Author Topic: Quiz3 Lec0101_B  (Read 281 times)

duoyizhang

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Quiz3 Lec0101_B
« on: October 09, 2020, 12:01:57 PM »
The question is :Compute the following line integral:

$\int_\gamma(z^2+3z +4) dz$,

where $\gamma$ is the circle ${|z|=2}$,oriented counterclockwise.

Here is my solution:

f(z)=$(z^2+3z +4)$,

$\gamma(t)=2e^{it}$ ,$0 \leq t \leq 2\pi$

${\gamma(t)}'$=$2ie^{it}$

$f(\gamma(t))=4e^{2it}+6e^{it}+4$

$\int_\gamma f(z) dz$=$\int_{0}^{2\pi} f(\gamma(t))\gamma(t)'dt$

                     =$\int_{0}^{2\pi} (4e^{2it}+6e^{it}+4)2ie^{it}dt$
                     
                     =2i$\int_{0}^{2\pi} (4e^{3it}+6e^{2it}+4e^{it})dt$
                     
                     =$2i(\frac{4}{3i}e^{3it}+\frac{6}{2i}e^{2it}+4e^{it})|_{0}^{2\pi}$
                     
                     =0
                     
Since $e^{it}|_{0}^{2\pi}$=$cos(2\pi)+isin(2\pi)-cos(0)-isin(0)$
                         
                          =0
                         
Similarly, $e^{3it}|_{0}^{2\pi}$= $e^{2it}|_{0}^{2\pi}$=0
« Last Edit: October 09, 2020, 12:47:59 PM by duoyizhang »