$\gamma_j(t)=(-1)^j+\frac{1}{2}\exp(it)$ for $j=1,2$ $\int_{\gamma_1}f(z) \mathrm{d}z + \int_{\gamma_2}f(z) \mathrm{d}z$ on $[0,2\pi]$

Question

$f(z)=\frac{1}{z+1}+\frac{1}{z-1}$

$\gamma_j:[0,2\pi]\rightarrow \mathbb{C} \ (j=1,2,3)$

$\gamma_j(t)=(-1)^j+\frac{1}{2}\exp(it)$ for $j=1,2$

$\gamma_3(t)=4\exp(it)$

I need to compute

$\int_{\gamma_1}f(z) \mathrm{d}z + \int_{\gamma_2}f(z) \mathrm{d}z$ and $\int_{\gamma_3}f(z) \mathrm{d}z$.

Are there any better ways to compute these than using

$$\int_{0}^{2\pi}f(\gamma(t))\gamma'(t) \mathrm{d}t$$?

Answer 1

You only have to know that $$\int_\gamma{dz\over z-a}=2\pi i\> n(\gamma,a)$$ for any $a\in{\mathbb C}$ and any closed curve $\gamma\subset{\mathbb C}\setminus\{a\}$. Here $n(\gamma, a)$ denotes the number of times the curve $\gamma$ winds around the point $a$.

It is not necessary to compute the line integrals "the hard way". This has been done for this special case once and for all in class.