1) Complex functions, derivatives and integrals

Question 1

What is the definition of an open disc in the complex plane

Answer 1

Question 2

What is an open set and closed set

Answer 2

• D is an open set if for every a ∈ D there exists ε > 0 such that D(a;ε) ⊂ D
• A set Z is closed if its complement 𝐶 \ Z is open

Question 3

What is the definition of a path between two points in 𝐶

Answer 3

A path from z0 to z1 in 𝐶 is a continuous function γ:[a,b]→C
such that γ(a)=z0 γ(b)=z1

Question 4

When is an open set connected

Answer 4

An open set D is connected if given any two point z1, z2 ∈ D, there exists a path from z1 to z2 entirely contained in D

Question 5

What is a domain

Answer 5

A non-empty set D ⊂ C is said to be a domain if it is open and connected

Question 6

Give some examples of domains

Answer 6

(i) C itself is a domain
(ii) Any open disc D(a; r ) is a domain
(iii) An annulus {z ∈ C | r1 < |z − z0| < r2} is a domain
(vi) A half-plane such as {z ∈ C | Re(z) > a} is a domain

Question 7

What does it mean for a function f:D→C to be continuous on a domain D

Answer 7

Question 8

What are some common special domains in complex analysis

Answer 8

Question 9

What does it mean for a function f:D→C to be differentiable at z0

Answer 9

Question 10

What does it mean for a function f:D→C to be holomorphic on a domain D

Answer 10

A function f is holomorphic on D if it is differentiable at every point in D

Question 11

What is the relationship between differentiability and continuity in complex analysis

Answer 11

If f is differentiable at z0, then f is also continuous at z0

Question 12

Describe the proof that a differentiable function is also continuous

Answer 12

Question 13

How is a complex function f:D→C expressed in terms of its real and imaginary parts

Answer 13

Question 14

What are the Cauchy-Riemann equations, and what are their significance

Answer 14

Question 15

What is closed path (or closed loop)

Answer 15

Let γ : [a,b] → C be a path. If γ(a) = γ(b) then we say that γ is a closed path or a closed loop

Question 16

What is a smooth path

Answer 16

A path γ is said to be smooth if γ : [a,b] → C is differentiable and γ′ is continuous

Question 17

What is the length of a smooth path

Answer 17

Question 18

What is the reverse path of a path

Answer 18

Let γ : [a,b]→ C be a path. Define the reverse path γ− : [a,b] → C to be the path γ−(t) = γ(a +b − t)
We say γ − has reverse orientation to γ

Question 19

What is a contour

Answer 19

Question 20

What is a closed contour

Answer 20

If the end-point of γn coincides with the start point of γ1 then we call γ a closed contour

Question 21

What is the length of a contour

Answer 21

The length of the contour γ (above) is defined to be length(γ) = length(γ1)+··· +length(γn)

Question 22

How is the integral of a function f along a path γ defined in complex analysis

Answer 22

Question 23

What happens to the integral of f along a path γ when the path is reparameterised

Answer 23

Question 24

What are the basic properties of contour integrals in complex analysis

Answer 24

Question 25

What is an antiderivative of a function f:D→C in complex analysis

Answer 25

Let D be a domain and f : D → C be a continuous function. We say that a function F : D → C is an antiderivative of f on D if, throughout D, F′ = f

Question 26

What is the Fundamental Theorem of Contour Integration

Answer 26

Question 27

What happens if a holomorphic function has a zero derivative on a domain

Answer 27

Suppose that F is holomorphic on a domain D and F′(z) = 0 for all z ∈ D. Then F is constant on D