Maths Maths

Question 1

Explain what is meant by the function f(z) is analytic in a region D of the complex plane

Answer 1

• f(z) is analytic at point z = z0 if the limit below exists
• i.e. indt of direction of approach of z-> z0 in the complex plane
• f(z) is analytic in region D if it is analytic at every point z0∈D (is an element of D)

Question 2

What is Cauchy’s integral formula, integral formula for derivatives and Cauchy’s theorem? State the conditions under which they hold.

Answer 2

• f(z) analytic in simply connected region D of the complex plane
  
  • for any closed curve C lying entirely inside D
  • and point w inside C
• Cauchy’s theorem
  
  • closed loop integral along C of f(z) dz = 0

Question 3

Isolated singularity

Answer 3

*

Question 4

Essential singularity

Answer 4

• If principal part of the Laurent expansion valid in the immediate vicinity of z0 is infinite (isolated singularity at z=z0 for function f(z))

Question 5

Pole of order m

Answer 5

• if principal part of the Laurent expansion valid in the immediate vicinity of z0 (isolated singularity at z = z0 for function f(z)) has a finite number of terms
  
  • and has its lowest non-zero coefficient for (z-z0)^-m
  • i.e. A_n = 0 for all n < -m

Question 6

Simple pole

Answer 6

Pole of order one

Question 7

Residue

Answer 7

• residue of an isolated singularity of an analytic fn f(z) at z=z0 is the coefficient (A_-1) of (z-z0)^-1 term in Laurent series valid in the immediate vicinity of the singularity

Question 8

What are the two methods of finding residues?

Answer 8

1. Find Laurent series for f(z) about z0
   
   1. read off coefficient of (z-z0)^-1 term (i.e. A_-1)
2. Use pole formula

Question 9

Laurent Series

Answer 9

• f(z) is analytic in some annular region R₁ ≤ |z - z0| ≤ R₂ may be expanded in a power series, uniformly convergent on R₁ ≤ |z - z0| ≤ R₂
• C is any simple closed contour enclosing the inner boundary of the annular region R₁ = |z - z0|

Question 10

What’s an annular?

Answer 10

• Region R₁ ≤ |z - z0| ≤ R₂

Question 11

What is the pole formula for order m and a simple pole?

Answer 11

Question 12

What is Cauchy’s Residue Theorem?

Answer 12

• if f(z) is analytic and single-valued everywhere in region R containing a closed contour C, except at a finite number of isolated singularities inside C (at z = z_j for j = 1, .., N) then
  
  • to integrate f(z) along C, just find all the singularities inside C and calculate residues of f(z)

Question 13

How to evaluate integrals of form

Answer 13

• semi-circular contour of radius R in upper half of complex plane
• consider limit where R->infinity
• path C_R goes to 0 in the limit where R->infinity
• use Jordan’s Lemma to show

Question 14

What is Euler-Lagrange eqn

Answer 14

Question 15

What is the Beltrami identity?

Answer 15

• special case F=F(y,y’)
  
  • integrand that does not depend explicity on x