1. ## Cauchy's integral formula

$\displaystyle \int_{C}\frac{f(z)}{z-z0}dz = 2\pi.i.f(z0)$

Im using cauchy's integral formula, see above to solve the following integral

$\displaystyle \int_{C}\frac{z}{z^{2}-1}dz$ where $\displaystyle C$ is a circle of radius 4 centred at 0.

I have deduced that the function $\displaystyle \frac{z}{z^{2}-1}$ is not analytic at $\displaystyle z = +i$ $\displaystyle z = -i$

I let $\displaystyle f(z) = \frac{z}{z+i}$ and $\displaystyle z0 = -i$

so filling into cauchy's integral formula i got

$\displaystyle \int_{C}\frac{z}{(z+i)(z-i)}dz = \frac{f(z)}{z-i} = 2\pi.i.f(i)$

Answer = $\displaystyle \pi.i$

My problem lies in the fact that both $\displaystyle z = +i$ $\displaystyle z = -i$ both lie in $\displaystyle C$. Therefore do i have to take both singularites into account when evaluating the integral using cauchy's integral formula???

Thanks,
Piglet

2. Originally Posted by piglet
$\displaystyle \int_{C}\frac{f(z)}{z-z0}dz = 2\pi.i.f(z0)$

Im using cauchy's integral formula, see above to solve the following integral

$\displaystyle \int_{C}\frac{z}{z^{2}-1}dz$ where $\displaystyle C$ is a circle of radius 4 centred at 0.

I have deduced that the function $\displaystyle \frac{z}{z^{2}-1}$ is not analytic at $\displaystyle z = +i$ $\displaystyle z = -i$

I let $\displaystyle f(z) = \frac{z}{z+i}$ and $\displaystyle z0 = -i$

so filling into cauchy's integral formula i got

$\displaystyle \int_{C}\frac{z}{(z+i)(z-i)}dz = \frac{f(z)}{z-i} = 2\pi.i.f(i)$

Answer = $\displaystyle \pi.i$

My problem lies in the fact that both $\displaystyle z = +i$ $\displaystyle z = -i$ both lie in $\displaystyle C$. Therefore do i have to take both singularites into account when evaluating the integral using cauchy's integral formula???

Thanks,
Piglet
Yes. See example 30.2 here: http://math.furman.edu/~dcs/courses/...lecture-30.pdf

3. $\displaystyle \frac{z}{z^{2}-1} \: = \:\frac{1}{2(z-1)} \: + \frac{1}{2(z+1)} \:$