# Getting Green's Function for Laplace's Equation in Cylindrical Coordinates

I am trying to understand a derivation for finding the Green's function of Laplace's eq in cylindrical coordinates.

Let the Green's function be written as: $$G(r,\theta,z,r',\theta',z') = G(\mathbf{r},\mathbf{r}')$$

We know that in cylindrical coordinates

$$\nabla^{2}G(\mathbf{r},\mathbf{r}')=\delta(\mathbf{r}-\mathbf{r}')=\frac{1}{r}\delta(r-r')\delta(\theta-\theta')\delta(z-z')$$

Using the cylindrical Laplacian we can then write:

$$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial G}{\partial r}\right)+\frac{1}{r^2}\frac{\partial^2 G}{\partial \theta^2} + \frac{\partial^2 G}{\partial z^2} = \frac{1}{r}\delta(r-r')\delta(\theta-\theta')\delta(z-z')$$

Using the identities:

$$\delta(\theta-\theta') = \frac{1}{2\pi}\sum^{\infty}_{m=-\infty}e^{im(\theta-\theta')}$$

$$\delta(z-z') = \frac{1}{2\pi}\int^{\infty}_{-\infty}e^{ik(z-z')}dk$$

We find that

$$G(\mathbf{r},\mathbf{r}') = \frac{1}{4\pi^2}\sum^{\infty}_{m=-\infty}\int^{\infty}_{-\infty}g_{m}(r,r')e^{ik(z-z')}e^{im(\theta-\theta')}dk$$

$$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial g_{m}}{\partial r}\right) - \left(k^2+\frac{m^2}{r^2} \right)g_{m} = \frac{1}{r}\delta(r-r')$$

I'm getting confused on the last step. It seems that the Green's function is assumed to be $$G(r,\theta,z,r',\theta',z') = R(r)Q(\theta)Z(z)$$ and this is plugged into the cylindrical Laplacian but I'm not seeing how to get $$G(r,\theta,z,r',\theta',z')$$ in the form shown or how to get that ODE in terms of $$g_{m}$$. Are there any ideas on how to do this intermediate step?

• You little g_m is written as a function r and r', but it also depends need to depend on k as a continuous parameter.
– hft
Mar 4 at 1:19

The expression

$$G(\mathbf{r},\mathbf{r}') = \frac{1}{4\pi^2}\sum^{\infty}_{m=-\infty}\int^{\infty}_{-\infty}g_{m}(r,r')e^{ik(z-z')}e^{im(\theta-\theta')}dk\tag{∗}$$

is nothing but a Fourier expansion of some function of $$z$$ and $$\theta$$, with $$g_m(r,r')$$ being the expansion coefficients. It's not something you should get: it's an ansatz.

Now, to get your ODE you can go the following way. Use your identities to express the whole RHS of your PDE (the product of deltas) in the form

$$\sum_{m=-\infty}^\infty\int_{-\infty}^\infty...\mathrm{d}k.\tag1$$

Then insert $$G(\mathbf r, \mathbf r')$$ in the form $$(*)$$ into the PDE. Apply each differential operator termwise, and then move all the summands-integrands under the single sum-integral sign.

You now get the equation

$$\sum_{m=-\infty}^\infty\int_{-\infty}^\infty\exp\big(ik(z-z')+im(\theta-\theta')\big)f_\mathrm{ODE}(k,m,r)\,\mathrm{d}k=0,\tag2$$

where $$f_\mathrm{ODE}(k,m,r)$$ is the $$\mathrm{LHS}-\mathrm{RHS}$$ of your ODE.

The LHS of $$(2)$$ is a Fourier expansion of some function of $$z-z'$$ and $$\theta-\theta'$$, where $$f_\mathrm{ODE}(k,m,r)$$ denotes the Fourier coefficients. By uniqueness of Fourier expansion, the only way for the whole LHS of $$(2)$$ to be zero is for $$f_\mathrm{ODE}$$ to be zero for all $$k$$ and $$m$$. Since we also want this equation to hold for all $$r$$, we finally get

$$f_\mathrm{ODE}=0\;\forall\, k,m,r,\tag3$$

which is equivalen to your ODE.

To make the typing easier I will set $$r'$$, $$\theta'$$, and $$z'$$ to zero (wlog). So, now G is considered to be a function of just $$r$$, $$\theta$$, and $$z$$ (wlog).

Any[1] function of $$r$$, $$\theta$$, and $$z$$ can be expanded in a double fourier series like: $$G(r,\theta, z) = \frac{1}{2\pi}\sum_m \frac{1}{2\pi}\int dk g_{mk}(r)e^{ikz}e^{im\theta}\;,$$ where $$k$$ is a continuous parameter and $$m$$ is a discrete parameter.

Plug this expansion into your equation for $$G$$ and find: $$\frac{1}{2\pi}\sum_m \frac{1}{2\pi}\int dk e^{ikz}e^{im\theta} \left[\frac{1}{r}\frac{d}{dr}r\frac{dg_{mk}(r)}{dr} -\frac{n^2}{r^2}g_{mk}(r) - k^2 g_{mk}(r)\right] =\frac{1}{r}\delta(r)\frac{1}{2\pi}\sum_{m'} \frac{1}{2\pi}\int d{k'} e^{ik'z}e^{im'\theta}$$

Because the $$e^{im\theta}$$ and $$e^{ikz}$$ are a complete orthonormal set, each side of the equation holds term by term for all $$m$$ and $$k$$. So we can "drop the sums" over the dummy variable on each side and we find: $$\left[\frac{1}{r}\frac{d}{dr}r\frac{dg_{mk}(r)}{dr} -\frac{m^2}{r^2}g_{mk}(r) - k^2 g_{mk}(r)\right] =\frac{1}{r}\delta(r)$$

[1] Some caveats may apply, but we don't care.