Complex Analysis

Question 1

What is de Moivres theorem

Answer 1

Check

Question 2

Write down the Cauchy Riemann equations to check for holomorphic functions.

Answer 2

Check in book.

Question 3

Write down the definition of a metric space and a Normed vector space and how you can prove that a Normed vector space is a metric space.

Answer 3

Check in book

Question 4

Write the equations to find
(I) image
(ii) path integral
(iii) speed
(iii) length
(Iv) complex fundamental theorem of calculus.

Answer 4

Check in book

Question 5

What are the steps to using Cauchy’s theorem?

Answer 5

1. Show it is holomorphic
2. Show it is a closed Jordan path
3. Then by Cauchy’s theorem the integral is 0 (use complex fundamental theorem of calculus to show this).

Question 6

What is Cauchy’s integral formula?

Answer 6

Check in book

Question 7

Write down how to find the modulus and argument of z.

Answer 7

Check

Question 8

Write down Cauchy’s integral formula.

And for derivatives

Answer 8

Check in book

Question 9

What is Liouville’s theorem?

Write down Cauchy’s inequality

Answer 9

Check in book

Question 10

What is the equation to find the radius of convergence?

Answer 10

1/L, L=lim n infinity of sup|cn|^1/n

Question 11

The formula for calculating the residue for a simple pole and a pole of order greater than 1.

Answer 11

Check in book