If I have a complex integral to solve using the Cauchy Integral formula with the same point but with different contours, in which the point used is inside both contours, is the result the same?
Say for example that I have $$ \int_C \frac{dz}{(z+i)(z-i)} $$
with two different contours:
- $\|z-i\|=1$
- A square of size 2 centered in $z=i$
In this case, one contour is a circunference and the other is a square, both centered in $z=i$, which I use when applying the Cauchy Integral formula.
Is it right that both will result in $\pi$ as the result? Or is there a difference when you use different contours?
It depends on the poles enclosed by each contour.