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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
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1.1
The Differential Equations of Physics . . . . . . . . . . . . . . . . . . . . . .
3
1.2
First-order Equations
4
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2 Separation of Variables in Second-order Linear PDE’s
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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 . . . . . . . . . . . .
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3 Solutions of the Associated Legendre Equation
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3.1
Series solution of the Legendre equation . . . . . . . . . . . . . . . . . . . .
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3.2
Properties of the Legendre polynomials
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18
3.3
Azimuthally-symmetric solutions of Laplace’s equation . . . . . . . . . . . .
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3.4
The generating function for the Legendre polynomials . . . . . . . . . . . .
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3.5
The associated Legendre functions . . . . . . . . . .