CREATESPACE Laplace and Helmholtz equations: detailed solution and orthogonality and completeness of solutions Volume I (Physics of Spacetime - Complete Collection)

The object of this book is the solution to the Laplace and Helmholtz equations with a special interest in the study of the solutions' orthogonality, normalization, and completeness. This book deals in detail and in a practical manner with the methods to solve differential equations in general, like separation of variables, power series, or Frobenius, and the particular methods to solve equations like Bessel's, Legendre's, Laguerre's, Schroedinger's, or Sturm-Liouville's. A series of mathematical tools are explained practically, from the most simple ones to the most complicated ones, like the binomial theorem, the changes of index in summations, the simplification of expressions by means of techniques from obvious to ingenious, the proof by induction, the convergence tests for series, including the power series and the radius of convergence, the Leibniz formula, the integration by parts, the inner product of functions, the analytic continuation of a function, the expansion of a function in terms of Taylor, MacLaurin, or Laurent series, the Laplace transform, the Fourier transform, the steepest descent method, the change of integration variable, the evaluation of integrals by complex variable methods, like the integrals of Euler, Hankel, Weber and Schafheitlin, or Barnes, along different integration contours, like Pochhammer's or Hankel's, studying the convergence of the integrals, and involving the residue theorem, or Cauchy's integral theorem, approaching the study of other functions, like the exponential and logarithmic functions, the Bernoulli polynomials, the Rodrigues formula, the Legendre polynomials, the Laguerre polynomials, the gamma function (involving the Stirling's formula and Euler-MacLaurin formula), the reciprocal gamma function, the digamma function, the beta function, the hypergeometric function (and the hypergeometric equation). The orthogonality, normalization, and completeness of Laplace, Lagrange, and Helmholtz equation solutions are dealt with in depth, involving the Sturm-Liouville theory, and Hilbert Space. The solution to the Laplace and Helmholtz equations, as well as the study of the orthogonality and completeness of the solutions is split into two volumes. The first volume deals with the Laplace and Helmholtz equations in spherical coordinates, and the orthogonality and normalization of the solutions. The second volume deals with the Laplace and Helmholtz equations in cylindrical, polar, and Cartesian coordinates, with a study on the completeness of all the solutions obtained in both volumes, based on Sturm-Liouville theory, and Hilbert Space. The second volume is complemented with a chapter dedicated to the solution to the Schroedinger equation for the hydrogen atom.