Cauchy Reimann & Complex functions

[phiby]

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) = G_x + jG_y ---------- (1)

From this, the next line says
"where

G_x = (σ + 1)/ ( (σ + 1)² + ω²)
G_y = (-ω)/( (σ + 1)² + ω²)
"

I don't see how you get G_x & G_y 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.

 

[HallsofIvy]

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

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

That is, the fraction reduces to
$$\frac{\sigma+ 1- j\omega}{(\sigma+1)^2+ \omega^2}$$
Gx and Gy are the real and imaginary parts of that.

 

[phiby]

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

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