Differentiability & AnalyticityTeam 2

MA3121 Complex Analysis

Department of Mathematics

Lecturer: Dr. Dimitrina Stavrova

Year2013

Alex Bell | Emily Thorne | George Mileham | Hugh Daman | Joel Duncan

Laura Mulligan | Manij Basnet | Robert Paul Sanders | Shamini Rajan | William Yong

Overview

• Complex Derivative

• Cauchy-Riemann Equations

• Analyticity

AlexWillManijShamini

JoelLauraHugh

RobertEmilyGeorge

Complex Derivative - Definition

Alternative Form of Definition

Implying ContinuityDifferentiability at a point implies continuity at that point.

Assume exists…

Rectangular Polar

lim𝑧→𝟎

𝑓 (𝑧 )↔ lim¿ 𝑧∨→0

𝑓 (𝑧 )

𝑥=𝑅𝑒(𝑧)

𝑦=𝐼𝑚(𝑧)

Conclusion: path dependence implies nowhere differentiable

Example 1

Example 2

We proceed to consider two cases…

Conclusion: differentiable only at the origin

1.

2.

Conclusion: differentiable everywhere

Sounds ‘entire’ to me…

Example 3 (𝐼𝑑𝑒𝑛𝑡𝑖𝑡𝑦 )

Familiar Rules

SUM

CHAIN

PRODUCT

QUOTIENT

Proof : Sum Rule

Proof : Product Rule

Applications – Particle Motion

𝑧 (𝑡 )=𝑥 (𝑡 )+𝑦 (𝑡 ) 𝑖

𝑅𝑒 (𝑧 )

𝐼𝑚(𝑧)

(𝑥 (𝑡 ) , 𝑦 (𝑡 ))

v 𝑎 (𝑡 )=𝑣 ′ (𝑡)=𝑥 ′ ′ (𝑡 )+𝑦 ′ ′ (𝑡 )𝑖

Circular Motion

𝑡=0

𝑘0 𝑡=𝜋2

𝑘0 𝑡=𝜋

𝑘0 𝑡=3𝜋2

𝑘0>0𝑡→ 𝜋

2𝑘0𝑡→ 𝜋𝑘0

𝑡→ 3𝜋𝑘0

𝑟0𝑥 (𝑡 )

𝑦 (𝑡)

𝑅𝑒(𝑧 )

𝐼𝑚(𝑧 )

𝑧 (𝑡 )

(𝑥 ′ (𝑡 ) , 𝑦 ′ (𝑡 ))

The Cauchy-Riemann Relations

Definition

The Cauchy-Riemann Relations are:

These give necessary conditions for the existence of a complex derivative. We also need the first order partial derivatives to be continuous to ensure differentiability.

Deriving The Cauchy-Riemann Relations

We must take in both the x and y direction. Firstly notice the change in

Let

where

Continued … when

Therefore, when we equate these from both directions, the following must hold

when

Example

Conclusion: Cauchy-Riemann equations are satisfied nowhere

Given that and find where the Cauchy-Riemann relations are satisfied

is satisfied nowhere

The Cauchy-Riemann Relations Theorem

Example 1

𝑢𝑥=2 𝑥 −𝑢 𝑦=0𝑣 𝑥=𝑦𝑣 𝑦=𝑥{ (𝑥 , 𝑦 )∈𝑅2|2𝑥=𝑥 , 𝑦=0 }={ (0,0 ) }

Partial derivatives all exist and are continuous so is only differentiable at the origin

𝑓 ′ (0 )=𝑢𝑥 (0,0 )+𝑣𝑥 (0,0 ) 𝑖=0

continuous at since it is differentiable there.

Example 2

Conclusion: Cauchy-Riemann equations are satisfied on the whole of

Given that and find where the Cauchy-Riemannrelations are satisfied

Definition of AnalyticityDefinitionA function is analytic at if there exists a neighbourhood such that every point within is differentiable.

We note that analyticity implies differentiability at a given point but the converse does not hold. This means analyticity is a stronger condition than differentiability.

ExampleThe function Here, And only when x and/or y = 0, i.e. on the coordinate axes

Another example is the function which is differentiable at every point.

Here, and

A complex function is entire if it is analytic .

Alternatively, we could say that it is differentiable at every point, since the plane is a neighbourhood of its points.

Any polynomial function is entire, which can be proved term-wise.The function is entire.

Entire FunctionsDefinitions

Example: is entire

A function is harmonic on a domain if it adheres to the “Laplace equation”

on , and all second order partial derivatives exist and are continuous on .

For an analytic function , and are also harmonic.

Harmonic FunctionsDefinitions

A Dirichlet Problem is to find a solution of partial differentiation equations that satisfy boundary conditions on the defined domain.

A function being harmonic is a fairly strict condition.Given the boundary values of a function on a domain, what possible harmonic functions exist?

Here’s an example inpolar co-ordinates:

Dirichlet Problems

Harmonic ConjugatesDefinition

For two functions and , is a harmonic conjugate of on a domain if;

and .

ExampleGiven the function For we need and i.e. and . Thus;

i.e.

Thanks for Listening

Any Questions?