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Largest set on which the function is analytic

  1. Oct 7, 2015 #1
    1. The problem statement, all variables and given/known data
    Determine the largest set on which the function is analytic.
    f(z) = (z2-2)e-xe-iy

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

    3. 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
     
  2. jcsd
  3. Oct 7, 2015 #2

    andrewkirk

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    Gold Member

    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##
     
  4. Oct 7, 2015 #3
    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
     
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