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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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