Week 5 Worksheet

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Winding Numbers

1. Calculate the winding number of the contour below around all points not on the path.

   

Cauchy's Theorem

2. Let ${\gamma }_1(t)=-1+\frac{1}{2}{e}^{it}$ on $[0,2\pi ]$, let ${\gamma }_2(t)=1+\frac{1}{2}{e}^{it}$ on $[0,2\pi ]$ and let $\gamma (t)=2e^{-it}$ on $[0,2\pi ]$. Let $f(z)=1/(z^2-1)$. Use Cauchy Theorem to deduce that $$-\underset{\gamma }{\int }f\mathsf{d}z=\underset{{\gamma }_1}{\int }f\mathsf{d}z+\underset{{\gamma }_2}{\int }f\mathsf{d}z$$

3. What is the contour integral of $f(z)=1/z$ over the contour $\gamma$ shown below?

4. Fix distinct complex numbers $z_1$ and $z_2$ and contours $\gamma ,{\gamma }_1,{\gamma }_2$ as in the figure. Suppose that $f:\mathbb{C}\setminus \{z_1,z_2\}\to \mathbb{C}$ is holomorphic.

   

   If ${\displaystyle \underset{{\gamma }_1}{\int }f=3+4i}$ and ${\displaystyle \underset{{\gamma }_2}{\int }f=5+6i}$ what is the value of ${\displaystyle \underset{\gamma }{\int }f}$?

5. Verify that $$f(z)=\{\begin{array}{ll}{\displaystyle \frac{\cos (z)-1}{z}}& z\ne 0\\ 0& z=0\end{array}$$ is holomorphic on $\mathbb{C}$. Then evaluate $$\underset{\gamma }{\int }{\displaystyle \frac{\cos (z)}{z}}\mathsf{d}z$$ where $\gamma (t)=e^{it}$ on $[0,2\pi ]$.