View source for Template:Separation of variables for the r and theta coordinates

== Separation of Variable for the <math>r</math> and <math>\theta</math> coordinates ==

For the solution of 
<center>
<math>
\frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial
Y}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 Y}{\partial
\theta^2} = k_m^2 Y(r,\theta),
</math>
</center>
we use the separation
<center>
<math>
\,\!Y(r,\theta) =: R(r) \Theta(\theta).
</math>
</center>
Substituting this into Laplace's equation yields
<center>
<math>
\frac{r^2}{R(r)} \left[ \frac{1}{r} \frac{\mathrm{d}}{\mathrm{d}r} \left( r
\frac{\mathrm{d} R}{\mathrm{d}r} \right) - k_m^2 R(r) \right] = -
\frac{1}{\Theta (\theta)} \frac{\mathrm{d}^2 \Theta}{\mathrm{d}
\theta^2} = \eta^2, 
</math>
</center>
where the separation constant <math>\eta</math> must be an integer, say <math>\nu</math>, 
in order for the potential to be continuous. <math>\Theta
(\theta)</math> can therefore be expressed as 
<center>
<math>
\Theta (\theta) = C \, \mathrm{e}^{\mathrm{i} \nu \theta}, \quad \nu \in \mathbb{Z}.
</math>
</center>
We also obtain the following expression
<center>
<math>
r \frac{\mathrm{d}}{\mathrm{d}r} \left( r \frac{\mathrm{d}
R}{\mathrm{d} r} \right) - (\nu^2 + k_m^2 r^2) R(r) = 0, \quad \nu \in
\mathbb{Z}.
</math>
</center>
Substituting <math>\tilde{r}:=k_m r</math> and writing <math>\tilde{R} (\tilde{r}) :=
R(\tilde{r}/k_m) = R(r)</math>, this can be rewritten as
<center>
<math>
\tilde{r}^2 \frac{\mathrm{d}^2 \tilde{R}}{\mathrm{d} \tilde{r}^2}
+ \tilde{r} \frac{\mathrm{d} \tilde{R}}{\mathrm{d} \tilde{r}}
- (\nu^2 + \tilde{r}^2)\, \tilde{R} = 0, \quad \nu \in \mathbb{Z},
</math>
</center>
which is the modified version of Bessel's equation. Substituting back,
the general solution is given by
<center>
<math>
R(r) = D_\nu \, I_\nu(k_m r) + E_\nu \, K_\nu(k_m r),\ \nu \in \mathbb{Z},
</math>
</center>
where <math>I_\nu</math> and <math>K_\nu</math> are the modified 
[http://en.wikipedia.org/wiki/Bessel_function Bessel functions] of the first
and second kind, respectively, of order <math>\nu</math>.

Note that <math>K_\nu (\mathrm{i} x) = \pi / 2\,\,
\mathrm{i}^{\nu+1} H_\nu^{(2)}(x)</math> with <math>H_\nu^{(2)}</math> denoting
the Hankel function of the second kind of order <math>\nu</math>.
Also, <math>I_\nu</math> does not satisfy the [[Sommerfeld Radiation Condition]]
since it becomes unbounded for increasing real argument and it
represents incoming waves.