skued.lorentzian

skued.lorentzian(coordinates, center, fwhm)

Unit integral Lorenzian function.

Parameters:

  • coordinates (array-like) – Can be either a list of ndarrays, as a meshgrid coordinates list, or a single ndarray for 1D computation

  • center (array-like) – Center of the lorentzian. Should be the same shape as coordinates.ndim.

  • fwhm (float) – Full-width at half-max of the function.

Returns:

out – Lorentzian function of unit integral.

Return type:

ndarray

Notes

The functional form of the Lorentzian is given by:

\[L(x) = \frac{1}{\pi} \frac{(\gamma/2)}{(x-c)^2 + (\gamma/2)^2}\]

where \(\gamma\) is the full-width at half-maximum, and \(c\) is the center.

For n dimensions, the functional form of the Lorentzian is given by:

\[L(x_1, ..., x_n) = \frac{1}{n \pi} \frac{(\gamma/2)}{(\sum_i{(x_i - c_i)^2} + (\gamma/2)^2)^{\frac{1+n}{2}}}\]

Example

>>> import numpy as np
>>> from skued import lorentzian
>>>
>>> span = np.arange(-15, 15, 0.1)
>>> xx, yy = np.meshgrid(span, span)
>>> l = lorentzian( coordinates = [xx,yy], center = [0,0], fwhm = 1)
>>> l.shape == xx.shape
True
>>> np.sum(l)*0.1**2          # Integral should be unity (spacing = 0.1)
0.9700030627398781