PHY310: Mathematical methods for physicists I

[Cr:4, Lc:3, Tt:1, Lb:0]

Complex algebra: Functions of complex variables, Cauchy-Riemann conditions, Cauchy’s integral theorem, Laurent expansion, singularities, mapping, conformal mapping. Calculus of residues, Dispersion relations, method of steepest descent.
Gamma and Beta functions: Gamma function, definition and properties, Stirling’s series, Beta function, Incomplete Gamma function.
Differential equations: Partial differential equations, First order differential equations, separation of variables, singular points, series solutions with Frobenius’ method, a second solution, nonhomogenneous equations, Green’s function, Heat flow and diffusion equations.
Sturm-Liouville theory: Self adjoint ordinary differential equations, Hermitian operators, Gram-Schmidt orthogonalization, completeness of eigenfunctions, Green’s function – eigenfunction expansion.
Special functions: Bessel functions of the first kind, orthogonality, Neumann functions, Hankel’s functions, Asymptotic expansions, Spherical Bessel functions. Legendre functions, generating function, recurrence relations, orthogonality, alternate definitions, associated Legendre functions, spherical harmonics, Hermite functions, Laguerre functions.