## Coursera Analysis Complex Kind Lesson 5.2 Complex integration Examples and properties

### Aug 1, 2016

## Slide 2 Resume

If is a mapping from real interval [a,b] to some smooth curve on the complex plane, C, and f is complex valued with domain somewhere in C. Then

So the integration along a path in the complex plane has been converted into a (complex-valued) function along a real-valued interval [a,b] for the real variable , t.

## Slide 3 Example. Linear path. Re(z)

and let f(z) = Re(z)

Since (1 – t(1 – i)) = 1 – t + it, so Re((1 – t(1 – i)) = 1 – t, and the integral becomes

## Slide 4 Example. Circular path. conjugate(z)

and let f(z) =

The path is a circle, centred origin, radius r, in complex plane. function is complex conjugate.

= (2i). Area unit disk centre origin

## Slide 5. standard integration by substitution

## Slide 6. Independence of parameterisation

<< diagram needed >>

## Slide 7 Piecewise smooth

If the path function is a succession of paths that connect up but the connections maybe sharp “points”, then the path integral is the sum of the individual path integrals

## Slide 8. “Reverse” path

<< Diagram required >>

If greates a path in C

then is the “reverse path” where

The (path) function that has these properties is

By differentiating:

and after substituting it turns out:

## Slide 9 Path integral properties

Properties are analogous usual integration properties of: addition of terms, multiplication by a constant factor, and the reverse property from previous slide – which is like reversing the limits requires putting a minus sign in front of the integral.

## Slide 10 and 11 Length of curve

## Slide 13 Integration with respect to arc length

This is different from Slide 10 & 11.

Notice the introduction of the modulus around the derivative of gamma.

<< Continue >>

## Slide 14,15,16 ML Estimate

<< to summarise >>

Aug 16, 2016 at 8:15 pm

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