Mumbai University > COMPS > Sem 4 > Applied Mathematics 4

Marks : 05

Year : MAY 2015

$|z|=1$ is unit circle with centre $(0, 0)$ & radius=1 $$Let\space \space I=\int\log z\space dz$$

Put $z=re^{i\theta}=1e^{i\theta}=e^{i\theta}$

$$ \therefore dz=e^{i\theta}.id\theta$$

$ \therefore I=\int\limits_0^{2\pi}\log(e^{i\theta}).e^{i\theta}.id\theta\\ =\int\limits_0^{2\pi}i\theta\log e.e^{i\theta}.id\theta\\ =i^2[\theta.\dfrac {e^{i\theta}}1-1.\dfrac {e^{i\theta}}{i^2}]_0^{2\pi}\\ =[i\theta e^{i\theta}-e^{i\theta}]_0^{2\pi}\\ =[e^{i\theta}(i\theta-1)]_0^{2\pi}\\ =e^{i2\pi} (i.2\pi-1)-e^0((i.0-1)\\ =(\cos2\pi+\sin2\pi)(2\pi-1)+1\\ =(1+0)(2\pi i-1)+1\\ =2\pi i-1+1\\ =2\pi i$