Statements Flashcards
Rules for computing derivatives
product, quotient, chain rule?
Convergent power series define …
holomorphic functions.
Cauchy-Riemann equations
∂u/∂x = ∂v/∂y
∂u/∂y = -∂v/∂x
Holomorphic function in a convex open set has …
a primitive.
Cauchy’s integral formula for a circle
(for all z in Circle C):
f(z) = 1/(2πi) * Integral over C of f(ζ)/(ζ−z) dζ
Holomorphic function f can be expanded in …
power series in any open disc in U.
Cauchy’s integral formula for f^(n) (z0)
f^(n) (z) = n!/(2πi) * Integral over C of f(ζ)/(ζ−z)^(n+1) dζ.
Liouville’s Theorem
If f is holomorphic on all of C and bounded then f is constant.
Non-constant holomorphic function on a connected set has …
isolated zeroes.
Analytic continuation
f and F analytic in Ω and Ω′, with Ω ⊂ Ω′. If f = F on Ω, F is an analytic continuation of f into the region Ω′. F is uniquely determined by f.
Convergence theorem for sequences of holomorphic functions that converge locally uniformly
If there exists a sequence of holomorphic functions that converges locally uniformly to f, then f is holomorphic?
Holomorphic functions defined by integrals
..
Removable singularities
If f is holomorphic except at z_0 and bounded on non empty open punctured disc around z_0 then f has an analytic continuation at z_0.
Residue formula for a circle
f holomorphic on open set containing circle C and its interior expect at poles, z_1,…,z_N:
Integral over Circle = 2πi* Sum from 1 to N of residues of f at z_k
1/(2πi) * Integral over γ of f’/f =
number of zeros with multiplicity inside γ − number of poles with multiplicity inside γ
Rouché’s Theorem
Suppose that f and g are holomorphic in an open set containing a circle C and its interior. If |f(z)| > |g(z)| for all z ∈ C,
then f and f + g have the same number of zeros inside the circle C.
Open Image Theorem
If f is holomorphic and non-constant in a region Ω, then f is open (maps open sets to open sets).
Maximum Modulus Principle
If f is a non-constant holomorphic function in a region Ω, then f cannot attain a maximum in Ω.
if sum over all|a_n(z)|converges locally uniformly then …
the product of all (1 + a_n(z)) is holomorphic.
Homotopy Theorem
If f is holomorphic in Ω then:
Integral over γ_0 of f = Integral over γ_1 of f
whenever the two curves γ_0 and γ_1 are homotopic in Ω
If f is holomorphic on U (simply connected open set) then …
f has a primitive
Cauchy’s formula in a simply connected open set
..
If U is simply connected there is …
a branch of the logarithm on U.
Residue formula with winding numbers for f meromorphic on U (simply connected)
..
Conformal maps have …
non-zero derivative everywhere.
Example of conformal maps H -> D_1(0)
F(z) = (i - z) / (i + z)
G(w) = i*(1 - w) / (1 + w)
Riemann’s Mapping Theorem
For Ω non-empty, not all of C and simply connected. If z_0 ∈ Ω, then there exists a unique conformal map F : Ω→D such that:
F(z_0) = 0 and F′(z_0) > 0
Automorphisms of D_1(0)
rotation by angle θ: z → ze^(iθ)
ψ_α(z) = (a - z) / (1 - α* z) with a* complex conjugate of a
Schwarz’s Lemma
f holomorphic D -> D with f(0) = 0 then:
1. |f(z)|≤|z|for all z∈D
2. If for some z_0 ≠ 0 we have |f(z_0)| = |z_0|, then f is a rotation.
3. |f′(0)| ≤ 1, and if equality holds, then f is a rotation.
If f_n is a sequence of injective holomorphic functions which converges locally uniformly, then …
the limit is constant or injective.
Montel’s Theorem
Family F of holomorphic function on Ω is normal if every sequence in F has a subsequence that converges uniformly on every compact subset of Ω (the limit can be outside of F).
Proof idea: consequence of uniform boundedness and equicontinuity
