Chapter 4: Contour Integration and Cauchy's Theorem Flashcards
Definition of Curve
A curve (also known as a path) is a continuous function
γ:[a,b]→C defined on some closed interval [a,b] in R
We write γ∗ ={γ(t):t∈[a,b]} for the image of the curve
Definition of Smooth Curve
A curve γ:[a,b]→C is smooth if γ is differentiable (with one-sided derivatives at the end-points a and b) and the derivative γ′ is continuous.
Parametrisation of Line Segment
If c,d∈C, the line segment from c to d is γ:[0,1]→C
given by γ(t)=c+(d−c)t for 0≤t≤1
Parameterisation of Circular Arc
If c ∈ C, r > 0 and θ1, θ2 are angles chosen in some appropriate range (with θ1 < θ2),
the (anti-clockwise) circular arc
γ : [θ1, θ2] → C
is given by
γ(t) = c + re^{it}
Definition of Piece-wise Smooth Curve
Definition of Closed Curve
Definition of Simple Curve
Definition of Contour
Definition of Curve Length
Line Segment Length
Circular Arc Length
Definition of Integral Along Curve
Properties of Piecewise Smooth Curve with a Continuous Function
Fundamental Theorem of Calculus for Integrals Along a Curve
Easy Version of Cauchy’s Theorem
Bound on modulus of an Integral
Crude Estimation Theorem
Jordan Curve Theorem
Cauchy’s Theorem
