# Laurent series of the function f(z)=1/z^2 for |z-a|>|a|

heman
Hi all,,

I have an Exam tommorow and this question is irritating me...Pls help

Laurent series of the function f(z)=1/z^2 for |z-a|>|a| .a is not equal to zero...
I am waiting for yours responses...I will be highly thankful to you.

## Answers and Replies

Staff Emeritus
Gold Member
Don't you have a formula for the coefficients, that involves integrating over the annulus |z - a| > |a|?

Or, maybe you could look at it as:

|1 / (z-a)| < 1/|a|

And try to rewrite 1/z^2 so it looks like the sum of a geometric series in 1/(z-a)?

heman
Omg,,,I did not use the formula for the coefficients,it did not seemed to strike me,,,yeah then its easy...and can be solved...

I was trying to do it by bringing in the form of geometric series..
Hurkyl,
Pls tell me how can i write 1/z^2 in that apt form as here the power of z is 2....
I seem to moving forward but then 2 series will be multiplied acc. to me,won't it be tedious..?

Staff Emeritus
Gold Member
Hrm, yah, writing it as a geometric series won't work... but maybe the sum of two geometric series?

I guess it would be easier to observe that d/dz (-1/z) = 1/z^2, so if you had a Laurent series for -1/z...

heman
Oh yes,,,i got it ....That will work....Thanks Hurkyl ,I feel much better now!!

But in writing the Laurentz Series for F(z)=exp(z+1/z) around zero ,i think i have got no escape ...i think i am bound to multiply the series for e^z and e^(1/z),,,But how will i write the coefficient for each term,,or there can be any better approach...

Staff Emeritus
Gold Member
I don't see the difficulty in multiplying the series... I guess you don't like the fact that each coefficient will be an infinite sum. (Though, it wouldn't surprise me if there's a clever way to do this that gives you a closed form)

heman
Hurkyl said:
I don't see the difficulty in multiplying the series... I guess you don't like the fact that each coefficient will be an infinite sum. (Though, it wouldn't surprise me if there's a clever way to do this that gives you a closed form)

"Each coefficient will be an infinite sum."....that i can see but where it will start that seems out of my reach.....limits are worrying me...i am able to see it mechanically only...
I think i sincerely need help on this!!

Staff Emeritus