T2. 3. Review of Complex analysis

Question 1

Define the complex derivative of the function f: D→C

Answer 1

f’(z0) = lim z→z0 f(z)-f(z0) / z-z0

Question 2

Define a holomorphic function on D

Answer 2

A function which has a derivative at every point in d; hence, it equals its taylor expansion around every z0∈d.

Question 3

Define entire for a function

Answer 3

If f: C→C and f is holomorphic

Question 4

What do we find if f is holomorphic on the closure of D and ∂D is piecewise C1.

Answer 4

INT_∂D f(z) dz = 0

Question 5

State Morera’s theorem

Answer 5

If f is continuous on D and INT_R f = 0 for any rectangle R⊂D with sides parallel to coord axis, then f is holomorphic on D.

Question 6

State Weirstrass theorem

Answer 6

Let D⊆C be a domain and f_n: D → C be a sequence of holomorphic functions on D. If f_n converges to f uniformly on comp subsets of D, then f is analytic on D.

Question 7

Define an isolated singularity

Answer 7

A function has an isolated singularity if it is analytic on the punctured neighborhood of the singularity.

Question 8

Give the Laurent expansion of a function with an isolated singularity Z0

Answer 8

f(z) = SUM -∞,∞ a_n (z-z0)^n

Question 9

Define a removable, isolated and essential singularity in terms of their Laurent expansion

Answer 9

1. Removable - if the Laurent expansion about the singular point has no singularities
2. Pole - if the Laurent expansion about the singular point maintains the singularity. The order is the n coefficient of the singularity
3. Essential - not removable or pole

Question 10

State the general form of the residue for an isolated singularity

Answer 10

a_1; the coefficient of the (z-z0)^-1 term in the Laurent expansion.

Question 11

State Cauchy’s residue theorem

Answer 11

The integral of a function in a bounded domain with f analytic except for a finite number of points be:
2πi SUM Residues