Analysis 3

Question 1

Goursat’s Theorem

Answer 1

Let γ be a triangular contour with its interior contained in some domain U and let f(z) be holomorphic on U, then

∫γ f(z) dz = 0

Question 2

Cauchy’s Theorem

Answer 2

If f is holomorphic on a simple curve c and its interior, then

∫c f(z) dX = 0

Question 3

Liouville’s Theorem

Answer 3

If f is an entire and bounded function, then f is constant

f = entire + bounded ⇒f = constant

Question 4

Morera’s

Answer 4

Let f be continuous on a region Ω and ∫γ f(z) dX

= 0 ∀ γ, closed contours in Ω with interior in Ω

Then f is holomorphic

Question 5

Define what it means for a function to be holomorphic at a point z₀

Answer 5

f is holomorphic at z₀ if lim (f(z) - f(z₀) / z- z₀) exists

If it does, we set f’(z) equal to it and call this the derivative of f at z₀

Question 6

Show that the function is not holomorphic

Answer 6

To show that the limit does not exist:

Approach z₀ first horizontally and then vertically

As the two answers are different; the limit does not exist and F is not differentiable at z₀

Question 7

Entire

Answer 7

If f is holomorphic on the whole complex plane

Question 8

Rouche’s theorem

Answer 8

Left two functions f(z) and g(z) be analytic inside and on a simple curve C, and suppose that |f(z)| > |g(z)| at each point on C.

Then f(z) and f(z) + g(z) have the same number of zeros, inside C

Question 9

convergence of power series

Answer 9

If the function is holomorphic in a disk, then it’s Taylor series converges in this disc

Question 10

Fundamental theorem of algebra

Answer 10

All polynomials of degree >0 with complex coefficients have a complex root

This is a corollary of Liouville’s theorem

Question 11

Use partial fractions

Answer 11

More than one pole and can’t compute the residue

Use partial fractions and then compute cauchy’s integral

Question 12

Isolated singularity

Answer 12

The point α is called an isolated singularity of the complex function f(z) if f is not analytic at z=α but there exists a real number R>0 St. f(z) is analytic everywhere in the punctured disk D(α,ε)

Log z does not have an isolated singularity as it is not analytic over the whole negative real line and the origin

Question 13

Isolated singularities

Answer 13

If there exists a deleted neighbourhood of z0 that contains no singularity

Includes poles, removable singularities, essential singularities and branch points

Question 14

Pole

Answer 14

An isolated singularity point z0 St. f(z) can be represented by an expression that is of the form

f(z) = F(z)/(z-z0)^n

Where f(z) is analytic at z0 and f(z0) ≠ 0
The integer n is the order of the pole
Poles of order 1 are SIMPLE POLES

Question 15

Removable singularity

Answer 15

And isolated singular point z₀ st. f can be defined or resigned at z₀ in such a way to be analytic at z₀. A singular point z₀ is removable if lim f(z) exists as z tends to z₀

Question 16

Cauchy’s integral formula for f and its derivatives

Answer 16

Where f is holomorphic on an open sent U containing the closed disc D(α, R) and C is the circle centred at α with radius R traversed anticlockwise and z₀ ∈D(α,R)

Question 17

Poles / zeros of denominator outside the circle

Answer 17

The denominator has zeroes at ..
These are outside the circle
Therefor the function f(z) is holomorphic in an open set containing the disc (contour)
By Cauchy’s theorem .. ∫c f(z) dz =0

Question 18

Cauchy’s inequality

Answer 18

|f^n(z)| ≤ [n! Max{|z=z₀|=R} |f(z)|]/R^n

Question 19

Antiderivative theorem

Answer 19

Let f be continuous on a region Ω and assume that ∃ a holomorphic function F on Ω with F’(z) = f(z).
Then f is called the antiderivative or primitive of f

Question 20

Conformal map

Answer 20

Geometrically the distance to ai is less than the distance to -ai iff. z is in the half-plane determined by the perpendicular bisect or of the segment from ai to -ai and containing ai

The bi sector is clearly the real axis

The map is holomorphic on as the root of the denominator is not in it

It is a rational function

The inverse map is given by

This is also holomorphic on D(0,1) as it is a rational function and 1∉D(0,1)

Finding the map z in terms of w shows that the map is an injection and the surjectivity has been shown

Question 21

Simple pole

Answer 21

Factorising to get a holomorphic no vanishing function in a neighbourhood of z₀
This makes the quotient holomorphic in the punctured disk D’(Z₀,r) and z₀is an isolated singularity

If a function is holomorphic and non-zero in a neighbourhood of z₀
Show 1/f has a simple pole and z₀ then so does f

Question 22

Integrating logs

Answer 22

Key hole contour

Outer circle of radius 1 and corridor along the real axis

Non-principal branch of the logarithm:
Log(z) = log|z| +iarg(z), arg(z) ∈[0,2π]

Question 23

Injective
Surjective
Bijective

Answer 23

• Every member of A has its own unique matching member in B
• -> one to one function
• -> function can be reversed
• every b has at least one matching A
• ->
• a perfect one to one correspondence
• -> both injective and surjective

Question 24

Maximum modulus principal

Answer 24

If f is holomorphic and non-constant in a region Ω - then |f(z)| has no maximum value inside Ω

Like opposite to Liouville’s

Question 25

Variation of the argument

Answer 25

Δc arg f(z) =1/i ∫c f’(z) / f(z) dz

Question 26

Keyhole

Answer 26

If discontinuous on x axis - multi variable function

Question 27

Indented semicircle

Answer 27

discontinuous as 0 eg. LogX doesn’t exist at 0

Question 28

Confirms Mapping

Answer 28

The mapping w=f(z) by an analytic function is conformal expect at critical points - points where the derivative f’(z) is zero