\part{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.

- \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}
\[ \ln (e^x)=x\]
\section{Series on $[1,2]$}

- \chapter{Gamma function}
- \section{Euler's reflection formula}
- \chapter{Beta function}
- \chapter{Bessel functions}
- \chapter{Modified Bessel functions}
- \chapter{Airy functions}
- \chapter{Weierstrass function}

\chapter{Riemann zeta function}
\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}

