \\ \int_0^x K_0(t) dt.\end{split}\]

  For more on \(I_0\) and \(K_0\) see `i0` and `k0`.

  Parameters:
    xarray_like

      Values at which to evaluate the integrals.

    outtuple of ndarrays, optional

      Optional output arrays for the function results.

  Returns:
    ii0scalar or ndarray

      The integral for `i0`

    ik0scalar 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 iti0k0
      >>> int_i, int_k = iti0k0(1.)
      >>> int_i, int_k
      (1.0865210970235892, 1.2425098486237771)

  Evaluate the functions at several points.

      >>> import numpy as np
      >>> points = np.array([0., 1.5, 3.])
      >>> int_i, int_k = iti0k0(points)
      >>> int_i, int_k
      (array([0.        , 1.80606937, 6.16096149]),
       array([0.        , 1.39458246, 1.53994809]))

  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 = iti0k0(x)
      >>> ax.plot(x, int_i, label=r"$\int_0^x I_0(t)\,dt$")
      >>> ax.plot(x, int_k, label=r"$\int_0^x K_0(t)\,dt$")
      >>> ax.legend()
      >>> plt.show()