There is a solution to the following problem that i don't understand. Find the antiderivative of $f(z)=1/(1+x^2)$ on $|z|<1$ and on $|z|>1$.
The problem is solved as follow:
First note that $(1+x^2) = (x+i)(x-i)$
Hence we can write $1\(z+i)=(1/i)\sum_{n=0}^{\infty}((-1)^nz^n)/i^n$
And we can also write $1\(z-i)=(-1/i)\sum_{n=0}^{\infty}(z^n)/i^n$
So far so good, but then the following claim is made:
Then $1/(1+x^2)=(1/2)\sum_{n=0}^{\infty}(1+(-1)^n)z^n/i^n$
How on earth is this true? Why have we just added the terms together? Should this not be the cauchy product of the two series?
The last expression is then simplified and integrated to give an antiderivative.
Why was this necessary to find an antiderivative? Could we not just have centered a taylor expansion at the origin for the original function f and integrated that(since we have analyticity on the unit circle)?
For the second part (find an antiderivative outside the unit circle) the following claim is made:
Here we can write $1\(z+i)=(1/z)\sum_{n=0}^{\infty}((-i)^n)/z^n$
And we can also write $1\(z-i)=(-1/z)\sum_{n=0}^{\infty}i^n/z^n$
The sums are then combined and integrated.
Why are these two laurent series now different from the previous part? Also, i thought that a laurent series was a way to represent a function analytic in some annalus around a singularity, since in this case, a taylor expansion is not well defined at the singularity. What is the annulus in this case, since we have two singularities? Which one are we centering around?
Finally, why does the problem exclude trying to find an antiderivative on the unit circle? Is it not well defined there? if so, why?
I am quite confused about this, despite reading over my textbook several times. Any insight into this solution would be great.
Hints. For the first part, note that $$\frac{1}{z+i}-\frac{1}{z-i}=\frac{(z-i)-(z+i)}{(z-i)(z+i)}=\frac{-2i}{z^2+1}.$$ As regards the second part, recall that the basic properties of convergence of the geometric series: $$\sum_{n=0}^{\infty}w^n=\frac{1}{1-w}$$ if and only $|w|<1$ where $w\in\mathbb{C}$.
In your case, what is $w$ when $|z|<1$? What is $w$ when $|z|>1$?