Bessel applications
\[
e^{x \left( z - 1/z \right) /2} = \sum_{n=-\infty}^{\infty} z^n J_n (x)
\]
Example 1: We consider Laplace's equation in cylindrical coordinates
\begin{align*}
\frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\,\frac{\partial u}{\partial r} + \frac{1}{r^2}\,\frac{\partial^2 u}{\partial \vartheta^2} + \frac{\partial^2 u}{\partial z^2} &= 0 \qquad \mbox{for} \quad 0 \le r < a, \quad 0 < z < h,
\\
u(r, \vartheta , 0 ) = u(r, \vartheta , h ) &= 0 ,
\\
u(a, \vartheta , z) &= g(vartheta , z) .
\end{align*}
By separation of variables, u(r, θ, z) = R(r) v(θ, z), we obtain
\[
\left[ R' + \frac{1}{r}\,R' \right] v(\theta , z) + R(r) \left[ \frac{1}{r^2}\,\frac{\partial^2 v}{\partial \vartheta^2} + \frac{\partial^2 v}{\partial z^2} \right] = 0 \qquad \Longrightarrow \qquad \frac{r^2 R'' + r\,R'}{R} = - \frac{1}{v} \left[ \frac{\partial^2 v}{\partial \theta^2} + r^2 \frac{\partial^2 v}{\partial z^2} \right] = \lambda \,r^2.
\]
So we get a differential equation for R(r):
\[
r^2 R'' (r) + r\,R' (r) - \lambda\,r^2 R(r) = 0 , \qquad 0 \le r < a.
\]
We separate variables for v(θ, z) = Θ(ϑ) Z(z) and get
\[
\Theta'' (\vartheta ) \,Z(z) + r^2 \Theta (\vartheta )\,Z'' (z) + \lambda \,r^2 \Theta (\vartheta )\,Z (z) = 0
\qquad \Longrightarrow \qquad \frac{\Theta'' (\vartheta )}{\Theta (\vartheta )} = - r^2 \frac{Z'' (z)}{Z(z)} - \lambda = -\mu .
\]
So we obtain the Sturm--Liouville problem for Z(z):
\[
Z'' (z) + \lambda\,Z(z) = 0, \qquad Z(0) =0, \quad Z(h) = 0.
\]
Its eigenvalues are λn = (n π/h)², to which correspond eigenfunctions
\[
Z_n (z) = \sin \left( \frac{n\pi z}{h} \right) , \qquad n=1,2,3,\ldots .
\]
For function Θ(ϑ) we also get a Sturm--Liouville problem, but with periodic boundary conditions:
\[
\Theta'' (\vartheta ) + \mu\,\Theta (\vartheta ) = 0 , \qquad \Theta (\vartheta ) = \Theta (\vartheta + 2\pi ) .
\]
Its solution is well-known and we have eigenvalues and eigenfunctions:
\[
\Theta_k (\vartheta ) = a_k \cos (k\vartheta ) + b_k \sin (k\vartheta ) , \qquad k=0,1,2,\ldots .
\]
For function R(r), we get
\[
r^2 R'' + r\, R' - \left( k^2 + r^2 \frac{n^2 \pi^2}{h^2} \right) R(r) = 0.
\]
This equation is singular at r = 0. Actually, it is a modified Bessel equation, so it has two linearly independent solutions \( I_k \left( \frac{n\pu r}{h} \right) \quad \mbox{and} \quad K_k \left( \frac{n\pu r}{h} \right) . \) However, the function Kk is unbounded at the origin, so we dismiss it. The solution in r variable becomes
\[
R_{n,k} (r) = \frac{I_k (n\pi r/h)}{I_k (n\pi a/h} ,
\]
which is 1 at r = 𝑎. We write the solution in the form
\[
u(r, \vartheta , z) = \frac{1}{2} \sum_{n\ge 1} a_{0,n} \,\frac{I_k (n\pi r/h)}{I_k (n\pi a/h} \,\sin \left( \frac{n\pi z}{h} \right) + \sum_{k\ge 1} \sum_{n\ge 1} \frac{I_k (n\pi r/h)}{I_k (n\pi a/h} \,\sin \left( \frac{n\pi z}{h} \right) \left[ a_{n,k} \cos (k\vartheta ) + b_{n,k} \sin (k\vartheta ) \right] ,
\]
where
\begin{align*}
a_{n,k} &= \frac{2}{\pi h} \int_0^h {\text d} z \int_{-\pi}^{\pi} {\text d}\vartheta \,g(\vartheta , z)\,\sin \left( \frac{n\pi z}{h} \right) \cos (k\vartheta ), \qquad n=1,2,\ldots ; \quad k=0,1,2,\ldots ;
\\
b_{n,k} &= \frac{2}{\pi h} \int_0^h {\text d} z \int_{-\pi}^{\pi} {\text d}\vartheta \,g(\vartheta , z)\,\sin \left( \frac{n\pi z}{h} \right) \sin (k\vartheta ), \qquad n=1,2,\ldots ; \quad k=1,2,\ldots .
\end{align*}
End of Example 1
- Bowman, Frank Introduction to Bessel Functions (Dover: New York, 1958). ISBN 0-486-60462-4.
- Dutka, J., On the early history of Bessel functions, Archive for History of Exact Sciences, volume 49, pages 105–134 (1995). https://doi.org/10.1007/BF00376544
- Watson, G.N., A Treatise on the Theory of Bessel Functions,
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