complex analysis

Question 1

define a disconnected set

Answer 1

A set is disconnected if you can split it into two separate open pieces that don’t touch but their union is the disconnected set

Question 2

true or false: A sequence of complex numbers converges if and only if it
is a Cauchy sequence.

Answer 2

true

Question 3

what does the fact that (xn, yn) is a Cauchy sequence in R2 directly imply

Answer 3

xn and yn are Cauchy sequences in R

Question 4

what does it mean for a set to be path connected

Answer 4

if every two points z0, z1 ∈ A can be joined by a
continuous path with values in A.

Question 5

Let Ω ⊆ C. A complex-valued function f : Ω → C is said to be continuous at z0 ∈ Ω if limz→z0 f(z) = f(z0),

Answer 5

fax

Question 6

what is a way to rewrite the partial derivative of f with respect to z

Answer 6

1/2 (∂f/∂x − i ∂f/∂y)

Question 7

now what about for the partial derivative of f with respect to the conjugate of z

Answer 7

∂f/∂z¯:=1/2 (∂f/∂x + i ∂f/∂y)


Question 8

what is a path

Answer 8

a continous map from [a,b] to the complex numbers

Question 9

what is a simple path

Answer 9

A path is said to be simple if it does not cross itself, except possibly at the end points