## Coursera Analysis Complex Kind Lesson 5.3 Fundamental Theorem Calculus Analytic Functions

### Aug 2, 2016

## Slide 2 Definition

Definition for real variable

D C. f: D C, a continuous function. Primitive of f on D is the analytic function F: D C such that F ‘ = f on D. Note f and F have same domain, D.

## Slide 3 Theorem

f is continuous on domain D, and f has primitive F in D, then for **any** curve (path) D (remember [a,b] is a real interval and maps this to some path in the complex plane) then

The value of the integral depends only only the initial, and final values of the path, not the intermediate values or route.

f needs to **have** a primitive

What are the condition for f to **have** a primitive?

## Slide 4 Example

Say we have a function along a line segment from 0 to 1+i.

To determine value of definite integral just perform integration as if real function then substitute ends values of path as the limit values.

## Slide 5 Integrating reciprocal function, 1/z

Evaluate

Now F(z) = Log (z) then F ‘ (z) = 1/z but not all of C.

F is analytic on complex plane excluding the negative real axis, C\

<< Diagram >>

If the path starts just **below** negative x-axis on unit circle, and turns round until it almost reaches the negative real x-axis from **above **on unit circle, , we can merge the primitive function , Log (z) and these “almost” limits

Log() – Log() = 2πi .

## Slide 6 Example exponential function

Path any path in C, from i to i/2,

=

=

<< continue >>