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

scipy.special.voigt_profile(x, sigma, gamma, out=None) = <ufunc 'voigt_profile'>#

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:

xarray_like

Real argument

sigmaarray_like

The standard deviation of the Normal distribution part

gammaarray_like

The half-width at half-maximum of the Cauchy distribution part

outndarray, 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

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

\[z = \frac{x + i\gamma}{\sqrt{2}\sigma}\]

where \(w(z)\) is the Faddeeva function.

References

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 * x2/sigma2)/(sigma * np.sqrt(2np.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: GGG", c='b') ax.plot(x, cauchy_profile, label="Cauchy: CCC", c='y', ls="dashed") xx = 0.5(x[1:] + x[:-1]) # midpoints ax.plot(xx, convolved[1:], label="Convolution: G∗CG * CG∗C", ls='dashdot', ... c='k') ax.plot(x, voigt, label="Voigt", ls='dotted', c='r') ax.legend() plt.show()