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Fundamental Solution of N-Dimensional Laplace equation

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The fundamental solution of the n-dimensional Laplace equation solves

, (1)

where is the n-dimensional delta function.

a. Show that if , the solution of the above equation (1) for is

,

where is a constant.

b. Use the n-dimensional Gauss theorem to evaluate the left hand side of the equation (1) and show that

where is the surface area of the n-dimensional unit sphere defined by

Note the singularity for n = 2.

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The fundamental solution of the n-dimensional Laplace equation solves

, ( 1)

where is the n-dimensional delta function.

a. Show that if , the solution of the above equation (1) for is

,

where is a constant.

b. Use the n-dimensional Gauss theorem to evaluate the left hand side of the equation (1) and show that

where is the surface area of the n-dimensional unit sphere defined by

Note the singularity for n = 2.

Solution:
a) Since we are looking for solution for r > 0, by multiplying (1) with (r2) we will get:
( 2)
But there is a property of Dirac function (distribution) which states that:
( 3)
For we will have
( 4)
and (2) becomes:
( 5)
This is an Euler ordinary differential equation (ODE) which can be solved by looking for solutions of form:
( 6)
By differentiation, we get:
( 7)
and by introducing in (5), one yields:
( ...

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