# Largest set on which the function is analytic

## Homework Statement

Determine the largest set on which the function is analytic.
f(z) = (z2-2)e-xe-iy

## Homework Equations

z=x+iy
f(x+iy) = U(x,y) + iV(x,y)
Ux=Vy
Uy=-Vx

## The Attempt at a Solution

I think I'm right in saying that f(z) is analytic if the CR equations (provided above) are satisfied.
So I would write f(z) as f(x+iy) to find U and V.

f(x+iy) = (x2-y2+i2xy-2)e-xeiy
This is where I am up to. I can't seem to think of how to split this into parts with and without i. (The exponential with i is what throws me).

Any advice on where to go next?

Thanks

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andrewkirk
Homework Helper
Gold Member
The exponential with i is what throws me
Your text or notes should have given you a formula for ##e^{i\theta}##. If not, look up Euler's Formula. Or better still, expand the Taylor series for ##e^{i\theta}## and compare it to the Taylor series for ##\cos\theta## and ##\sin\theta##

oh wow I actually can't believe I didn't see that.
I am familiar with Euler's Formula and don't know why I didn't think of it in this situation.
When I'm home I'll try using that and then post how I get on.