The presentation of the Bessel functions are based on a articles in [49] [50]. The Bessel functions are solutions \( y(x) \) : $$ \begin{equation} y(x) = c_1 J_{n}(x) + c_2 Y_{n}(x) \tag{8.46} \end{equation} $$ of Bessel's differential equation : $$ \begin{equation} \frac{d^2 y}{dx^2} + \frac{1}{x}\frac{dy}{dx} +\left (1-\frac{n^2}{x^2} \right ) y = 0 \tag{8.47} \end{equation} $$ Bessel functions are also known as cylinder functions or cylindrical harmonics because they are found in the solution to Laplace's equation in cylindrical coordinates. Bessel functions of the first kind, denoted as \( J_n(x) \), are solutions of Bessel's differential equation that are finite at the origin (x = 0) for non-negative integer \( n \), and diverge as x approaches zero for negative non-integer \( n \). The solution type (e.g. integer or non-integer) and normalization of \( J_n(x) \) are defined by its properties below. It is possible to define the function by its Taylor series expansion around x = 0: $$ \begin{equation} J_n(x) = \sum_{m=0}^{\infty} \frac{(-1)^m}{m!\Gamma (m+n+1)} (\frac{x}{2})^{2m+n} \tag{8.48} \end{equation} $$ where \( \Gamma(z) \) is the gamma function, a generalization of the factorial function to non-integer values.
The Bessel functions of the second kind, denoted by \( Y_n(x) \), are also solutions of the Bessel differential equation. However, they are singular (infinite) at the origin (x = 0). Such solutions are not relevant for the applications in the this presentation, and thus we consider \( c_2=0 \) in equation (8.46).
Some useful properties of the Bessel functions of first kind are: $$ \begin{align} J_{-n} (x) & = (-1)^n J_{n} (x) \tag{8.49} \\ \tag{8.50} \frac{d}{dx} \left (x^n J_n(x) \right ) & = x^n J_{n-1} (x) \end{align} $$
The modified Bessel equation is very similar to equation (8.47) and may be presented: $$ \begin{equation} \frac{d^2 y}{dx^2} + \frac{1}{x} \frac{dy}{dx} - \left (1-\frac{n^2}{x^2} \right ) y = 0 \tag{8.51} \end{equation} $$ The solutions are the modified Bessel functions of the first and second kinds, and can be written: $$ \begin{align} y &= a_1 J_n(-ix) + a_2 Y_n(-ix) \tag{8.52}\\ &= c_1 I_n(x) + c_2 K_n(x) \tag{8.53} \end{align} $$ where \( I_n(x) \) are \( K_n(x) \) modified Bessel functions of order \( n \) of the first and second kind, respectively.
The modified Bessel function of order \( n \) of first kind may be defined by the relation: $$ \begin{align} I_n(x) &= i^{-n} J_n(ix) \tag{8.54} \end{align} $$
5: Note that \( n \) may in general be a complex number. Here we assume \( n \) to be an integer.