scipy.special.iti0k0#

scipy.special.iti0k0(x, out=None) = <wrapped_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

Notes

Array API Standard Support

iti0k0 has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	⛔
PyTorch	✅	⛔
JAX	✅	⛔
Dask	⛔	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but typically none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable. iti0k0 does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

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()

../../_images/scipy-special-iti0k0-1.png