Show that for any \(a\in \mathbb C\) the curve \(\gamma(t)=e^{\alpha t}\) satisfies \[\gamma'(t) = \alpha e^{\alpha t}(=\alpha \gamma(t)).\] Conclude that for any \(-\infty<a<b<\infty\) and any \(\alpha\not=0\) \[\int_{a}^b e^{\alpha t} dt= \frac{e^{\alpha b} - e^{\alpha a}}{\alpha}.\]
(Assignment sheet 9 problem 5) Calculate \(\int_\gamma\frac{1}{z}\,dz\), where \(\gamma(t)=(1+2t)e^{4\pi it}\) for \(0\le t\le 1\).
(Assignment sheet 9 problem 6) Let \(\gamma_\rho\) be the curve \(\gamma_\rho(\theta) := \rho e^{i \theta}\) with \(0 \leq \theta \leq \pi\). Let \(z^{\frac{1}{2}}\) be the branch of square root corresponding to the branch of \(\log\) with argument in \((-\pi/2, 3\pi/2)\), that is, if \(z= \rho e^{i\theta}\) with \(\theta\in (-\pi/2, 3\pi/2)\) then \(z^{\frac{1}{2}}= \sqrt{\rho}e^{i\theta/2}\). Show that \[\lim_{\rho \rightarrow \infty} \int_{\gamma_\rho} \frac{z^{1/2}}{z^2 +1}dz = 0.\]
(Assignment sheet 9 problem 7) Let \(\gamma\) be any piecewise \(C^1\)-curve from \(-3\) to \(3\) such that, except for the end points, lies entirely in the upper half plane. Calculate the integral \[\int_{\gamma} f(z) \, dz,\] where \(f(z)\) is the branch of \(z^{\frac{1}{2}}\) defined by \(\sqrt{r} e^{i \theta/2}\) with \(0 < \theta < 2 \pi\).