scipy.special.fresnel — SciPy v1.15.2 Manual (original) (raw)

scipy.special.fresnel(z, out=None) = <ufunc 'fresnel'>#

Fresnel integrals.

The Fresnel integrals are defined as

\[\begin{split}S(z) &= \int_0^z \sin(\pi t^2 /2) dt \\ C(z) &= \int_0^z \cos(\pi t^2 /2) dt.\end{split}\]

See [dlmf] for details.

Parameters:

zarray_like

Real or complex valued argument

out2-tuple of ndarrays, optional

Optional output arrays for the function results

Returns:

S, C2-tuple of scalar or ndarray

Values of the Fresnel integrals

References

Examples

import numpy as np import scipy.special as sc

As z goes to infinity along the real axis, S and C converge to 0.5.

S, C = sc.fresnel([0.1, 1, 10, 100, np.inf]) S array([0.00052359, 0.43825915, 0.46816998, 0.4968169 , 0.5 ]) C array([0.09999753, 0.7798934 , 0.49989869, 0.4999999 , 0.5 ])

They are related to the error function erf.

z = np.array([1, 2, 3, 4]) zeta = 0.5 * np.sqrt(np.pi) * (1 - 1j) * z S, C = sc.fresnel(z) C + 1j*S array([0.7798934 +0.43825915j, 0.48825341+0.34341568j, 0.60572079+0.496313j , 0.49842603+0.42051575j]) 0.5 * (1 + 1j) * sc.erf(zeta) array([0.7798934 +0.43825915j, 0.48825341+0.34341568j, 0.60572079+0.496313j , 0.49842603+0.42051575j])