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Homework 7 (MATH 1199 Fall 2019)
1. Draw the contour given by the formula...
(a) $\left\{ \begin{array}{ll}
z(t) = 4e^{it} \\
0 \leq t \leq \pi
\end{array} \right.$
(b) $z(t) = \left\{ \begin{array}{ll}
t-i & \quad 0 \leq t \leq 2 \\
t+(t-3)i & \quad 2 \leq t \leq 3.
\end{array} \right.$
2. Draw the contour $C$ and calculate the integral...
(a) $\displaystyle\int_C \dfrac{1}{z} \mathrm{d}z, \quad \quad C: \left\{ \begin{array}{ll}
z(t) = e^{it} \\
0 \leq t \leq 2\pi
\end{array} \right.$
(b) $\displaystyle\int_C \dfrac{1}{z^2} \mathrm{d}z, \quad \quad C: \left\{ \begin{array}{ll}
z(t) = e^{it} \\
0 \leq t \leq 2\pi
\end{array} \right.$
(c) $\displaystyle\int_C \dfrac{z+2}{z} \mathrm{d}z, \quad \quad C: \left\{ \begin{array}{ll}
z(t)=1+e^{it} \\
0 \leq t \leq \pi
\end{array} \right.$
(d) $\displaystyle\int_C z^2+2z \mathrm{d}z, \quad \quad C: \left\{ \begin{array}{ll}
z(t)=e^{it} \\
0 \leq t \leq 2\pi \end{array} \right.$
3. Let $C$ be the boundary of the square with vertices at the points $0$, $1$, $1+i$, and $i$ oriented counterclockwise. Calculate $\displaystyle\int_C \pi \exp(\pi \overline{z}) \mathrm{d}z$. (hint: break $C$ into four parts, parametrize each separately, and add up in the integrals over each part)
4. Let $C$ be the unit circle. Calculate $\displaystyle\int_C z^n \mathrm{d}z$, when $n \neq -1$. (hint: parametrize $C$ in the "usual way")
