I was assigned this problem, and quite honestly I do not know where to begin.
If I could get some help and an explanation of the Bessel function, also?
Thank you.
I know my conditions are: \begin{align} V(a, z) &= 0\\ V(r, 0) &= 1 \end{align} $V\to 0$ as $z\to \infty$
To show that solution is of the form presented, we first need to know what the Laplace equation in cylindrical coordinates is. $$ \nabla^2u = \frac{1}{r}\partial_r(ru_r) + \frac{1}{r^2}u_{\phi\phi} + u_{zz} = 0 $$ We have no angular dependence so $$ \nabla^2u = \frac{1}{r}\partial_r(ru_r) + u_{zz} = 0 $$ Let $u(r,z) = R(r)Z(z)$. \begin{alignat}{2} \frac{1}{r}(R'Z + rR''Z) + RZ'' &= \frac{R' + rR''}{rR} + \frac{Z''}{Z}\\ \frac{R' + rR''}{rR} &= -\frac{Z''}{Z} &&{}= -k^2\\ rR'' + R' +rk^2R &=0\tag{1}\\ Z''-k^2Z &=0 \end{alignat} Therefore, we have $Z\sim\{e^{k z}, e^{-k z}\}$. Since the equation is bounded at infinity, $Z\sim e^{-k z}$. Now let's multiple equation (1) by $r$. $$ r^2R'' + rR' + r^2k^2 R = 0 $$ which is the Bessel equation of order zero since $m^2 = 0$. The general form of the Bessel equation is $$ r^2R'' + rR' + (r^2k^2 - m^2) R = 0 $$ Thus, $R(r) = \mathcal{J}_0(k r)$. Then $R(a) = \mathcal{J}_{0n}(k_n a) = 0$. Let $\lambda_n = k_na$ be the zeros of the Bessel equatoin. Then $k_n = \frac{\lambda_n}{a}$. $$ V(r, z) = \sum_{n=1}^{\infty}A_{0n}\mathcal{J}_{0n}\Bigl(\frac{\lambda_{0n}r}{a}\Bigr)e^{-\frac{\lambda_{0n}z}{a}} $$