Analysis 3 Flashcards
Goursat’s Theorem
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
Cauchy’s Theorem
If f is holomorphic on a simple curve c and its interior, then
∫c f(z) dX = 0
Liouville’s Theorem
If f is an entire and bounded function, then f is constant
f = entire + bounded ⇒f = constant
Morera’s
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
Define what it means for a function to be holomorphic at a point z₀
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
Show that the function is not holomorphic
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
Entire
If f is holomorphic on the whole complex plane
Rouche’s theorem
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
convergence of power series
If the function is holomorphic in a disk, then it’s Taylor series converges in this disc
Fundamental theorem of algebra
All polynomials of degree >0 with complex coefficients have a complex root
This is a corollary of Liouville’s theorem
Use partial fractions
More than one pole and can’t compute the residue
Use partial fractions and then compute cauchy’s integral
Isolated singularity
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
Isolated singularities
If there exists a deleted neighbourhood of z0 that contains no singularity
Includes poles, removable singularities, essential singularities and branch points
Pole
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
Removable singularity
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
Cauchy’s integral formula for f and its derivatives
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)
Poles / zeros of denominator outside the circle
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
Cauchy’s inequality
|f^n(z)| ≤ [n! Max{|z=z₀|=R} |f(z)|]/R^n
Antiderivative theorem
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
Conformal map
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
Simple pole
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
Integrating logs
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]
Injective
Surjective
Bijective
- 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
Maximum modulus principal
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
Variation of the argument
Δc arg f(z) =1/i ∫c f’(z) / f(z) dz
Keyhole
If discontinuous on x axis - multi variable function
Indented semicircle
discontinuous as 0 eg. LogX doesn’t exist at 0
Confirms Mapping
The mapping w=f(z) by an analytic function is conformal expect at critical points - points where the derivative f’(z) is zero
