Methods Of Theoretical Physics 1

Methods of Theoretical Physics: I ABSTRACT First-order and second-order differential equations; Wronskian; series solutions; ordinary and singular points. Orthogonal eigenfunctions and Sturm-Liouville theory. Complex analysis, contour integration. Integral representations for solutions of ODE’s. Asymptotic expansions. Methods of stationary phase and steepest descent. Generalised functions. Books E.T. Whittaker and G.N. Watson, A Course of Modern Analysis. G. Arfken and H. Weber, Mathematical Methods for Physicists. P.M. Morse and H. Feshbach, Methods of Theoretical Physics. Contents 1 First and Second-order Differential Equations 3 1.1 The Differential Equations of Physics . . . . . . . . . . . . . . . . . . . . . . 3 1.2 First-order Equations 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Separation of Variables in Second-order Linear PDE’s 7 2.1 Separation of variables in Cartesian coordinates . . . . . . . . . . . . . . . . 7 2.2 Separation of variables in spherical polar coordinates . . . . . . . . . . . . . 10 2.3 Separation of variables in cylindrical polar coordinates . . . . . . . . . . . . 12 3 Solutions of the Associated Legendre Equation 13 3.1 Series solution of the Legendre equation . . . . . . . . . . . . . . . . . . . . 13 3.2 Properties of the Legendre polynomials . . . . . . . . . . . . . . . . . . . . 18 3.3 Azimuthally-symmetric solutions of Laplace’s equation . . . . . . . . . . . . 24 3.4 The generating function for the Legendre polynomials . . . . . . . . . . . . 27 3.5 The associated Legendre functions . . . . . . . . . .