1) Complex functions, derivatives and integrals Flashcards
What is the definition of an open disc in the complex plane
What is an open set and closed set
- 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
What is the definition of a path between two points in 𝐶
A path from z0 to z1 in 𝐶 is a continuous function γ:[a,b]→C
such that γ(a)=z0 γ(b)=z1
When is an open set connected
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
What is a domain
A non-empty set D ⊂ C is said to be a domain if it is open and connected
Give some examples of domains
(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
What does it mean for a function f:D→C to be continuous on a domain D
What are some common special domains in complex analysis
What does it mean for a function f:D→C to be differentiable at z0
What does it mean for a function f:D→C to be holomorphic on a domain D
A function f is holomorphic on D if it is differentiable at every point in D
What is the relationship between differentiability and continuity in complex analysis
If f is differentiable at z0, then f is also continuous at z0
Describe the proof that a differentiable function is also continuous
How is a complex function f:D→C expressed in terms of its real and imaginary parts
What are the Cauchy-Riemann equations, and what are their significance
What is closed path (or closed loop)
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
What is a smooth path
A path γ is said to be smooth if γ : [a,b] → C is differentiable and γ′ is continuous
What is the length of a smooth path
What is the reverse path of a path
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
What is a contour
What is a closed contour
If the end-point of γn coincides with the start point of γ1 then we call γ a closed contour
What is the length of a contour
The length of the contour γ (above) is defined to be length(γ) = length(γ1)+··· +length(γn)
How is the integral of a function f along a path γ defined in complex analysis
What happens to the integral of f along a path γ when the path is reparameterised
What are the basic properties of contour integrals in complex analysis
What is an antiderivative of a function f:D→C in complex analysis
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
What is the Fundamental Theorem of Contour Integration
What happens if a holomorphic function has a zero derivative on a domain
Suppose that F is holomorphic on a domain D and F′(z) = 0 for all z ∈ D. Then F is constant on D
