Projection matrix in cylindrical geometry

In polar coordinates equation (B.6) becomes

$$F_{nm} = \frac{1}{\pi R_1^2}\int\limits_{0}^{R_1} \int\limits_{0}^{2\pi} \psi_{n}^{(1)}\psi_{m}^{(2)} r d\theta dr.$$ (B.15)

Substituting in equation (2.48) for $\psi_n (r)$ and performing the integration with respect to $\theta$ gives:

$$F_{nm}=\frac{2}{R_1^2 J_0(\gamma_n) J_0(\gamma_m)}\int\limits_{0}^{R_1} r J_0(\gamma_n r/R_1) J_0(\gamma_m r/R_2) dr.$$ (B.16)

This is in the form of the standard integral from equation (A.1) of appendix A. Substituting in the variables: $x=r$, $p=q=0$, $\alpha=\gamma_n/R_1$ and $\beta=\gamma_m/R_2$ gives

$${F_{nm} = \left(\frac{2}{R_1^2 J_0(\gamma_n) J_0(\gamma_m)}\right) \times}$$

$$\left[ \frac{(\gamma_m r/R_2) J_0(\gamma_n r/R_1) J_{-1}(\gamma_m... ...0(\gamma_m r/R_2)} {(\gamma_n/R_1)^2 - (\gamma_m/R_2)^2} \right]_{r=0}^{r=R_1}.$$

When the evaluation is carried out the contribution when $r=0$ is zero giving:

$${F_{nm} = \left(\frac{2}{R_1^2 J_0(\gamma_n) J_0(\gamma_m)}\right) \times}$$

$$\frac{(\gamma_m R_1/R_2) J_0(\gamma_n) J_{-1}(\gamma_m R_1/R_2) -... ... J_{-1}(\gamma_n) J_0(\gamma_m R_1/R_2)} {(\gamma_n/R_1)^2 - (\gamma_m/R_2)^2}.$$ (B.17)

Now noticing from equation (A.2) that $J_{-1}(x) = - J_1(x)$ and using the fact that $\gamma_n$ is a zero of $J_1$ the second term vanishes:

$$F_{nm} = \left(\frac{2}{R_1^2 J_0(\gamma_n) J_0(\gamma_m)}\... ...J_1(\gamma_m R_1/R_2)} {(\gamma_m/R_2)^2 - (\gamma_n/R_1)^2}.$$ (B.18)

Expressing this in terms of the ratio of the radii, $\beta = R_1/R_2$ we get

$$F_{nm} = \frac{2 \beta \gamma_m J_1(\beta \gamma_m)} {(\beta^2 \gamma_m^2 - \gamma_n^2) J_0(\gamma_m)}$$ (B.19)

hence we have proved equation (2.85).

The integration used to obtain the analytical expression for $V_{nm}$ is identical to that for $F_{nm}$ except that the labels are interchanged for surface 1 and surface 2. Interchanging $R_1$ and $R_2$ means that $\beta = R_1/R_2$ will be replaced with $1/ \beta = R_2/R_1$ giving $V_{nm}(\beta) = F_{nm}(1/\beta)$.