- #1

- 152

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## Main Question or Discussion Point

By taking the Fourier transform of the fundamental Helmholtz equation

[tex](\nabla^2+k^2)G(\vec{x})=-\delta(\vec{x})[/tex],

one finds that

[tex]G(\vec{x})=\frac{e^{ikr}}{r}[/tex]

and

[tex]\tilde{G}(\vec{\xi})=\frac{1}{k^2-\xi^2}[/tex].

However, I can't figure out how to directly confirm that this Fourier transform pair is correct. I tried directly transforming [itex]e^{ikr}/r[/itex] as if it were a regular function, but I ended up with something which doesn't converge:

[tex]\frac{4\pi}{\xi}\int\limits_0^\infty e^{ikr}sin(\xi r)dr[/tex]

I didn't really expect that to work, since distributions are involved. So I tried doing it with distributions:

[tex]

\begin{align*}

&=\int\mathcal{F}\left\{\frac{e^{ikr}}{r}\right\}u(\vec{\xi})d\vec{\xi}\\

&\equiv\int \frac{e^{ikr}}{r}\mathcal{F}\left\{u(\vec{\xi})\right\}d\vec{x}\\

&=\int \frac{e^{ikr}}{r}\int u(\vec{\xi})e^{-i\vec{\xi}\cdot\vec{x}}d\vec{\xi}d\vec{x}

\end{align*}

[/tex]

At this point I'm thinking about using a Gaussian convergence factor, but I'm not sure exactly how to do it. Can anyone help out?

[tex](\nabla^2+k^2)G(\vec{x})=-\delta(\vec{x})[/tex],

one finds that

[tex]G(\vec{x})=\frac{e^{ikr}}{r}[/tex]

and

[tex]\tilde{G}(\vec{\xi})=\frac{1}{k^2-\xi^2}[/tex].

However, I can't figure out how to directly confirm that this Fourier transform pair is correct. I tried directly transforming [itex]e^{ikr}/r[/itex] as if it were a regular function, but I ended up with something which doesn't converge:

[tex]\frac{4\pi}{\xi}\int\limits_0^\infty e^{ikr}sin(\xi r)dr[/tex]

I didn't really expect that to work, since distributions are involved. So I tried doing it with distributions:

[tex]

\begin{align*}

&=\int\mathcal{F}\left\{\frac{e^{ikr}}{r}\right\}u(\vec{\xi})d\vec{\xi}\\

&\equiv\int \frac{e^{ikr}}{r}\mathcal{F}\left\{u(\vec{\xi})\right\}d\vec{x}\\

&=\int \frac{e^{ikr}}{r}\int u(\vec{\xi})e^{-i\vec{\xi}\cdot\vec{x}}d\vec{\xi}d\vec{x}

\end{align*}

[/tex]

At this point I'm thinking about using a Gaussian convergence factor, but I'm not sure exactly how to do it. Can anyone help out?