5) Cauchy’s Residue Theorem

Question 1

What are the three types of isolated singularities for a holomorphic function

Answer 1

Question 2

What is an isolated singularity of a function f

Answer 2

If a function f is holomophic on a domain D with the exception of a point w ∈ D then we say w is an isolated singularity of f

Question 3

What does it mean for a function to be meromorphic

Answer 3

Let D be a domain. A function f : D → C is said to be meromorphic if f is holomorphic on D except for a set of isolated singularities, and these are either removable singularities or poles

Question 4

What is an isolated zero of a function f

Answer 4

A function f defined on a domain D has an isolated zero at z0 if f (z0) = 0 and there exists ε > 0 such that f (z) /= 0 for all z ∈ D(z0;ε) \ {z0}

Question 5

What does it mean for a function f to have a zero of order m at z0

Answer 5

Question 6

What is the condition for a holomorphic function to have a zero of order m at z0

Answer 6

Question 7

What is the condition for f(z)= p(z) / q(z) to have a pole of order m at z0

Answer 7

Question 8

What is the residue of a function at an isolated singularity z0

Answer 8

Question 9

What is a simple closed loop

Answer 9

A closed contour γ is said to be a simple closed loop if, for every point z not on γ, the winding number is either w(γ; z) = 0 or w(γ; z) = 1. If w(γ; z) = 1 then we say that z is inside γ

Question 10

What does Cauchy’s Residue Theorem state

Answer 10

Question 11

How can residues be calculated for different types of poles

Answer 11