The Geometry of Holomorphic Maps Flashcards
Define conformal at z0.
A (real differentiable) map f : π β β on a region π β β is conformal at z0 if it is angle and orientation preserving at z0.
Finish the following proposition
Finish the corollary: Conformal maps take orthogonal grids (in the x-y plane) to β¦ ?
Orthogonal grids (in the u-v plane).
Let f = u + iv be a holomorphic function and let Ζ and Ζ Μ be two paths given by the level curves u(x, y) = C1 and v(x, y) = C2 respectively (for C1, C2 real constants). Then what sets (whenever fβ²(z) ΜΈ= 0)do the paths Ζ and Ζ Μ form?
Orthogonal set in the x-y plane
Define a biholomorphic map from π to π β.
We call a holomorphic map f on a region π a biholomorphic map from π to π β = f(π ) if:
- π β is a region
- f is one-to-one and the inverse map f-1:π β β π is also holomorphic
What symbol do you use to show two regions π and π β are biholomorphic.
f : π β₯² π β
What is the Lemma about the automorphism group of π .
Define simply connected.
A region π is simply-connected if every closed path Ζ in π (that is Ζ(0) = Ζ(1)) can be βcontinuosuly shrunkβ to a point in π . Heuristically this means π has no holes.
Is βΩ or β\ββ€0 simply connected?
β\ββ€0
What is the Riemann mapping theorem.
Every simply connected region π β β, there exists a biholomorphic map f:π β₯² B1(0)
Define a MΓΆbius transformation.
Why do we exclude matrices for which det(T) = 0 in MΓΆbius transformations?
MT is constant if det(T) = 0
If k = sqrt(|det(T)|) show that we can scale any MΓΆbius transformation so that det(T) = Β± 1.
What is the Lemma about the set of MΓΆbius transformation forming a group under composition.
Prove the following Lemma.
Need to do - see sheet 5
What is the symbol for the extended complex plane?
What are three algebraic operations on β that are extended to ββΌ ?
What do these four properties show.
That the
What is the propositiong about circles, line and MΓΆbius transformations?
MΓΆbius transformations map circles or lines to circles or lines
What are the three different maps between circles and lines with MΓΆbius transformations?
- Circles not through the origin go to tcircles not through the origin
- Circles through the origin go to lines not through the origin
- Lines through the origin go to lines through the origin
How can you find a map from β to β by writing β in a clever way?
What is the Mob transformation to take the upper half plane to the unit disc?
What is the inverse of this map called?
Caley Map
How is the Caley Map denoted?
