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# Theory of Functions of a Complex Variable

Select from the following:
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 low,
electrostatics, heat flow. Potential functions. Conservative,
irrotational and solenoidal fields. Equipotential lines,
streamlines, isothermal lines and flux lines.

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