Largest set on which the function is analytic

[Jon.G]

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)
U_x=V_y
U_y=-V_x

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^-xe^iy
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

 

[andrewkirk]

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 $\sin\theta$

 

[Jon.G]

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.

Thanks for your time