4) Taylor series and Laurent series

Question 1

What does the Cauchy-Taylor Theorem state for holomorphic functions

Answer 1

Question 2

What does it mean for a function f:D→C to be analytic

Answer 2

A function f:D→C is analytic if, for each z ∈D, f is equal to its Taylor series at z0 on some open set containing z0

Question 3

What does Cauchy’s Estimate say about the derivatives of a holomorphic function

Answer 3

Question 4

What does it mean for the Taylor series of a holomorphic function to converge to the function itself

Answer 4

Let f ∈ O(D), for some domain D, and let z0 ∈ D. Let ρ be the distance from z0 to C\D (and if D = C then ρ = ∞). Then the Taylor series for f about z0 converges to f in D(z0,ρ)

Question 5

What is Liouville’s Theorem

Answer 5

Suppose that the entire holomorphic function f is bounded on the whole of C. Then f is constant

Question 6

What does it mean if a subset U of a domain D is both open and closed

Answer 6

Let D be a domain, and suppose U ⊂ D is both open and closed. Then either U = ∅ or U = D

Question 7

What conditions guarantee that a holomorphic function f is identically zero on a domain D

Answer 7

Question 8

What does it mean if two holomorphic functions f and g defined on C agree on a set R

Answer 8

Then f = g

Question 9

What us the Fundamental Theorem of Algebra

Answer 9

Question 10

How can a polynomial of degree n ≥1 with complex coefficients be factorised

Answer 10

Question 11

What is a Laurent series

Answer 11

Question 12

What is Laurent’s theorem

Answer 12

Question 13

What does the uniqueness of Laurent series state

Answer 13