Search problems

5 problems match “contour integration”. Clear

Visitors search a sample. Sign in for free to search every problem. Sign in

1. 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 $$\int_\gamma e^{2z+i}\, dz.$$
2. 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.
3. 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 $\frac{1}{2\pi i}\oint_{\Gamma} \frac{f'(z)}{f(z)}\,dz$.
4. 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.
5. 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.