Math Challenges

Submissions for Problem #70

Problem #70

Use a parametric representation for the contour C to evaluate the integral.

\[ \displaylines{\int_{C}f\left(z\right)dz\text{ where }f\left(z\right)=z-1\text{ and }C\text{ is the arc from }z=0\text{ to }z=2\text{ consisting of}\\ 1)\text{the semicircle }z=1+e^{i\theta}\left(\pi\le\theta\le2\pi\right)\\ 2)\text{the segment }0\le x\le2\text{ of the real axis.}} \]

← Back to problem

zapwai

Solution: \( \displaylines{2)\int_0^2x-1\differentialD x=\frac12x^2-x\left|_0^2\right|=0\\ \text{The residue theorem will yield 0 as well, implying that (1) equals 0.}\\ \text{If you do it directly:}\\ 1)\int_{\pi}^{2\pi}\left(\left(1+e^{i\theta}\right)-1\right)ie^{i\theta}d\theta=\frac12e^{i2\theta}\left|_{\pi}^{2\pi}\right.=0} \)