How Do You Expand Power Series for Complex Functions?

[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

 

[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.

 

[HallsofIvy]

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

 

[g_edgar]

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

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.

 

[HallsofIvy]

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.

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

 

[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?