Contour integrals

Solve the first problem 15 problems

Integral of an exponential over a triangle ●○○○○ 10 ptsLet $\gamma$ be the triangle in $\mathbb C$ whose vertices are the three cube roots of $1+i$ , oriented clockwise. Find the value of $\int_\gamma e^{2z+i}\, dz.$
Contour integral over upper half circle ●●○○○ 20 ptsFind $\int_C \left(1+\frac{2}{z}\right)\,dz$ , where $C$ is the upper unit half-circle $C(\theta)=e^{i\theta}$ , $0 \le \theta \le \pi$ , traversed from $z=1$ to $z=-1$ . Give the real part and the imaginary part of the result separately. Round the imaginary part to two decimal places.
Contour integral of 1/(z^2+1) ●●○○○ 20 ptsCompute the complex contour integral $\displaystyle\int_{|z|=2}\frac{1}{z^2+1}\,dz$ , where the circle $|z|=2$ is traversed once counterclockwise.
Argument principle on a doubly traversed circle ●●○○○ 20 ptsLet $f(z)=z^3e^{z^2}$ for $z\in\mathbb{C}$ , and let $\Gamma$ be the circle $z=e^{i\theta}$ , where $0\leq\theta\leq4\pi$ , so the contour goes around the unit circle twice counterclockwise. Find $\frac{1}{2\pi i}\oint_{\Gamma} \frac{f'(z)}{f(z)}\,dz$ .
Contour integral of a square-root branch ●●●○○ 50 ptsLet $C$ be the circle $|z|=R$ with $R>1$ , centred at the origin. Use the branch of $(z^2-1)^{1/2}$ with a straight branch cut joining $z=-1$ and $z=1$ , chosen so that $(z^2-1)^{1/2}>0$ for real $x>1$ . Let $I=\oint_C (z^2-1)^{1/2}\,dz$ , taken counterclockwise. Write $I=b\,i$ with $b$ real. Find $b$ , rounded to two decimal places.
Principal value with real poles ●●●○○ 50 ptsFind the Cauchy principal value of $\int_{-\infty}^{\infty}\frac{x\,dx}{(x-1)^4-1}$ , where the poles at $0$ and $2$ lie on the real axis. Round to two decimal places.
Complex line integral of 1/(1+z^2) ●●●○○ 50 ptsEvaluate $\int_1^{1+i}\frac{1}{1+z^2}\,dz$ along the straight line segment from $z=1$ to $z=1+i$ . Give the real part and the imaginary part of the result separately. Round each to two decimal places.

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