Furthermore, this book covers many functions that arise naturally in the study of statistics and physics, and can serve as a mathematical reference to such functions and their properties


\chapter{Special real functions}

As consequences of the many definitions introduced in real analysis (and therefore differential equations), many special functions that cannot be expressed as finite combinations of the 4 arithmetic operations arise.

Many of these functions exhibit interesting properties that either arise from their definition or can be proven using the methods of real analysis.


Note that I will define the functions for the most general domain possible (such as the complex plane), however the results we look at are applicable for real numbers.

\chapter{Sine function}
\[\sin(0)=0, \sin''=-\sin\]
\[ \sin(x)= \sum^{\infty}_{n=0} (-1)^{n}\frac{x^{2n+1}}{(2n+1)!} \]
\chapter{Cosine function}
\[\cos(0)=1, \cos''=-\cos\]
\[ \cos(x)= \sum^{\infty}_{n=0} (-1)^{n}\frac{x^{2n}}{(2n)!} \]
\section{}
\chapter{Exponential function}
\[ \exp(0)=1, \exp'=\exp \]
\[ \exp(x)= \sum^{\infty}_{n=0} \frac{x^n}{n!} \]
\chapter{Natural logarithm}
\[ \ln (e^x)=x\]
\[ \ln (e^x)=x\]
\section{Series on $[1,2]$}

\chapter{Gamma function}
\section{Euler's reflection formula}
\chapter{Beta function}
\chapter{Bessel functions}
\section{Bessel functions}

Bessel functions

THe following proposition essentially serves as a shortcut for the series method; the series method is indeed the reasoning for such a proposition.
\begin{propositition}[Bessel function substitution method]
\[x^2 y'' + (1-2s)xy' + [(s^2 -r^2 \alpha^2 )  + a^2 r^2 x^{2r}]y = 0\]
\[ y(x) = c_1 x^s J_{\alpha}(ax^r) + c_2 x^s Y_{\alpha}(ax^r)\]
\end{proposition}
\chapter{Modified Bessel functions}
\chapter{Airy functions}
\chapter{Weierstrass function}

\chapter{Riemann zeta function}
\[\zeta(s)\]
\section{Euler's product formula}
The following property due to Euler connects this function intimately to number theory.


\chapter{Riemann xi function}
\chapter{Dirichlet eta function}


\chapter{Lambert W function}

\chapter{Spherical harmonics}
