Toronto Math Forum

MAT334-2018F => MAT334--Tests => Quiz-3 => Topic started by: Victor Ivrii on October 12, 2018, 06:12:36 PM

Title: Q3 TUT 0101
Post by: Victor Ivrii on October 12, 2018, 06:12:36 PM

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.

Title: Re: Q3 TUT 0101
Post by: Meng Wu on October 12, 2018, 06:23:13 PM

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