- #1

- 158

- 0

[tex]\int_{\gamma}\frac{dz}{z^2+1}[/tex]

Around a circle of radius 2 centered at the origin oriented counterclockwise.

[tex]\frac{i}{2}\left[\int_0^{2\pi}\frac{1}{z+i}dz-\int_0^{2\pi}\frac{1}{z-i}dz\right][/tex]

[tex]\gamma(t)=2e^{it}, \quad \gamma'(t)=2ie^{it}[/tex]

The answer is zero. I am supposed to get each integral to be [itex]2\pi i[/itex] which is 0 when subtracted.

I know it is related to the fact that [itex]\int_{\gamma}\frac{1}{z}dz = 2\pi i[/itex].

And using u-sub isn't correct since any closed path would be zero when that isnt true. The integral of 1/z shows that not all closed paths will be zero.