Complex Methods For The Sciences

Complex numbers, Argand diagrams, modulus and argument, De Moivre's theorem. Functions of a complex variable, elementary examples, Cauchy-Riemann equations. Contour integrals, Cauchy's theorem and Cauchy's integral formula. Taylor series, Laurent series, zeros, poles and essential singularities, residues. Fourier transform, inversion, convolution, Parseval's theorem, delta function, applications. Elementary partial differential equations, methods of separation. Brief introduction to special functions, e.g., gamma function, beta function, Bessel's function, Legendre's function.