MAT334-2018F > Quiz-3

Q3 TUT 0101

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Victor Ivrii:
Directly compute the following line integral:
$$
\int_\gamma \frac{dz}{z+4},
$$
where $\gamma$ is the circle of radius $1$ centered at $-4$, oriented counterclockwise. Draw the picture.

Meng Wu:
Let $$\gamma(t)=p+Re^{it}=-4+e^{it}, \text{ where } 0\leq t \leq 2\pi.$$
$$f(z)=\frac{1}{z+4}$$
Thus $$\gamma'(t)=ie^{it}$$
$$\begin{align}\int_\gamma f(z)dz&=\int_{0}^{2\pi}f(\gamma(t))\gamma'(t)dt\\&=\int_{0}^{2\pi}\frac{1}{-4+e^{it}+4}(ie^{it})dt\\&=\int_{0}^{2\pi}{e^{-it}}(ie^{it})dt\\&=\int_{0}^{2\pi}idt\\&=it\Big|_0^{2\pi}\\&=2\pi i\end{align}$$

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