I need to verify part of the proof involve Green's function.
This is actually a subset of proofing . Where G is the Green's function. I don't want to present the whole thing, just the part I have question.
Let D be an open solid region with surface S. Let where both are green function at point a and b resp. inside D. This means Q is defined at point a ( harmonic at point a ) and P is defined at point b. Both P and Q are defined in D except at a and b resp. Both equal to zero on surface S.
Green function defined:
In this proof, I need to make two spherical cutout each with radius = with center at a and b. I call the spherical region of this two sphere A and B resp and the surface resp. Then I let so both P and Q are defined and harmonic in .
Now come to the step I need to verify:
I want to prove:
This is my work:
. in sphere region A.
Form (1) I break into 3 parts:
Because Q is harmonic and
From second identity:
because both H and Q are harmonic in A and on surface .
The proof of the Strauss's book is very funky to put it politely. This is the way I proof it and please bare with the long explaination and tell me whether I am correct or not.