複變函數論

Topology of complex plane: open and closed sets, limits of functions, continuity, path, change of parameter, subpaths and sums of paths, the paving lemma.

Topology of complex plane: connectedness, space-filling curves Power series: sequence, series, power series, product of series

Differentiation: basic results, The Cauchy-Riemann equations, connected set and differentiability, hybrid functions, power series

The exponential function: the exponential function, real exponentials and logrithms, trigonometric functions, an analytic definition of $\pi$, the behavior of real trigonometric functions

The exponential function: explanation of Euler's formula, complex exponential and trigonometric functions are periodic, other trigonometric functions, hyperbolic functions

Integration: The real case, complex integration along a smooth path, length of a path, regular paths and curves, regular and singular points

Integration: countour integration, the fundamental theorem of countour integration, an integration depends on the path, Gamma function

Integration: The estimation lemma, consequence of the fundamental theorem Angles, Logrithms, and the widing number: radian measure of angle, the argument of a complex number, the complex logrithm, the winding number

Angles, Logrithms, and the widing number: the winding number as an integral, components of the complement of a path, computing the winding number

Cauchy's Theorem: Cauchy's Theorem for a triangle, existence of an intiderivative in a star domain, local existence of an antiderivative

Cauchy's Theorem: Cauchy's Theorem, applications of Cauchy's Theorem, simply connected domain Homotopy version of Cauchy's Theorem

Taylor series: Cauchy integral formula, Taylor series Morera's Theorem, Cauchy's estimate, zero, extension function

Laurent series: series involving negative powers, isolated singularities, behavior near an isolated singularity, the extended complex plane,

Laurent series: behavior of a differentiable function at infinity, meromorphic function Residue: Cauchy residue theorem, counting residues

Residue: evaluation of definite integrals, summation of series, counting zeros