# Frustrations while solving Poisson's equation

Recently I faced several moral issues while I was solving Poisson's equation in a simple case of point charge in $\overline{r}=\overline{0}$. They concern Fourie transform and improper integral calculation.

Here is the deal. I have $\Delta\phi(\overline{r})=4\pi e\delta(\overline{r})$.
Let's take Fourier transform as $F(\overline{p})=\int{f(\overline{r})\exp(-i\overline{r}\cdot\overline{p})} \, d\overline{r}$ and the inverse transform as

$$f(\overline{r})=\frac{1}{(2\pi)^3}\int{F(\overline{p})\exp(i\overline{r}\cdot\overline{p})} \, d\overline{p}.$$

So I can rewrite my equation as

$$\frac{1}{(2\pi)^3}\Delta \left[\int{\Phi(\overline{p})\exp(i\overline{r}\cdot\overline{p})} \, d\overline{p}\right]=\frac{4\pi e}{(2\pi)^3}\left[\int{\exp(i\overline{r}\cdot\overline{p})} \, d\overline{p}\right]$$

or $$\int{[-p^2\Phi(\overline{p})]\exp(i\overline{r}\cdot\overline{p})d\overline{p}}=\int{[4\pi e]\exp(i\overline{r}\cdot\overline{p})d\overline{p}}.$$

This gives me algebraical equation for $\Phi(\overline{p})$:
$\Phi(\overline{p})=-\frac{4\pi e}{p^2}$.
After that I can find the potential as
$\phi(\overline{r})=-\frac{4\pi e}{(2\pi)^3}\int{\frac{\exp(i\overline{r}\cdot\overline{p})}{p^2}d\overline{p}}=-\frac{e}{2\pi^2}\int{\frac{\exp(i\overline{r}\cdot\overline{p})}{p^2}d\overline{p}}$.
And here comes my first confusion: as we don't use the inverse transform the choise of coefficients in straight and invers Fourie transforms results the solution. If we take $F(\overline{p})=\frac{1}{(2\pi)^3}\int{f(\overline{r})\exp(-i\overline{r}\cdot\overline{p})}d\overline{r}$ and $f(\overline{r})=\int{F(\overline{p})\exp(i\overline{r}\cdot\overline{p})}d\overline{p}$, it gives us
$\phi(\overline{r})=-4\pi e\int{\frac{\exp(i\overline{r}\cdot\overline{p})}{p^2}d\overline{p}}$.
What am I doing wrong?

Let's consider, that everything is OK with the above. Now I have to calculate the integral $\int{\frac{\exp(i\overline{r}\cdot\overline{p})}{p^2}d\overline{p}}$. First of all I pass on to spherical coordinate system:
$\int\limits_{0}^{2\pi}d\phi\int\limits_{0}^{+\infty}dp\int\limits_0^\pi\exp(ipr\cos\theta)\sin\theta d\theta=\\ |t=\cos\theta|= -2\pi\int\limits_{0}^{+\infty}dp\int\limits_1^{-1}\exp(iprt)dt= \frac{2\pi}{ir}\int\limits_{0}^{+\infty}\frac{[\exp(ipr)-\exp(-ipr)]dp}{p}$
Then a little trick:
$\int\limits_{0}^{+\infty}\frac{[\exp(ipr)-\exp(-ipr)]dp}{p}= \int\limits_{0}^{+\infty}\frac{\exp(ipr)dp}{p}-\int\limits_{0}^{+\infty}\frac{\exp(-ipr)dp}{p}=\\ |p\rightarrow-p\ in\ second\ integral|= \int\limits_{-\infty}^{+\infty}\frac{\exp(ipr)dp}{p}$.
So we have $\phi(\overline{r})=-\frac{e}{i\pi r}\int\limits_{-\infty}^{+\infty}\frac{\exp(ipr)dp}{p}$ and last thing to do is to calculate $\int\limits_{-\infty}^{+\infty}\frac{\exp(ipr)dp}{p}$.
To do this I extend integration curve to the top complex half-plane with curve $C_\infty$, which goes counter-clockwise. It gives me

$$\int\limits_{-\infty}^{-\epsilon}[\cdots]+\int\limits_{C_\epsilon}[\cdots]+ \int\limits_{+\epsilon}^{+\infty}[\cdots]+\int\limits_{C_\infty}[\cdots]$$

which is equal to $0$ as $\int\limits_{C_\epsilon}[...]$ goes over the pole $p=0$ ($p=\epsilon\exp(i\alpha)$, $\alpha=[\pi;0]$) and we have no poles in closed curve.
$\int\limits_{C_\infty}[...]=0$ so $\int\limits_{-\infty}^{-\epsilon}[...]+ \int\limits_{+\epsilon}^{+\infty}[...]=-\int\limits_{C_\epsilon}[...]$.
With $\epsilon\rightarrow0$ we have the sought-for integral in the left part of the last equation. So the goal is to find
$\int\limits_{C_\epsilon}\frac{\exp(ipr)dp}{p}$.
And there is my second problem. I suppose, that I should somehow bring it to Cauchy's integral formula, but I just can't do it. My flair tells me that it is just half the result, that we have if we take the full circle instead of half of it, but I would like to find something stricter.

P.S. I expect someone to advice me using Green's function, but I prefer to find out, what is wrong with my solution. Basically because I hardly understand the essense of Green's function :)

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## migrated from mathoverflow.netAug 12 '13 at 21:06

This question came from our site for professional mathematicians.

Moral issues with a math problem makes it sound like you think some types of math are sinful. :P And using a Green's function to solve this problem would be fruitless, as the solution is the Green's function. –  Muphrid Aug 12 '13 at 23:09