Coursera Analysis Complex Kind Lesson 5.3 Fundamental Theorem Calculus Analytic Functions

Aug 2, 2016

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 >>


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