scipy.special.itairy

scipy.special.itairy(x, out=None) = <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].

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="$\int_0^x\, Ai(t)\, dt$")
>>> ax.plot(x, bpt, ls="dashed", label="$\int_0^x\, Bi(t)\, dt$")
>>> ax.plot(x, ant, ls="dashdot", label="$\int_0^x\, Ai(-t)\, dt$")
>>> ax.plot(x, bnt, ls="dotted", label="$\int_0^x\, Bi(-t)\, dt$")
>>> ax.set_ylim(-2, 1.5)
>>> ax.legend(loc="lower right")
>>> plt.show()