Coursera Analysis Complex Kind Lesson 5.2 Complex integration Examples and properties

Aug 1, 2016

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 5. standard integration by substitution

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 >>

 

 

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