## Coursera Analysis Complex Kind Lesson 5.4 Cauchy’s Theorem

### Aug 3, 2016

The **BIG** one !!

## Slide 2 Cauchy’s theorem statement and proof strategy

D is a simply connected domain in C.

(function ) f is analytic in D.

is a piecewise , smooth, **closed** curve in D i.e.

the

For example f(z) = is analytic (everywhere) in C and C is simply connected.

Now it is difficult (impossible?) to find a primitive of

But for **any closed** curve, , in C,

Proof approach

Since D has no holes we can deform to a point in D.

We have to show the integral does not change as this deformation occurs.

## Slide 3 Corollary of Cauchy theorem

## Slide 4 example

<< diagram >>

R is rectangle centred .

Claim , where is the boundary of the rectangle. Notice z = is excluded from the domain.

Proof: Look at the circle that surrounds the rectangle and touches it at some points (is this necessary?), | z – | = r

<< diagram >>

Now

** learn following **

First parameterise the path,

Well this is circle centre

So the integrand becomes:

Further

Merging:

and by the Cauchy Theorem this is also the value of the integral along the boundary of the rectangle within the circle.

## Slide 5 Example

** learn following example **

First, partial fractions = (*)

<< diagram >>

|z| = 1 is circle centre 0, radius = 1.

1/z has excluded point z = 0 in |z| = 1, and we calculated previously =

1\z + 2, excluded point z = -2, is NOT inside path |z| = 1 therefore integral around closed loop = 0

So (*) becomes

## Slide 6 Cauchy Integral Formula (important technique for calculating integrals)

<< Diagram >>

D simply connected domain

boundary of D is piecewise smooth (closed) curve .

f is analytic in U which contain the closure of D (i.e. D and )

>> then f(w) = <<

## Slide 7 Proof of Cauchy Integral Formula

## Slide 8 Example

Note integrand is of form

Next question, does w=1 lie inside the path |z| = 2? YES.

So from integral formula :

Rearranging

## Slide 9 Example similar to previous

Note, integrand is of “correct” form but w=2 is not inside the curve |z| = 1 and also the rational function is analytic everywhere inside disk B1.5(0) therefore integral around the closed curve |z| = 1 is zero, by Cauchy theorem.

## Slide 10 example using Log function

<< continue >>