Week

Topics covered

1

The Algebra of Complex Numbers; The Geometry of Complex Numbers; The Topology of Complex Numbers.

2

Functions and Linear Mappings; The Mappings w=zⁿ and w=z^1/n; Limits and Continuity; Branches of Functions; The Reciprocal Transformation w=1/z.

3

Differentiable and Analytic Functions; The Cauchy-Riemann Equations; Harmonic Functions.

4

Sequences and Series; Geometric Series and Convergence Theorems; Power Series Functions.

5

The Complex Exponential Function; The Complex Logarithm; Complex Exponents.

6

Trigonometric and Hyperbolic Functions; Inverse Trigonometric and Hyperbolic Functions.

7

Complex Integrals; Contours and Contour Integrals; The Cauchy-Goursat Theorem.

8

The Fundamental Theorems of Integration; Integral Representations for Analytic Functions; The Theorems of Morera and Liouville, and Extensions.

9

Uniform Convergence; Taylor Series Representations; Laurent Series Representations.

10

Singularities, Zeros, and Poles; Applications of Taylor and Laurent Series.

11

The Residue Theorem.

12

Trigonometric Integrals.

13

Improper Integrals of Rational Functions; Intended Contour Integrals.

14

Integrands with Branch Points; The Argument Principle and Rouché’s Theorem.

15

Applications to Evaluating of Sums of Series. Inverse Laplace Transforms.