Integration of Complex Variables and Power Series

These flashcards cover definitions, examples, and key concepts related to complex variable integration and contours.

What is the equation for the smooth curve C in the z-plane joining the two points 2(a) and (b)?

z(t) = x(t) + iy(t), where 0 ≤ t ≤ 1.

What conditions must x(t) and y(t) satisfy for the curve to be considered smooth?

x(t) and y(t) must have continuous derivatives such that dy/dt ≠ 0 or dy(t)/dt ≠ 0 in the interval a < t < b.

How is the length of a smooth curve C calculated?

The length is given by the integral of the form ∫√((dx/dt)^2 + (dy/dt)^2) dt from t=a to t=b.

What is a piecewise or conditionally smooth curve?

A curve composed of a finite number of smooth curves, also known as a contour.

Define a simple contour in the context of curves.

A curve or contour which does not intersect itself, meaning for any two points t₁ and t₂, z(t₁) ≠ z(t₂) if t₁ ≠ t₂.

What is an example of a simple closed contour?

A simple closed contour can be represented by r(t) = a cos(t), y(t) = a sin(t), for 0 ≤ t ≤ 2π.

How do you differentiate between continuous and discrete points when calculating lengths?

For continuous points, you use integration, and for discrete points, you use summation.