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

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SUMMARY

The discussion focuses on deriving Green's function for the Laplacian operator in three dimensions, specifically under infinite boundary conditions. Key references include Arfken's “Mathematical Methods for Physicists” and Haberman's book on Partial Differential Equations (PDE), particularly Section 9.5.6. The relationship between the Green's function and unit excitation is established in equation 9.173, while equations 9.174 and 9.175 detail the gradient operator in spherical coordinates and its indefinite integral, respectively.

PREREQUISITES
  • Understanding of Green's functions in mathematical physics
  • Familiarity with the Laplacian operator
  • Knowledge of spherical coordinates and surface integration
  • Basic concepts of Partial Differential Equations (PDE)
NEXT STEPS
  • Study the derivation of Green's functions in Haberman's “Partial Differential Equations”
  • Explore the application of Green's functions in solving PDEs
  • Review the properties of the Laplacian operator in various coordinate systems
  • Investigate surface integration techniques in spherical coordinates
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Physicists, mathematicians, and students studying mathematical methods in physics, particularly those focused on solving partial differential equations and understanding Green's functions.

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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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