3) Cauchy’s theorem & Cauchy’s formula Flashcards
How can the winding number of a closed path γ around a point z0 be calculated
Draw a ray from z0 to outside any disc enclosing γ. Count the number of intersections of γ with this ray:
* +1 for crossings from right to left (anticlockwise)
* -1 for crossings from left to right (clockwise)
* The total count gives the winding number
What are the properties of the winding number of a closed path γ around a point z0
What can be said about the parametrisation of a path γ in C∖{0} with respect to the argument function
What is the relationship between the winding number of a closed path around z0 and an integral
What does Cauchy’s Theorem state about a holomorphic function and a closed contour
What is a simply connected domain
A domain D is simply connected if for all closed contours γ in D and for all z ∉ D, we have w(γ; z) = 0
What does Cauchy’s Theorem state for holomorphic functions on simply connected domains
What does the generalised Cauchy theorem state for holomorphic functions on a domain
What does the inequality for the integral of a continuous function ϕ:[a,b]→C state
What is the Estimation Lemma
What does Cauchy’s Integral Formula for a circle state for a holomorphic function
