## Slide 2 Resume

If $\gamma$ 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

$\int_{\gamma}f(z)\;dz = \int_a^b\;f(\gamma(t)) \; \gamma \; '(t) \; dt$

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)

$\gamma = 1 - t(1 - i), 0 \leq t \leq 1$ and let f(z) = Re(z)

$\int_{\gamma}f(z)\;dz = \int_0^1\;Re(1 - t(1 - i)) \; (-1)(1-i) \; dt$

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

$(i - 1) \int_0^1 \; 1 -t \; dt = \frac{i-2}{2}$

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

$\gamma = re^{it}, 0 \leq t \leq 2\pi$ and let f(z) = $\overline{z}$

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

$\int_{\gamma}f(z)\;dz =\int_{\gamma} \overline{z} \;dz$

$\int_0^{2\pi}\;\overline{\gamma(t)} \; \gamma \; '(t) \; dt$

$\int_0^{2\pi}\;re^{-it} \; rie^{it} \; dt$

$r^2i\int_0^{2\pi}\;e^{-it} \; e^{it} \; dt = r^2i\int_0^{2\pi}\;dt = 2 \pi i r^2$

= (2i). Area unit disk centre origin

## Slide 6. Independence of parameterisation

<< diagram needed >>

## Slide 7 Piecewise smooth

If the path function $\gamma$ is a succession of paths $\gamma_1, \gamma_2, \gamma_3, .. ,\gamma_n$  that connect up but the connections maybe sharp “points”, then the path integral is the sum of the individual path integrals $\int_{\gamma} = \int_{\gamma_1} + \int_{\gamma_2} + \int_{\gamma_3} + ... + \int_{\gamma_n}$

## Slide 8. “Reverse” path $-\gamma$

<< Diagram required >>

If $\gamma:[a,b] \rightarrow C$ greates a path in C

then $-\gamma:[a,b] \rightarrow C$ is the “reverse path” where $-\gamma(a) = \gamma(b) \textrm{ and } -\gamma(b) = \gamma(a)$

The (path) function that has these properties is $-\gamma(t) = \gamma(a+b-t)$

By differentiating: $-\gamma\; ' \;(t) = (-1)\gamma \; ' \;(a+b-t)$

and after substituting it turns out: $\int_{-\gamma} = -\int_{\gamma}$

## 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

$\int_a^b |\gamma \; ' \; (t) |\;dt$

## Slide 13 Integration with respect to arc length

This is different from Slide 10 & 11.

$\int_{\gamma} f(z) \; |dz| = \int_a^b f(\gamma(t))|\gamma \; ' (t) |\;dt$

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

<< Continue >>

## Slide 14,15,16 ML Estimate

<< to summarise >>