nlsq.core.functions.lorentzian¶

nlsq.core.functions.lorentzian(x, amp, x0, gamma)[source]¶

Lorentzian (Cauchy) peak function: y = amp / (1 + ((x - x0) / gamma)²)

Common for spectral line shapes (NMR, IR, Raman), resonance curves, and natural line broadening.

Parameters:

x (array_like) – Independent variable
amp (float) – Amplitude (peak height)
x0 (float) – Center position (peak location)
gamma (float) – Half-width at half-maximum (HWHM, positive)

Returns:

y – Dependent variable

Return type:

array_like

Notes

Peak position: x = x0
Peak height: y = amp
FWHM (Full Width at Half Maximum): FWHM = 2 * gamma
Integral: ∫ lorentzian dx = amp * gamma * π
Heavier tails than Gaussian (decays as 1/x² vs exp(-x²))

Examples

>>> from nlsq import curve_fit
>>> from nlsq.core.functions import lorentzian
>>> import numpy as np
>>>
>>> # Spectral peak at x=5 with FWHM = 2
>>> x = np.linspace(0, 10, 200)
>>> y = 10 / (1 + ((x - 5) / 1)**2) + np.random.normal(0, 0.2, 200)
>>> popt, pcov = curve_fit(lorentzian, x, y, p0='auto')
>>> print(f"Peak at {popt[1]:.2f}, FWHM = {2*popt[2]:.2f}")