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Module « scipy.special »

Fonction voigt_profile - module scipy.special

Signature de la fonction voigt_profile

def voigt_profile(*args, **kwargs) 

Description

help(scipy.special.voigt_profile)

voigt_profile(x1, x2, x3, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])

voigt_profile(x, sigma, gamma, out=None)

Voigt profile.

The Voigt profile is a convolution of a 1-D Normal distribution with
standard deviation ``sigma`` and a 1-D Cauchy distribution with half-width at
half-maximum ``gamma``.

If ``sigma = 0``, PDF of Cauchy distribution is returned.
Conversely, if ``gamma = 0``, PDF of Normal distribution is returned.
If ``sigma = gamma = 0``, the return value is ``Inf`` for ``x = 0``,
and ``0`` for all other ``x``.

Parameters
----------
x : array_like
    Real argument
sigma : array_like
    The standard deviation of the Normal distribution part
gamma : array_like
    The half-width at half-maximum of the Cauchy distribution part
out : ndarray, optional
    Optional output array for the function values

Returns
-------
scalar or ndarray
    The Voigt profile at the given arguments

See Also
--------
wofz : Faddeeva function

Notes
-----
It can be expressed in terms of Faddeeva function

.. math:: V(x; \sigma, \gamma) = \frac{Re[w(z)]}{\sigma\sqrt{2\pi}},
.. math:: z = \frac{x + i\gamma}{\sqrt{2}\sigma}

where :math:`w(z)` is the Faddeeva function.

References
----------
.. [1] https://en.wikipedia.org/wiki/Voigt_profile

Examples
--------
Calculate the function at point 2 for ``sigma=1`` and ``gamma=1``.

>>> from scipy.special import voigt_profile
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> voigt_profile(2, 1., 1.)
0.09071519942627544

Calculate the function at several points by providing a NumPy array
for `x`.

>>> values = np.array([-2., 0., 5])
>>> voigt_profile(values, 1., 1.)
array([0.0907152 , 0.20870928, 0.01388492])

Plot the function for different parameter sets.

>>> fig, ax = plt.subplots(figsize=(8, 8))
>>> x = np.linspace(-10, 10, 500)
>>> parameters_list = [(1.5, 0., "solid"), (1.3, 0.5, "dashed"),
...                    (0., 1.8, "dotted"), (1., 1., "dashdot")]
>>> for params in parameters_list:
...     sigma, gamma, linestyle = params
...     voigt = voigt_profile(x, sigma, gamma)
...     ax.plot(x, voigt, label=rf"$\sigma={sigma},\, \gamma={gamma}$",
...             ls=linestyle)
>>> ax.legend()
>>> plt.show()

Verify visually that the Voigt profile indeed arises as the convolution
of a normal and a Cauchy distribution.

>>> from scipy.signal import convolve
>>> x, dx = np.linspace(-10, 10, 500, retstep=True)
>>> def gaussian(x, sigma):
...     return np.exp(-0.5 * x**2/sigma**2)/(sigma * np.sqrt(2*np.pi))
>>> def cauchy(x, gamma):
...     return gamma/(np.pi * (np.square(x)+gamma**2))
>>> sigma = 2
>>> gamma = 1
>>> gauss_profile = gaussian(x, sigma)
>>> cauchy_profile = cauchy(x, gamma)
>>> convolved = dx * convolve(cauchy_profile, gauss_profile, mode="same")
>>> voigt = voigt_profile(x, sigma, gamma)
>>> fig, ax = plt.subplots(figsize=(8, 8))
>>> ax.plot(x, gauss_profile, label="Gauss: $G$", c='b')
>>> ax.plot(x, cauchy_profile, label="Cauchy: $C$", c='y', ls="dashed")
>>> xx = 0.5*(x[1:] + x[:-1])  # midpoints
>>> ax.plot(xx, convolved[1:], label="Convolution: $G * C$", ls='dashdot',
...         c='k')
>>> ax.plot(x, voigt, label="Voigt", ls='dotted', c='r')
>>> ax.legend()
>>> plt.show()

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