fermi-dirac-calculator

A script written in C++ to compute the various Fermi-Dirac quantities for a given species.

While writing these scripts, I try to use my own functions as much as I can, all the way to basic numerical integration. I resort to using the MATLAB engine for symbolic integration. I operate almost solely in the complex domain to ensure compliance with the initial definitions of the functions.

Definitions of functions

Polylogarithm

The polylogarithm is defined by a power series in a complex parameter $z$, given by

$$ Li_s(z) = \sum_{n=1}^{\infty}\frac{z^n}{n^s} $$

The polylogarithm is defined for all complex orders $s$ and all complex arguments where $|z| < 1$.

Derivative of the polylogarithm

The derivative of the polylogarithm is simple, and is given by

$$ \frac{d}{ds} Li_s(z) = \sum_{n=1}^{\infty} \frac{\partial}{\partial s} \frac{z^n}{n^s} = -\sum_{n=1}^{\infty} \frac{z^n}{n^s}\ln(n) $$

Gamma function

The gamma function is a continuation of the factorial function to complex numbers. For any complex number $z$ with $\mathcal{R}(z) > 0$, the gamma function is defined as

$$ \Gamma(z) = \int_{0}^{\infty}t^{z-1}e^{-t}dt $$

Digamma function

The digamma function appears multiple times in the relativistic expansion of Fermi-Dirac integrals. It is commonly expressed as

$$ \psi(z) = \frac{d}{dz}\ln \Gamma(z) = \frac{\Gamma'(z)}{\Gamma(z)} $$

I use the integral representation to allow for easier computation.

$$ \psi(z) = -\gamma + \int_{0}^{1}\frac{1-x^{z-1}}{1-x}dx $$

where $\gamma$ represents the Euler-Mashceroni constant, $\gamma \approx 0.57721$

Modified Bessel function of the second kind

Modified Bessel functions of the second kind are often used in non-relativistic expansions of Fermi-dirac integrals. With some simplification relevant to the work,

$$ K_2(z) \simeq \sqrt{\frac{\pi}{2z}}\sum_{n=0}^{\infty}\frac{\Gamma(5/2 + n)}{\Gamma(5/2 - n)n! (2z)^n} $$

A more exact form of the modified Bessel function (the one, in fact, used for computation) is given by

$$ K_2(z) = \frac{2}{z^2} - \frac{1}{z} + \sum_{n=0}^{\infty}\frac{1}{n!(n+2)!} \left(\frac{z}{2}\right)^{2n+2} \left[\frac{\psi(n+1) + \psi(n+3)}{2} - ln \frac{z}{2}\right] $$