# Complex Function Calculus

1. Let C denote the positively oriented circle |z|=2 and evaluate the integral

.. see attachment

2. a) Fine the bi-linear transformation w=S(z) that maps the crescent-shaped region that lies inside the disk D:|z-2|<2 and outside the circle |z-1|=1 onto a horizontal strip

b) Graph

3... see attachment

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#### Solution Preview

Let be analytic function everywhere in a domain D except at some points where there is a singularity.

We can expand the function a Laurent series around a singularity in the form:

(1.1)

The residue of at that point is defined as

(1.2)

Or in other words, the residue at is the coefficient of the term in the Laurent series expansion.

If the domain is defined by a simple curve C, then Cauchy's residue theorem state that:

(1.3)

1.

(a)

we need to find the integral where is the circle of radius 2 centered at the origin.

We can write

(1.4)

The function is analytic everywhere in the domain while the function is also analytic everywhere but it equals zero at hence the function has poles at

The Taylor series of the trigonometric functions around functions are:

(1.5)

The Taylor series of the trigonometric functions around functions are:

(1.6)

Since are simple poles, we can write:

(1.7)

By equating coefficients we obtain at

(1.8)

Therefore

(1.9)

And at

(1.10)

So again:

...

#### Solution Summary

The 16 pages file contains detailed step-by-step explanations how to solve various problems in complex function calculus - How to find Lauren'ts expansion, how to calculate a contour integral and how to use bilinear mapping.