Greens function has me blue ....

[ognik]

Not following this example (PDE for Greens function) in my book:

Book states: $ \left( {\nabla}^{2} +{k}^{2}\right)G(r, r_2)=-\delta(r-r_2) = -\int \frac{e^{ip.(r-r_2)}}{\left(2\pi\right)^3} {d}^{3}p$

I recognised this as the Hemlmholtz eqtn, but cannot find where the 3rd term comes from? It looks like it could be the 3D Fourier Transform representation of the dirac-delta function? (if so a link to, or a derivation would be nice)

Then they say they solve the PDE in terms of a Fourier Integral $ G(r, r_2)=\int \frac{1}{\left(2\pi\right)^3}g_0(p) e^{ip.(r-r_2)} d^3p $

I know the Fourier Integral in 3D is $F(k)=\frac{1}{\left(2\pi\right)^{\frac{3}{2}}} \int f(r)e^{ik.r} \,d^3r $, (sqrt in denominator?) so I'm not sure what they are doing here?

 

[Jhoneine]

The third term in your example is indeed the 3D Fourier transform of the Dirac delta function. The 3D Fourier transform of a function $f(r)$ is given by $$\mathcal{F}[f(r)] = \frac{1}{(2\pi)^{\frac{3}{2}}} \int f(r) e^{i k \cdot r} d^3 r$$where $k \cdot r = k_x x + k_y y + k_z z$. This expression can be thought of as a generalization of the 2D Fourier transform, which uses a square root in the denominator instead of a cube root.The solution to the PDE that they give is a general solution to the homogeneous equation $\left( {\nabla}^{2} +{k}^{2}\right)G(r, r_2)=0$, which can be written as$$G(r, r_2) = \int \frac{1}{(2\pi)^3} g_0(p) e^{i p \cdot (r - r_2)} d^3 p.$$This is just the 3D Fourier transform of a function $g_0(p)$, which is some arbitrary function of momentum $p$.

 

[math771]

It seems like the person who wrote the book is using a different notation for the Fourier Transform. In their notation, the Fourier Transform is denoted by $g_0(p)$ instead of $F(k)$. Also, they are using a different normalization factor in the denominator, which is why there is a square root.

To understand where the third term comes from, it might help to think about the inverse Fourier Transform. In the book, they are essentially taking the inverse Fourier Transform of the function $\frac{1}{\left(2\pi\right)^3} e^{ip.(r-r_2)}$ to get the Dirac delta function. This is why the third term appears in the PDE.

As for a derivation of the Fourier Transform representation of the Dirac delta function, you can refer to any standard textbook on Fourier analysis. The key idea is that the Dirac delta function is a distribution, not a function, so it cannot be represented in the usual sense. The Fourier Transform provides a way to represent it as a limit of a sequence of functions.

I hope this helps clarify things for you. If you have any further questions, feel free to ask.