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Sketch the domain where it is analytic

  1. Sep 2, 2009 #1
    1. The problem statement, all variables and given/known data
    determine if

    a) f(z) = e^z / (z^2 + 4)

    b) f(z) = conj(z) / |z|^2

    c) f(z) = sum from 0 to inf. [ (e^z / 3^n) * (2z - 4)^n ]


    is analytic and sketch the domain where it is analytic.

    2. Relevant equations



    3. The attempt at a solution

    a) i don't know how to separate the function into a real and imaginary part. I have a feeling that the denominator needs to be manipulated but I have no clue how.

    f(z) = e^(x + yi) / ((x+yi)^2 + 4)

    f(z) = (e^x * cosy) / ((x+yi)^2 + 4) + (ie^x * siny) / ((x+yi)^2 + 4)

    (e^x * cosy)* ((x-yi)^2 + 4) / ((x+yi)^2 + 4)((x-yi)^2 + 4)


    I'm guessing it is analytic everywhere except

    z^2 = -4

    b) f(z) = conj(z) / |z|^2

    f(z) = 1 / z

    1/z = 1 / ( x+yi)

    (x - yi) / (x^2 + y ^2)

    = x / (x^2 + y ^2) - yi / (x^2 + y ^2)

    since du/dx u(x,y) = dv/dy u(x,y)

    and dv/dx u(x,y) = - dv/dx u(x,y)

    the function is analytic everywhere except at the origin.

    c) I used the ratio test and got lim n-> inf |1/3 (2z - 4)| = |1/3 (2z - 4)|

    it's analytic on 0 < |2z - 4| < 3 ?
     
    Last edited: Sep 2, 2009
  2. jcsd
  3. Sep 2, 2009 #2
    Re: analytic

    some hint to start:
    e^z is analytic.

    by theroem: A rational function (the quotient of two polynomials) is analytic, except at zeroes of the denominator.

    So take care of Z^2 + 4 - > find the values at which the term become zero.
    Sketch it on the complex plane.

    By the way, should be bounded by some region when u say if you would like to take integral... if not you would know if the whole function is analytic anot. In my opinion, z^2+4 will fail to be analytic when it is set to zero.

    hence cauchy integral formula should be used.

    hope it helps
     
  4. Sep 3, 2009 #3
    Re: analytic

    helps a bit, yet. Im having trouble transforming the original equation from part a) into f(z) = u(x,y) + i v(x,y) before the cauchy equations can be used.
     
  5. Sep 3, 2009 #4
    Re: analytic

    (1) replace z = x + jy.
    (2) expand and simplify.

    :)

    You'll get it. It may gets abit complicated. Be careful on the steps.
     
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