nlsq.core.functions.gaussian

nlsq.core.functions.gaussian(x, amp, mu, sigma)[source]

Gaussian (normal distribution) function: y = amp * exp(-(x-mu)² / (2*sigma²))

Common for spectral peaks, chromatography peaks, probability distributions.

Parameters:

  • x (array_like) – Independent variable

  • amp (float) – Amplitude (peak height)

  • mu (float) – Mean (center position, peak location)

  • sigma (float) – Standard deviation (width parameter, positive)

Returns:

y – Dependent variable

Return type:

array_like

Notes

  • Peak position: x = mu

  • Peak height: y = amp

  • FWHM (Full Width at Half Maximum): FWHM = 2.355 * sigma

  • Integral: ∫ gaussian dx = amp * sigma * sqrt(2π)

Examples

>>> from nlsq import curve_fit
>>> from nlsq.core.functions import gaussian
>>> import numpy as np
>>>
>>> # Spectral peak at x=5 with FWHM ≈ 2.355
>>> x = np.linspace(0, 10, 200)
>>> y = 10 * np.exp(-(x-5)**2 / (2*1**2)) + np.random.normal(0, 0.2, 200)
>>> popt, pcov = curve_fit(gaussian, x, y, p0='auto')
>>> print(f"Peak at {popt[1]:.2f}, FWHM = {2.355*popt[2]:.2f}")