Proposed Syllabus

Sections

Comments/Topics

I.1

Complex Numbers

I.2

Polar representation

I.4

The square and square root functions

I.5 and I.6

The exponential and logarithm functions

II.1

Review of basic analysis

II.2

Analytic functions

II.3

The Cauchy-Riemann equations

IV.1

Complex line integrals

IV.2

Fundamental Theorem of Calculus for Analytic Functions

IV.3

Cauchy's Theorem

IV.4

The Cauchy Integral Formula

IV.5

Liouville's Theorem

IV.6

Morera's Theorem

IV.7

Goursat's Theorem

V.1-4

Power series

V.4-5

Power series (cont.)

V.6

Manipulation of power series

V.7

The zeros of an analytic function

V.8

Analytic continuation

VI.1

The Laurent decomposition

VI.2

Isolated Singularities of an analytic function

VI.3

Isolated singularity at infinity

VII.1

The residue theorem