MAT334--2020S > Quiz 3

Q3: TUT 0201

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Ziyi Wang:
Question: 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.
Answer: Let $\gamma (t) = -4 + e^{it}$, where $t \in [0, 2\pi]$.
Then, $\gamma ' (t) = ie ^{it}$.
Then, we compute the line integral:
$$\int _\gamma \frac{dz}{z+4} = \int _0^{2\pi}\frac{ie ^{it} dt}{-4 + e^{it} + 4}$$
$$= \int _0^{2\pi}\frac{ie ^{it}}{ e^{it}}dt$$
$$= \int _0^{2\pi}i dt$$
$$= it \Big|_{t = 0}^{t = 2\pi}$$
$$= 2\pi i$$