scipy.special.fresnel

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

See also

fresnel_zeros

zeros of the Fresnel integrals

References

dlmf

NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/7.2#iii

Examples

>>> 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])