\chapter{Bessel functions and related}


Frequently arise in physics as solutions to a specific ODE encountered when solving PDEs that exhibit polar or cylindrical symmetry.

The specific ODE is the following
\[x^2 y''+xy'+(x^2-\nu^2)y=0\].





\section{Bessel functions (first kind)}


From an analytical perspective, the first kind of Bessel functions are those that are nonsingular at the origin (the second kind of Bessel function "blows up" near $0$).

Although Bessel functions of the first kind are deeply rooted in physics, they also appear in the Fourier series coefficients for functions of the form $\sin(at+b\sin(ct))$, which have significance in signal processing, namely frequency modulation (FM).




Bessel's equation is a second order homogeneous linear ODE, and therefore according to the theory of ODEs, one has a linear subspace $V \leq C^2 (\mathbb{R}_{+})$ with $\mathrm{dim}(2)$ that represents all the solutions to the ODE.

This immediatly provokes the question of finding a basis for $V$; this leads to the notion of Bessel functions of the first kind and Bessel functions of the second kind. One can employ the Frobenius method to arrive at the first of these basis functions.

\begin{definition}[Bessel functions (first kind)]
	\[J_{\nu}(x) = \sum^{\infty}_{n=0} \frac{(-1)^n}{n! \Gamma(n+\nu+1)} (\frac{x}{2})^{2n+\nu}\]
\end{definition}

There has been (and continues to wage) extensive research on these functions, possibly second only to the gamma function.

\[J_{-n}(x)=(-1)^n J_n(x)\]

\[J'_{n}(x) = J_{n-1}(x)-\frac{n}{x}J_{n}(x)\]
\[J'_{n}(x) = \frac{1}{2}[J_{n-1}(x)-J_{n+1}(x)\]



\subsection{Alternative definitions}











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]
\[x^2 y'' + (1-2s)xy' + [(s^2 -r^2 \nu^2 )  + a^2 r^2 x^{2r}]y = 0\]
\[ y(x) = c_1 x^s J_{\nu}(ax^r) + c_2 x^s Y_{\nu}(ax^r)\]
\end{proposition}

\section{Bessel functions (second kind)}



\section{Modified Bessel functions}

\section{Airy functions}


Related to the Bessel functions are the \emph{Airy functions}

\[y''-xy=0\]


\[\mathrm{Ai}(x) = 3^{-2/3} \sum^{\infty}_{n=0} \frac{x^{3n}}{n! \Gamma(n+2/3)9} -3^{-4/3} \sum^{\infty}_{n=0} \frac{x^{3n+1}}{n! \Gamma(n+4/3)9}\]