Find parametrisations for the following curves:
(a) The line segment from \(-1\) to
\(1-i\),
(b) The circle of radius 2 centred at \(1+i\),
(c) The ellipse \(x^2+4y^2=1\).
Let \(\gamma : [a,b] \rightarrow \mathbb C\) be a \(C^1\)-curve, and define \((-\gamma) : [-b,-a] \rightarrow \mathbb C\), by \((-\gamma)(t):=\gamma(-t)\). Show that for any \(f\) such that \(\int_{\gamma} f(z) \, dz\) is well defined we have that \[\int_{-\gamma} f(z) \, dz = -\int_{\gamma} f(z)\, dz\]
Let \(\gamma: [0,4]\to\mathbb C\) be the curve given by \[\gamma(t) = \left\{ \begin{array}{ll} t ,& 0\le t\le 1, \cr 1+i(t\!-\!1), & 1\le t\le 2, \cr 3\!-\!t+i ,\quad & 2\le t\le 3, \cr i(4\!-\!t), & 3\le t\le 4. \cr \end{array} \right.\] Draw this curve in the complex plane and directly compute \(\int_\gamma e^z dz\) (without using the Fundamental Theorem of Calculus).
Calculate \(\int_{\gamma}| z |\,dz\) when \(\gamma\) is the straight line from \(-i\) to \(i\), and when \(\gamma\) is the segment of the unit circle which joins \(-i\) to \(i\) in the right hand half-plane.
Calculate \(\int_\gamma\frac{1}{z}\,dz\), where \(\gamma(t)=(1\!+\!2t)e^{4\pi it}\) for \(0\le t\le 1\).
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 \[\lim_{\rho \rightarrow \infty} \int_{\gamma_\rho} \frac{z^{1/2}}{z^2 +1} = 0.\]
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 \(0 < \theta < 2 \pi\).