Module « scipy.special »

Fonction fresnel - module scipy.special

Signature de la fonction fresnel

Description

fresnel.__doc__

fresnel(x[, out1, out2], / [, out=(None, None)], *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

fresnel(z, out=None)

Fresnel integrals.

The Fresnel integrals are defined as

.. math::

   S(z) &= \int_0^z \sin(\pi t^2 /2) dt \\
   C(z) &= \int_0^z \cos(\pi t^2 /2) dt.

See [dlmf]_ for details.

Parameters
----------
z : array_like
    Real or complex valued argument
out : 2-tuple of ndarrays, optional
    Optional output arrays for the function results

Returns
-------
S, C : 2-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])