Complex Analysis - Differentiability and Analyticity (Team 2) - University of Leicesterr
Transcript of Complex Analysis - Differentiability and Analyticity (Team 2) - University of Leicesterr
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?