## Coursera: Analysis of a Complex Kind. 5.1 Introduction to Complex Integration

### Jul 30, 2016

## Slide 2 Real integration

Start with Riemann integral as sum of rectangles between curve and x-axis.

## Slide 3 Fundamental Theorem of Calculus

## Slide 4 Properties of Integrals

Definition of antiderivative

## Slide 5 Integral f(z) based on real integrals

Real integral is over an **interval** [a,b].

Complex integral is over a **curve** in C.

One part of the process is to create the curve from the (real) interval.

A (Complex) curve is a (piecewise) smooth function, γ, that maps a real interval into the complex plane

In symbols: γ [a,b] -> C, γ = x(t) + i y(t)

Notice t is a real value that lies in the interval [a,b] and x and y are functions of a real variable (t).

Next we have (are given) a function f that has domain γ.

=

{to remember: think about as being the y-ordinate or function value and being the }

## Slide 6. Path integral

is read as the integral of the function f(z) along the path γ.

γ is the path of complex values by applying the function γ to the real interval [a,b]

If t is the real variable that runs from a to b along the interval [a,b] then it turns out (See later)

So, if we are given a function, that maps the real interval [a,b] to some path of complex values in the complex plan , C, where the end points of the path are respectively, then **a and b** become the lower and upper limits of the transformed integral.

The integral with respect to z has become an integral with respect to t, where t has real end limit values, but the transformed function may include “i” e.g. . Then integrate the transformed function w.r.t. variable t and substitute the limit values.

<< rest of slide is sketch of proof >>

## Slide 7 Integrals over complex valued functions

Function g has domain a real interval [a,b] and creates complex outputs based on the functions , u v, with real domain interval [a,b]

Define:

g(t) = u(t) + iv(t)

then

Examples

= [sin t] – i [cos t] = 2i

or directly

= 2i

Another example (direct)

## Slide 8 Example of path integral

## Slide 9 Example of path integral – changing path to a function of t

Represent |z| = 1 as path

So

## Slide 10 Example Path integral

We are given the path function , and told the initial and final values of the real variable, t, . This is equivalent to the representation of the path as |z| = 1. gamma will map to 1 to 1 (a closed path). Differentiate gamma to create

So,

## Slide 11 example

## Slide 12 General solution