Complex Analysis

Math 312  MWF 10:30am - 11:25am
\begin{matrix}c \\ ◯ \end{matrix}1998 Ron Buckmire  Fowler 112

Class 32 (Monday April 13)

SUMMARY Review for Second Exam Here is a brief summary of all the topics that we have considered in this second half of the course:

• complex functions of a real variable

• Arcs, curves, contours

• Parametrization

• Differentiation and antidifferentiation of complex functions of a real variable

• Contour integration

• ∮ (z − z_{0})^{n}dz = 2πiδ_{n,−1}

• Antiderivatives of analytic functions

• Cauchy-Goursat Theorem

• Deformation Invariance Theorem

• Path Independence Theorem

• Simply and Multiply Connected Domains

• Cauchy Integral Formula

• Generalized Cauchy Integral Formula

• Liouville’s Theorem

• Cauchy’s Inequality

• Maximum and minimum modulus theorem(s)

• Analytic functions are in�nitely differentiable

• Zeroes and poles

• Classi�cation of singularities (isolated, removable, branch and essential)

• Residues

• Cauchy’s Residue Theorem

• Laurent Series

• Applications of Laurent Series to Contour Integration

• Cauchy’s Second Residue Theorem

• Applications of Residues to Real Trigonometric Integrals

• Applications of Residues to Real Improper Integrals (Jordan’s Lemma)