(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Calculate the following line integrals from point z'=(0,-1) to z"=(0,1) along three different contours, [itex]C_j=(0,1,2)[/itex].

[tex]\int_{C_j}|z|dz[/tex]

where [itex]C_0[/itex] is the straight line along the y-axis, [itex]C_1[/itex] is the right semi-circular contour of radius 1, and [itex]C_2[/itex] is the left semi-circular contour of radius 1.

3. The attempt at a solution

(i) Along [itex]C_0[/itex], [tex]z=iy \implies dz = idy[/tex] and the integral is

[tex]\int_{C_0}|z|dz=i^2 \int_{-1}^1ydy=-\frac{y^2}{2}|_{-1}^1=-\frac{1}{2}+\frac{1}{2}=0[/tex]

(ii) Along [itex]C_1[/itex], [itex]z=re^{i \theta} \implies dz = ire^{i \theta}d \theta[/tex] with [itex]\theta:\frac{3 \pi}{2} \rightarrow \frac{\pi}{2}[/itex]. Note that r=1.

So, [tex]\int_{C_1}|z|dz = ir^2\int_{\frac{3 \pi}{2}}^{\frac{\pi}{2}}e^{2i \theta}d \theta=\frac{1}{2}e^{2 i \theta}|_{\frac{3 \pi}{2}}^{\frac{\pi}{2}}=\frac{1}{2} ( e^{i \pi}-e^{3i \pi})=0[/tex]

(iii) Along [itex]C_2[/itex], [tex]z=re^{i \theta} \implies dz = ire^{i \theta}d \theta[/tex] with [itex]\theta:-\frac{\pi}{2} \rightarrow \frac{\pi}{2}[/itex].

The integral is similar to (ii), and one obtains:

[tex]\frac{1}{2} ( e^{i \pi}-e^{-i \pi})=0[/tex]

Did I do these integrals correctly (correct limits in ii and iii)? If so then geometrically, why are these integrals equal to zero?

Thanks for your comments.

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# Homework Help: Complex Contour Integral

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