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

(168) | |

(169) | |

(170) | |

(171) | |

(172) |

Spherical Laplacian Solution

(173) | |

Legendre Polynomials

Orthogonality relationship: | |

(174) |

Gaussian Integrals

(175) |

Square the full integral over a second variable, pass this to polar coordinates, and perform the (now easy!) integral.

etc.

(176) |

Scale . Easy.

(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.

Of the type used in paths, density matrices, etc.

(178) |

complete the square in the exponent. | |

Now this is a Gaussian integral of the 'easy' type from the previous section.

(179) |

Just sub in the previous solution.

(180) |

Sub in the previous solution.