Functions of a Complex Variable
Classic Examples
Polynomials
They are entire.
Power Series
Holomorphic in radius of convergence,
Laurent Series
If
Pole Terminology
Order is largest
Rational Functions
Exponential
Trigonometric
Logarithmic
Multivalued Function and Branches
Just like the square root gives two values, additional context lets you know which value is returned for that problem.
We can choose
In this way, we choose a branch cut from 0 to negative infinity along the negative real axis. Branch point, you cannot move a branch point from a branch cut.
Roots
So,
Derivatives
Derivatives of a complex function are the same as typical derivatives.
Motivation
Suppose we have two curves going from
Cauchy-Gorsat Theorem
If
Conventions
The integrals are typically oriented in a CCW fashion.
Cauchy Integral Representation
Proof. Because
Implications
Note,
Cauchy Residue Theorem Implications from Integral Representation
From this, we have,
Aside
Notation:
If
A bounded entire function is constant. Thus, all interesting functions either have singularities if they are bound at infinity or are unbound.
Thus, a function is either constant or unbound at infinity.
What about Regions in which is not Holomorphic everywhere?
I.e. where we have singularities.
Excise the singularity from the domain… Lauraunt expansion.
Laurent Series
Theorem
Suppose that
This is also written as
The
Proof. Similar to Taylor series. Let
Finding Coefficients
Calculating Laurent series coefficients is typically done by: guess and check, partial fractions, leverage the binomial expansion, taylor series.
Example
Examples
Example 1
So, Res
Example 2
Example 3
Res
Res
Example 4
Then,
Res
Res
Example 5
Res
Example 6
An integral that arises in optics.
The exponential approaches zero in the upper half of the plane but explodes in the bottom half of the plane. Thus, we see that the top semicircle contributes nothing and closes the contour.
Our integral traverses through the singularity - our theorems don’t apply. So the answer is ambiguous.
We may choose the path
For the top path, we see that we are holomorphic in the entire region, thus we get 0.
For the bottom path, we have a simple pole and in the limit of z to zero, thus we have zero.
Thus, either way we find the overall contour integral to be zero.
Therefore,