# Green's function for Helmholtz Equation

## Homework Statement

Arfken & Weber 9.7.2 - Show that

$\frac{exp(ik|r_{1}-r_{2}|)}{4\pi |r_{1}-r_{2}|}$

satisfies the two appropriate criteria and therefore is a Green's function for the Helmholtz Equation.

## Homework Equations

The Helmholtz operator is given by

$\nabla ^{2}A+k^{2}A$

Symmetricity of Green's functions.

## The Attempt at a Solution

Right off the bat I am not sure what is mean't by "the two appropriate criteria" phrase. What exactly are the two appropriate criteria that they ask for in Arfken & Weber problem 9.7.2? Where can I find this criteria so that I know how to answer this question?

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## Homework Statement

Arfken & Weber 9.7.2 - Show that

$\frac{exp(ik|r_{1}-r_{2}|)}{4\pi |r_{1}-r_{2}|}$

satisfies the two appropriate criteria and therefore is a Green's function for the Helmholtz Equation.

## Homework Equations

The Helmholtz operator is given by

$\nabla ^{2}A+k^{2}A$

Symmetricity of Green's functions.

## The Attempt at a Solution

Right off the bat I am not sure what is mean't by "the two appropriate criteria" phrase. What exactly are the two appropriate criteria that they ask for in Arfken & Weber problem 9.7.2? Where can I find this criteria so that I know how to answer this question?
I answered my own question. I read the material 6 hours ago, looked at the assignment and did it but kept coming up with the solution that $(\nabla ^{2}+k^2)G=0$, but I kept claiming that was incorrect. Well, looking back for the 100th time I realized this has to be true based on page 598 of Arfken & Weber, not only because it says so but also because the Helmholtz equation indicates a Green's function corresponding to an outgoing wave, which means that G(r1,r2) must satisfy a homogenous differential equation. Wow, that was a lot but I think it makes sense.

afkern and weber problem 9.7.3

Hey I am still struggling with the solution of the problem and trying to figure out your explanation. Can you explain it more elaborately.

The solutions are given by hankel functions of first kind and second kind.A time dependence of exp(-iwt) is assumed.In physical cases only outgoing wave is present so only one is chosen and not the other.