## Verifying the Helmholtz Green’s function.

Posted by peeterjoot on December 9, 2011

# Motivation.

In class this week, looking at an instance of the Helmholtz equation

We were told that the Green’s function

that can be used to solve for a particular solution this differential equation via convolution

had the value

Let’s try to verify this.

# Guts

Application of the Helmholtz differential operator on the presumed solution gives

## When .

To proceed we’ll need to evaluate

Writing we start with the computation of

We see that we’ll have

Taking second derivatives with respect to we find

Our Laplacian is then

Now lets calculate the derivatives of . Working on again, we have

So we have

Taking second derivatives with respect to we find

So we find

or

Inserting this and into 2.8 we find

This shows us that provided we have

## In the neighborhood of .

Having shown that we end up with zero everywhere that we are left to consider a neighborhood of the volume surrounding the point in our integral. Following the Coulomb treatment in section 2.2 of [1] we use a spherical volume element centered around of radius , and then convert a divergence to a surface area to evaluate the integral away from the problematic point

We make the change of variables . We add an explicit suffix to our Laplacian at the same time to remind us that it is taking derivatives with respect to the coordinates of , and not the coordinates of our integration variable . Assuming sufficient continuity and “well behavedness” of we’ll be able to pull it out of the integral, giving

Recalling the dependencies on the derivatives of in our previous gradient evaluations, we note that we have

so with , we can rewrite our Laplacian as

This gives us

To complete these evaluations, we can now employ a spherical coordinate change of variables. Let’s do the volume integral first. We have

To evaluate the surface integral we note that we’ll require only the radial portion of the gradient, so have

Our area element is , so we are left with

Putting everything back together we have

But this is just

This completes the desired verification of the Green’s function for the Helmholtz operator. Observe the perfect cancellation here, so the limit of can be independent of how large is made. You have to complete the integrals for both the Laplacian and the portions of the integrals and add them, *before* taking any limits, or else you’ll get into trouble (as I did in my first attempt).

# References

[1] M. Schwartz. *Principles of Electrodynamics*. Dover Publications, 1987.

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