Statements

Question 1

Rules for computing derivatives

Answer 1

product, quotient, chain rule?

Question 2

Convergent power series define …

Answer 2

holomorphic functions.

Question 3

Cauchy-Riemann equations

Answer 3

∂u/∂x = ∂v/∂y
∂u/∂y = -∂v/∂x

Question 4

Holomorphic function in a convex open set has …

Answer 4

a primitive.

Question 5

Cauchy’s integral formula for a circle

Answer 5

(for all z in Circle C):
f(z) = 1/(2πi) * Integral over C of f(ζ)/(ζ−z) dζ

Question 6

Holomorphic function f can be expanded in …

Answer 6

power series in any open disc in U.

Question 7

Cauchy’s integral formula for f^(n) (z0)

Answer 7

f^(n) (z) = n!/(2πi) * Integral over C of f(ζ)/(ζ−z)^(n+1) dζ.

Question 8

Liouville’s Theorem

Answer 8

If f is holomorphic on all of C and bounded then f is constant.

Question 9

Non-constant holomorphic function on a connected set has …

Answer 9

isolated zeroes.

Question 10

Analytic continuation

Answer 10

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.

Question 11

Convergence theorem for sequences of holomorphic functions that converge locally uniformly

Answer 11

If there exists a sequence of holomorphic functions that converges locally uniformly to f, then f is holomorphic?

Question 12

Holomorphic functions defined by integrals

Answer 12

..

Question 13

Removable singularities

Answer 13

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.

Question 14

Residue formula for a circle

Answer 14

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

Question 15

1/(2πi) * Integral over γ of f’/f =

Answer 15

number of zeros with multiplicity inside γ − number of poles with multiplicity inside γ

Question 16

Rouché’s Theorem

Answer 16

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.

Question 17

Open Image Theorem

Answer 17

If f is holomorphic and non-constant in a region Ω, then f is open (maps open sets to open sets).

Question 18

Maximum Modulus Principle

Answer 18

If f is a non-constant holomorphic function in a region Ω, then f cannot attain a maximum in Ω.

Question 19

if sum over all|a_n(z)|converges locally uniformly then …

Answer 19

the product of all (1 + a_n(z)) is holomorphic.

Question 20

Homotopy Theorem

Answer 20

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

Question 21

If f is holomorphic on U (simply connected open set) then …

Answer 21

f has a primitive

Question 22

Cauchy’s formula in a simply connected open set

Answer 22

..

Question 23

If U is simply connected there is …

Answer 23

a branch of the logarithm on U.

Question 24

Residue formula with winding numbers for f meromorphic on U (simply connected)

Answer 24

..

Question 25

Conformal maps have …

Answer 25

non-zero derivative everywhere.

Question 26

Example of conformal maps H -> D_1(0)

Answer 26

F(z) = (i - z) / (i + z)
G(w) = i*(1 - w) / (1 + w)

Question 27

Riemann’s Mapping Theorem

Answer 27

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

Question 28

Automorphisms of D_1(0)

Answer 28

rotation by angle θ: z → ze^(iθ)
ψ_α(z) = (a - z) / (1 - α* z) with a* complex conjugate of a

Question 29

Schwarz’s Lemma

Answer 29

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.

Question 30

If f_n is a sequence of injective holomorphic functions which converges locally uniformly, then …

Answer 30

the limit is constant or injective.

Question 31

Montel’s Theorem

Answer 31

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