Group Properties:
- Closure: A, B in G, then AB is in G.
- Associative: (AB)C = A(BC)
- Identity:
A = A
- Inverse: AG, =B;
Cylindrical Laplacian Solution
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(168) |
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(169) |
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(170) |
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(171) |
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(172) |
Spherical Laplacian Solution
Legendre Polynomials
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Orthogonality relationship: |
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(174) |
Gaussian Integrals
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(175) |
Square the full integral over a second variable, pass this to polar coordinates, and perform the (now easy!) integral.
etc.
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(176) |
Scale
. Easy.
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(177) |
The trick here is to differentiate under the integral sign (a professed favorite trick of a certain someone).
Advanced Gaussian Integrals
Of the type used in paths, density matrices, etc.
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(178) |
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complete the square in the exponent. |
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Now this is a Gaussian integral of the 'easy' type from the previous section.
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(179) |
Just sub
in the previous solution.
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(180) |
Sub
in the previous solution.