How to Derive Green's Function for the Laplacian in 3D?

centry57
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Here are some pages of Arfken's “Mathematical Methods for Physicists ”
I don't How to work out the Green's function!
attachment.php?attachmentid=21046&stc=1&d=1255190271.png

attachment.php?attachmentid=21049&stc=1&d=1255190488.png

attachment.php?attachmentid=21048&stc=1&d=1255190271.png

Can anyone explain (9.174)and(9.175) for me ?
I'm hoping for your help, Thank you !
 

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This is about finding the Green's function for the Laplacian operator under infinite boundary condition. Basically, 9.173 gives the relation between the Green's function and the unit excitation. In 3D, the unit excitation is assumed located at the center of a sphere. If you write out the gradient operator in 9.173 in the spherical coordinate on the radial component, and perform surface integration, you will get 9.174. 9.175 is the indefinite integral of 9.174.

Haberman's book on PDE Section9.5.6 has the explicit derivation.
 
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