scipy.special.itairy#

scipy.special.itairy(x, out=None) = <wrapped_ufunc 'itairy'>#

Integrals of Airy functions.

Calculates the integrals of Airy functions from 0 to x.

Parameters:

xarray_like: Upper limit of integration (float).
outtuple of ndarray, optional: Optional output arrays for the function values

Returns:

Aptscalar or ndarray: Integral of Ai(t) from 0 to x.
Bptscalar or ndarray: Integral of Bi(t) from 0 to x.
Antscalar or ndarray: Integral of Ai(-t) from 0 to x.
Bntscalar or ndarray: Integral of Bi(-t) from 0 to x.

Notes

Wrapper for a Fortran routine created by Shanjie Zhang and Jianming Jin [1].

Array API Standard Support

itairy 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. itairy does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

References

[1]

Zhang, Shanjie and Jin, Jianming. “Computation of Special Functions”, John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f_src/special_functions/special_functions.html

Examples

Compute the functions at x=1..

>>> import numpy as np
>>> from scipy.special import itairy
>>> import matplotlib.pyplot as plt
>>> apt, bpt, ant, bnt = itairy(1.)
>>> apt, bpt, ant, bnt
(0.23631734191710949,
 0.8727691167380077,
 0.46567398346706845,
 0.3730050096342943)

Compute the functions at several points by providing a NumPy array for x.

>>> x = np.array([1., 1.5, 2.5, 5])
>>> apt, bpt, ant, bnt = itairy(x)
>>> apt, bpt, ant, bnt
(array([0.23631734, 0.28678675, 0.324638  , 0.33328759]),
 array([  0.87276912,   1.62470809,   5.20906691, 321.47831857]),
 array([0.46567398, 0.72232876, 0.93187776, 0.7178822 ]),
 array([ 0.37300501,  0.35038814, -0.02812939,  0.15873094]))

Plot the functions from -10 to 10.

>>> x = np.linspace(-10, 10, 500)
>>> apt, bpt, ant, bnt = itairy(x)
>>> fig, ax = plt.subplots(figsize=(6, 5))
>>> ax.plot(x, apt, label=r"$\int_0^x\, Ai(t)\, dt$")
>>> ax.plot(x, bpt, ls="dashed", label=r"$\int_0^x\, Bi(t)\, dt$")
>>> ax.plot(x, ant, ls="dashdot", label=r"$\int_0^x\, Ai(-t)\, dt$")
>>> ax.plot(x, bnt, ls="dotted", label=r"$\int_0^x\, Bi(-t)\, dt$")
>>> ax.set_ylim(-2, 1.5)
>>> ax.legend(loc="lower right")
>>> plt.show()

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