Complex Analysis: prove the function is entire

[dh363]

Hey guys, i just started a complex analysis course this semester and we just went over CR-equations, and various ways to show that a function is holomorphic. I'm a bit stuck on this one homework question where we have to prove the function is entire.

Homework Statement

so we have f(z)=(z^2-2)(e^-x)(e^-iy)

The Attempt at a Solution

First I tried expanding the e^-iy out to cos(-y)+isin(-y) and tried to multiply it out to get a u and a v. That required a lot of tedious algebraic manipulation for one of the earlier problems in the pset for this prof so then I tried looking for a more elegant solution. I turned (e^-x)(e^-iy) into e^(-z), and tried to show that the partial derivative with respect too the conjugate of z = 0, but I'm still not to clear on that whole method. Was wondering if anyone can point me along the right direction/explain something maybe I don't understand.

Thanks

 

[lanedance]

I think multplying out as you started with is not a bad way to approach this

 

[dh363]

Hmmm, maybe I did not put enough work in that direction before trying other stuff. I'll work on it and get back.

EDIT: I was wondering also, for the criteria that partial(f)/partial(zbar)=0, is zbar treated like any other varaible? In other words, since I can express the first function entirely in terms of z, can I say that partial(f)/partial(zbar) = 0 and so say that they are holomorphic?

 

[Dick]

Hmmm, maybe I did not put enough work in that direction before trying other stuff. I'll work on it and get back.

Sure, your function is e^(-z)*(z^2-2). It seems a little painful to show CR applies directly. You don't know things like the product of analytic functions is analytic??

 

[dh363]

Sure, your function is e^(-z)*(z^2-2). It seems a little painful to show CR applies directly. You don't know things like the product of analytic functions is analytic??

mmm, i think this method and the partial(f)/partial(zbar)=0 method work to make this problem very easy. We never proved product of analytic functions is analytic, maybe he expected us to know this from analysis I. I skipped it because I wouldn't have otherwise been able to fit complex analysis into my 4 years here :( Been keeping up fine teaching myself stuff I'm not familiar with though.

lol I am an idiot, I wasn't understanding the conjugate of z equation properly. Looked back at the derivation and i get it now.