# Expansion of power series

1. Jul 25, 2009

### caramello

Hi,

I have 2 questions regarding how to expand power series.

1). Find the power series expansion of Log z about the point z= i - 2

2). Expand the function 1/(z^2 + 1) in power series about infinity

Any help will be greatly appreciated. This is because I am totally unsure about what to do when they asked for an expansion of complex function or power series. And if possible, can you show me a somewhat detailed step by step explanation? I'm really sorry for the trouble. This is because I'm really clueless on how to even start.

Thank you so much

2. Jul 25, 2009

### John Creighto

Do it the same way as you would if z wasn't complex. Just watch your algebra with regards to any simplifications you might make.

3. Jul 25, 2009

### HallsofIvy

Staff Emeritus
For the second one, write 1/(z^2+ 1) as
$$\frac{1}{1- (iz)^2}$$
and express it as a geometric series.

4. Jul 25, 2009

### g_edgar

I would do that to expand at zero. But at infinity, probably I would do
$$\frac{1}{z^2+ 1} = \frac{1}{z^2}\left(\frac{1}{1+(1/z^2)}\right)$$,
then expand as a geometric series.

5. Jul 25, 2009

### HallsofIvy

Staff Emeritus
Absolutely right. I did not see that "about infinity". Thanks.

6. Jul 25, 2009

### caramello

thank you so much for all of your help! :) i really appreciate that..

Does anyone of you know how to do number 1 though?