Definitions

Question 1

Open Set

Answer 1

For all point there exists a radius such that ball around point is in set
Examples: empty set, all C, Disc

Question 2

Closed Set

Answer 2

Complement of an open set <=> all series with all elements in set converge in that set
Examples: empty set, all C, closed Disc

Question 3

Compact Set

Answer 3

Compact and bounded <=> all series with elements in set have a convergent subsequence
Examples: empty set, closed Disc, Cr(z)?

Question 4

Connected Set

Answer 4

Cannot be the union of two open sets?
Examples: empty set, all C, (closed) Disc. all R
Counter-examples: Z, Q

Question 5

f: U -> C is holomorphic on U if …

Answer 5

for all z_0 in U the limit z->z_0 of [f(z) - f(z_0)] / [z - z_0] exists. This limit is called the complex derivative.

Question 6

(Smooth) curves

Answer 6

..

Question 7

Line Integrals

Answer 7

..

Question 8

Primitive

Answer 8

holomorphic function F such that F’ = f, then F is primitive of f.

Question 9

Order of vanishing of holomorphic function at z_0

Answer 9

Order of the zero of f at z_0, which derivative is unequal to 0

Question 10

Poles of holomorphic function

Answer 10

there exists a non-vanishing holomorphic function h and unique positive integer n such that
f(z) = (z − z0)^{−n}*h(z)

Question 11

Order of a pole

Answer 11

rate at which function goes to zero, the multiplicity of the zero

Question 12

Principal part of function at pole

Answer 12

Sum of a_{-n}/(z-z_0)^n

Question 13

Residue of a function at pole

Answer 13

coefficient a_{-1}

Question 14

Meromorphic function

Answer 14

if there exists a sequence of points z_n that has no limit points in Ω such that:
1. f is holomorphic on Ω - {z_n|all natural n}
2. f has poles at {z_n|all natural n}

Question 15

Infinite product of complex numbers

Answer 15

Given sequence a_n, the product of 1 to infinity of (1 + a_n) converges if the limit of N->infinity of the product form 1 to N converges.

Question 16

Homotopy of smooth curves

Answer 16

for each 0≤s≤1 there exists a curve γ_s ⊂ Ω, parametrized by γ_s(t) defined on [a, b], such that for every s
γ_s(a) = α and γ_s(b) = β,
and for all t ∈ [a,b]
γ_s(t) [|s=0] = γ_0(t) and γ_s(t)[|s=1] = γ_1(t).

Question 17

Simply connected open set

Answer 17

if any pair of curves with the same end-point are homotopic

Question 18

Branch of logarithm on open set

Answer 18

log holomorphic on U such that exp(log(z)) = z

Question 19

Principal branch of logarithm

Answer 19

log z = log r + iθ
Ω = C − {(−∞, 0]}
where z = r*e^iθ with |θ| < π
log(1) = 0

Question 20

z^a for z in simply connected open set

Answer 20

branch of log exists so z^a = exp(a*log(z))

Question 21

n-th root of z

Answer 21

z^(1/n) = exp((1/n)*log(z))

Question 22

Winding number of a curve around a point

Answer 22

W_γ(z) = 1/(2πi) * Integral over γ of 1/(c-z) dc

Question 23

Conformal map

Answer 23

bijective holomorphic function

Question 24

Conformal equivalence

Answer 24

there exists a conformal map between two sets

Question 25

Automorphism

Answer 25

conformal map from an open set to itself