## Coursera Analysis Complex Kind Lesson 5.5 Consequences of Cauchy’s Theorem and Integral Formula

### Aug 3, 2016

## Slide 2 Cauchy Theorem and Integral formula

Relating the kth derivative and the integral over a **closed** curve of a function of the form

D is a simply connected domain (no holes), F is analytic in a region that contains D, then,

for all w in D, k > 0.

## Slide 3 Cauchy’s estimate

Slide 4 Liouville’s Theorem

f is analytic in the complex plane (i.e. entire function)

and if f is bounded

then f must be constant.

converse : There are no bounded, varying analytic functions in C.

c.f. Sin x in reals where function value is bounded between -1 and +1 but is (definitely) not constant.

Proof: uses Cauchy’s estimate

## Slide 5 Example of Liouvile Theorem

## Slide 6 Liouville Theorem to prove fundamental theorem of algebra

## Slide 7 Consequence of fundamental theorem of algebra

A polynomial can be factored into n factors.

## Slide 8 Maximum Principle *Important *

Let f be analytic in domain D

and suppose there exists a point z_0 in D such that for all z in D

i.e. there is a maximum value at some point in D,

then f is constant in D.

Consequence if f is analytic on D then |f(z)| reaches its **maximum on the boundary**

## Slide 9 Maximum Principle Example *Important *

<< diagram >>

Domain is square where one corner is origin other corners are 1, i and 1+i, so sides of length 1. z = x + **i**y, so both x and y are real values between 0 and 1.

(which is analytic but not constant)

What is the maximum of |f(z)|?

Boundary

so |f(z)| = |f(x)|= |x^2 – 2x| = |x(x-2)|

for the domain of x, maximum occurs when x = 1, so

Boundary

so |f(z)| = |f(1+iy)|=

for the domain of y zero to 1 and fixed value of x=1, maximum occurs when y = 1, so

<< provide detailed working for further two boundaries >>

By considering the modulus of the function applied to x + i (which is one way of describing this boundary) and using pythag to get expression for the modulus of f(x+i)

e.g. wolfram gives