## Elements of Green's Functions and Propagation: Potentials, Diffusion, and WavesThis text takes the student with a background in undergraduate physics and mathematics towards the skills and insights needed for graduate work in theoretical physics. The author uses Green's functions to explore the physics of potentials, diffusion, and waves. These are important phenomena in their own right, but this study of the partial differential equations describing them also prepares the student for more advanced applications in many-body physics and field theory. Calculations are carried through in enough detail for self-study, and case histories illustrate the interplay between physical insight and mathematical formalism. The aim is to develop the habit of dialogue with the equations and the craftsmanship this fosters in tackling the problem. The book is based on the author's extensive teaching experience. |

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### Contents

Introduction i | 2 |

Diffusion m | 3 |

Appendices | 4 |

Ordinary differential equations | 41 |

A preview | 70 |

Potentials | 89 |

II Dirichlet problems | 117 |

III Neumann problems | 142 |

III Examples | 290 |

The Helmholtz equation | 329 |

A Notations and formulary | 371 |

B The Dirichlet integral | 384 |

E Degeneracy and reality properties of complex | 397 |

Dilemmas with notations for boundary and initial | 410 |

Greens functions for circle | 412 |

the variational method and | 426 |

### Common terms and phrases

Accordingly Appendix apply appropriate arbitrary argument boundary conditions Cauchy closed surface coefficients consider constant contrast d/dt DBCs defined delta-function density derivatives determine differential equation diffusion equation dipole Dirichlet Dirichlet problem dnys eigenfunctions eigenvalue energy entails evaluate Exercise expansion explicitly expression f dV field point finite fixed Fourier Fredholm alternative GD(r given grad Green's function Green's theorem halfspace harmonic functions Helmholtz equation Hermitean homogeneous BCs homogeneous problem hyperboundary infinite infinity inhomogeneous initial conditions instance integrand Laplace equation limit linear linearly independent magic rule mathematical NBCs Neumann Neumann problem obeys physical plane point charge point source Poisson's equation potential prescribed Proof propagator radius region representation result satisfies simply singular solution solve source distribution sphere strong definition surface integral symmetry term theorem unbounded space unique vanishes velocity Verify wave equation whence write yields zero