## Slide 2 Definition

Definition for real variable

D $\in$ C. f: D $\to$ C, a continuous function. Primitive of f on D is the analytic function F: D $\to$ 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) $\gamma \; [a,b] \to$ D (remember [a,b] is a real interval and $\gamma$ maps this to some path in the complex plane) then $\int_{\gamma} \; f(z) \; dz \; = \; F(\gamma(b)) - F(\gamma(a))$

The value of the integral depends only only the initial, $\gamma(a)$ and final $\gamma(b)$ 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 $\int_{|z|=1}\frac{1}{z} \;dz$

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\ $(\infty,0]$

<< Diagram >>

If the path $\overset{~}{\gamma}$ starts just below negative x-axis on unit circle, $e^{-i \pi}$ and turns round until it almost reaches the negative real x-axis from above on unit circle, $e^{+i \pi}$, we can merge the primitive function , Log (z) and these “almost” limits

Log( $e^{+i \pi}$) – Log( $e^{-i \pi}$) = 2πi .

## Slide 6 Example exponential function

Path any path $\gamma$ in C, from i to i/2, $\int_{\gamma}e^{\pi z} dz$

= ${[{\frac{1}{\pi}} e^{\pi z}]}_i^{\frac{i}{2}}$

= $\frac{1}{\pi}(i + 1)$

<< continue >>