# Derivation of FWHM and Height for Peak Fitting Functions

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## Gaussian (Normal distribution)

A = amplitude, $\mu$ = center, and $\sigma$ = sigma (see Wikipedia for more info)

### Gaussian Height

Max height occurs at x = $\mu$

### Gaussian FWHM

FWHM is found by finding the values of x at 1/2 the max height. For simplicity $\mu$ can be set to 0.

## Lorentzian

A = amplitude, $\mu$ = center, and $\sigma$ = sigma (see Wikipedia for more info)

### Lorentzian Height

Max height occurs at x = $\mu$

### Lorentzian FWHM

FWHM is found by finding the values of x at 1/2 the max height. For simplicity $\mu$ can be set to 0.

## Voigt

A = amplitude, $\mu$ = center, $\sigma$ = sigma, and $\gamma$ = gamma (see Wikipedia for more info)

where

erfc is the complimentary error function. As above,

### Voigt Height

Max height occurs at x = $\mu$

### Voigt FWHM

FWHM is found by finding the values of x at 1/2 the max height. For simplicity $\mu$ can be set to 0.

Hmmm IDK where to go from here.

## Pseudo Voigt

A = amplitude, $\mu$ = center, $\sigma$ = sigma, and $\alpha$ = fraction (see Wikipedia for more info)

where $\sigma_g = {\sigma}/{\sqrt{2\ln{2}}}$ so that the full width at half maximum of each component and of the sum is $2\sigma$

### Pseudo Voigt Height

Max height occurs at x = $\mu$

### Pseudo Voigt FWHM

The definition of the function depends on FWHM=2$\sigma$.

Please see derivations of Gaussian and Lorentzian.

## Moffat

A = amplitude, $\mu$ = center, $\sigma$ = sigma, and $\beta$ = beta (see Wikipedia for more info)

### Moffat Height

Max height occurs at x = $\mu$

### Moffat FWHM

FWHM is found by finding the values of x at 1/2 the max height. For simplicity $\mu$ can be set to 0.

## Pearson 7

A = amplitude, $\mu$ = center, $\sigma$ = sigma, and $m$ = exponent (see wikipedia for more info)

where $\beta$ is the beta function

### Pearson 7 Height

Max height occurs at x = $\mu$

### Pearson 7 FWHM

FWHM is found by finding the values of x at 1/2 the max height. For simplicity $\mu$ can be set to 0.

## Students T

A = amplitude, $\mu$ = center, and $\sigma$ = sigma (see Wikipedia for more info)

where $\Gamma(x)$ is the gamma function.

### Students T Height

Max height occurs at x = $\mu$

### Students T FWHM

FWHM is found by finding the values of x at 1/2 the max height. For simplicity $\mu$ can be set to 0.

## Breit-Wigner-Fano

A = amplitude, $\mu$ = center, $\sigma$ = sigma, and $q$ = exponent (see Wikipedia for more info)

### Breit-Wigner-Fano Height

Max height occurs at x = $\mu$

### Breit-Wigner-Fano FWHM

FWHM is found by finding the values of x at 1/2 the max height. For simplicity $\mu$ can be set to 0.

## Lognormal

A = amplitude, $\mu$ = center, and $\sigma$ = sigma (see Wikipedia for more info)

### Lognormal Height

Max height occurs at x = $e^{\mu-\sigma^2}$

### Lognormal FWHM

FWHM is found by finding the values of x at 1/2 the max height.

## Damped Oscillator

A = amplitude, $\mu$ = center, and $\sigma$ = sigma (see Wikipedia for more info)

### Damped Oscillator Height

Max height occurs at x = $\mu$

### Damped Oscillator FWHM

FWHM is found by finding the values of x at 1/2 the max height.

Substitute $y=x_0^2$

Substitute back $y=x_0^2$

Two peaks each with a FWHM of:

## Damped Harmonic Oscillator

A = amplitude, $\mu$ = center, $\sigma$ = sigma, and $\gamma$ = gamma (see Wikipedia and DAVE/PAN for more info)

### DHO Height

Max height occurs at x = $\mu$.

### DHO FWHM

FWHM is found by finding the values of x at 1/2 the max height.

Hmmm IDK where to go from here.

But as defined in DAVE/PAN:

## Exponential Gaussian

A = amplitude, $\mu$ = center, $\sigma$ = sigma, and $\gamma$ = gamma (see [Wikipedia(http://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution)

where :func:erfc is the complimentary error function.

### Exponential Gaussian Height

Max height occurs at x = $\mu$

### Exponential Gaussian FWHM

FWHM is found by finding the values of x at 1/2 the max height. For simplicity $\mu$ can be set to 0.

Hmmm IDK where to go from here.

But by observation:

## Skewed Gaussian

A = amplitude, $\mu$ = center, $\sigma$ = sigma, and $\gamma$ = gamma (see Wikipedia for more info)

where :func:erf is the error function.

### Skewed Gaussian Height

Max height occurs at x = $\mu$

### Skewed Gaussian FWHM

FWHM is found by finding the values of x at 1/2 the max height. For simplicity $\mu$ can be set to 0.

Hmmm IDK where to go from here.

But by observation:

## Donaich Sunjic (photo-emission)

A = amplitude, $\mu$ = center, $\sigma$ = sigma, and $\gamma$ = gamma (see Casaxps for more info)

### Donaich Sunjic Height

Max height occurs at x = $\mu$

### Donaich Sunjic FWHM

FWHM is found by finding the values of x at 1/2 the max height. For simplicity $\mu$ can be set to 0.

Hmmm IDK where to go from here.