Cauchy Reimann & Complex functions

  • Thread starter phiby
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  • #1
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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.
 

Answers and Replies

  • #2
HallsofIvy
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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.
 
  • #3
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You have
[tex]\frac{1}{\sigma+ 1+ j\omega}[/tex]
and, essentially, you want to "rationalize the denominator".

Thank you very much. I was stuck on that for quite some time.
 

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