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Cauchy Reimann & Complex functions

  1. Aug 14, 2011 #1
    I am a mechanical engineer who hasn't done any mechanical engineering for close to 20 years & hence forgotten all Mechanical Engineering & all Engineering Mathematics.

    I need to revise on some Engineering Math now - Calculus, Laplace Transforms etc. I have been doing it for a couple of days.

    I am getting stuck in a discussion of Complex Variables & Complex functions in Laplace Transforms.

    G(s) = 1/(s+1)

    We are trying to check if this satisfies Cachy-Reimann & hence analytic.

    This is how the analysis goes in the notes I am referring to

    G(σ + jω) = 1/(σ + jω + 1) = Gx + jGy ---------- (1)

    From this, the next line says
    "where

    Gx = (σ + 1)/ ( (σ + 1)2 + ω2)
    Gy = (-ω)/( (σ + 1)2 + ω2)
    "

    I don't see how you get Gx & Gy from (1)

    Can someone help?

    If this is based on some other Math stuff which I need to study before getting here, do let me know.
     
  2. jcsd
  3. Aug 15, 2011 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    You have
    [tex]\frac{1}{\sigma+ 1+ j\omega}[/tex]
    and, essentially, you want to "rationalize the denominator".

    Multiply both numerator and denominator of that fraction by [itex]\sigma+ 1- j\omega[/itex], the "complex conjugate" of the denominator:
    [tex]\frac{1}{\sigma+ 1+ j\omega}\frac{\sigma+ 1- j\omega}{\sigma+ 1- j\omega}[/tex]
    In the numerator, we will have, of course, [itex]\sigma+ 1- j\omega[/itex]. In the denominator, we have a product of "sum and difference" which is the "difference of the squares"- [itex](\sigma+ 1+ j\omega)(\sigma+ 1- j\omega)= (\sigma+ 1)^2- (j\omega)^2[/itex][itex]= (\sigma+ 1)^2+ \omega^2[/itex] since [itex]j^2= -1[/itex]

    That is, the fraction reduces to
    [tex]\frac{\sigma+ 1- j\omega}{(\sigma+1)^2+ \omega^2}[/tex]
    Gx and Gy are the real and imaginary parts of that.
     
  4. Aug 16, 2011 #3
    Thank you very much. I was stuck on that for quite some time.
     
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