Bessel function, Generating function

[LagrangeEuler]

Generating function for Bessel function is defined by

$$G(x,t)=e^{\frac{x}{2}(t-\frac{1}{t})}=\sum^{\infty}_{n=-\infty}J_n(x)t^n$$
Why here we have Laurent series, even in case when functions are of real variables?

 

[Orodruin]

It is only a Laurent series if you look at it as an expansion around $t = 0$, at which the function is clearly singular for all $x \neq 0$. The more relevant point is to look at $f(r,\varphi) = G(r,e^{i\varphi})$ (i.e., $|t| = 1$), which solves the eigenvalue equation $(\nabla^2 + 1) f(r,\varphi) = 0$, where $\nabla^2$ is the Laplace operator written in polar coordinates. You will find that
$$
\nabla^2 e^{r(e^{i\varphi}- e^{-i\varphi})/2} = \nabla^2 e^{r\sin(\varphi)} = e^{r\sin(\varphi)}.
$$
Correspondingly, you will find that
$$
(\nabla^2 + 1) f(r,\varphi) = \sum_{n=-\infty}^\infty e^{in\varphi} \left( \frac{1}{r} \partial_r r \partial_r J_n(r) + \left(1 - \frac{n^2}{r^2}\right) J_n(r)\right) = 0.
$$
Since each $e^{in\varphi}$ is linearly independent, this can only be satisfied if
$$
\frac{1}{r} \partial_r r \partial_r J_n(r) + \left(1 - \frac{n^2}{r^2}\right) J_n(r) = 0,
$$
which is Bessel's differential equation.

 

[Orodruin]

It is only a Laurent series if you look at it as an expansion around $t = 0$, at which the function is clearly singular for all $x \neq 0$. The more relevant point is to look at $f(r,\varphi) = G(r,e^{i\varphi})$ (i.e., $|t| = 1$), which solves the eigenvalue equation $(\nabla^2 + 1) f(r,\varphi) = 0$, where $\nabla^2$ is the Laplace operator written in polar coordinates. You will find that
$$
\nabla^2 e^{r(e^{i\varphi}- e^{-i\varphi})/2} = \nabla^2 e^{r\sin(\varphi)} = e^{r\sin(\varphi)}.
$$
Correspondingly, you will find that
$$
(\nabla^2 + 1) f(r,\varphi) = \sum_{n=-\infty}^\infty e^{in\varphi} \left( \frac{1}{r} \partial_r r \partial_r J_n(r) + \left(1 - \frac{n^2}{r^2}\right) J_n(r)\right) = 0.
$$
Since each $e^{in\varphi}$ is linearly independent, this can only be satisfied if
$$
\frac{1}{r} \partial_r r \partial_r J_n(r) + \left(1 - \frac{n^2}{r^2}\right) J_n(r) = 0,
$$
which is Bessel's differential equation.