# Even function has a Laurent decomposition of even functions and even powers of z

## Main Question or Discussion Point

Hi,

if let's say that there's an even function f(z) then how do we know if its Laurent decomposition (i.e. f(z) = f0(z) + f1(z) ) will be even functions and have even powers of z?

Any help will be greatly appreciated.

Hurkyl
Staff Emeritus
Gold Member
Take a generic Laurent series and compute it's even and odd parts. What do you see?

(You know a formula for the even and odd part of any function, right?)

i know that for even function f(z) = f(-z) which means that the sum from n=0 to infinity of a_n z^n is equals to the sum from n=0 to infinity of a_n (-z)^n.
(note: a_n is the coefficient of the series)

Then this will give me an equation of a_n = a_n (-1)^n ---> conclusion: a_n can't be equal to 0, am I right?

But then after this I don't know what else to do in order to prove that f0(z) and f1(z) are even functions that only has powers of z.

Thank you so much

Must be Laurent series in powers of $z$.