Website owner:  James Miller

Theory of Functions of a Complex Variable

Number systems. Rational, irrational, real and complex numbers. Open and closed intervals. Fields. Absolute values, conjugates of complex numbers. Laws.

Products, quotients and roots of complex numbers in polar form. De Moivre's theorem. Roots of unity.

Point sets in one, two, three and n-dimensional Euclidean spaces. Neighborhoods, closed sets, open sets, limit points, isolated points. Interior, exterior and boundary points. Derived set. Closure of a set. Arcwise connected sets. Regions.

The elementary functions

Functions, mappings, limits, continuity, sequences, series

Frequently used formulas

Multiple-valued functions, branch points, branch lines, Riemann surfaces

Singular points. Isolated, removable, essential singularities. Poles.

Complex differentiation. Cauchy-Riemann equations. Analytic functions. Harmonic functions. Indeterminate forms. L'Hopital's rule. Orthogonal trajectories. Curves. Elementary functions.

Differentiation rules and formulas

Dot and cross products. Complex conjugate coordinates. Complex differential operators. Gradient, divergence, curl and Laplacian of complex functions.

Complex integration. Complex and real line integrals, Green's theorem in the plane, Cauchy's integral theorem, Morera's theorem, indefinite integral, simply and multiply-connected regions, Jordan curve

Integrals of special functions

Cauchy's integral formulas, Cauchy's inequality, Liouville's theorem, Gauss' mean value theorem, maximum modulus theorem, minimum modulus theorem

Zeros, poles, Argument principle, Rouche's theorem

Sequences, series. Absolute and uniform convergence. Region of convergence. Power series. Taylor's theorem. Laurent's theorem. Classification of singular points. Entire, meromorphic functions.

Analytic continuation

Method of Residues. Residue theorem. Evaluation of real definite integrals. Cauchy principal value. Summation of series.

Point, one-to-one, inverse transformations. Jacobian. Conformal mapping. Riemann's mapping theorem. Successive transformations. Linear, bilinear, fractional, Mobius, Schwartz-Christoffel transformations.

Boundary-value problems of potential theory. Dirichlet and Neumann problems. Poisson's integral formula for the unit circle and upper half plane.

Complex variable methods applied to fluid flow, electrostatics, heat flow. Potential functions. Conservative, irrotational and solenoidal fields. Equipotential lines, streamlines, isothermal lines and flux lines.