Understanding Complex Func., Laplace Transforms & Cauchy Riemann

[phiby]

I am reading a chapter on Complex Functions, Laplace Transforms & Cauchy Riemann (as part of Control theory)

And I don't understand how they arrive at a particular part.
[ I tried to type it out in tex, but it takes way too much time so uploaded a screenshot to flickr]

[PLAIN]http://www.flickr.com/photos/66943862@N06/6093176535/

Here is a http://www.flickr.com/photos/66943862@N06/6093176535/"

I understand how you get to Eqn1 & Eqn2.
But how does it add up to Equation3?

Can someone explain?

Also, I don't understand why it's not analytic at s = -1?

 

[micromass]

Hi phiby!

\begin{eqnarray*}
\frac{\partial G_x}{\partial \sigma}+j\frac{\partial G_y}{\partial \sigma}
& = & \frac{\omega^2-(\sigma+1)^2+2j\omega(\sigma+1)}{[(\sigma+1)^2+\omega^2]^2}\\
& = & \frac{\omega^2+2j\omega(\sigma+1)+j^2(\sigma+1)^2}{[\omega^2-j^2(\sigma+1)^2]^2}\\
& = & \frac{[\omega+j(\sigma+1)]^2}{[(\omega-j(\sigma+1))(\omega+j(\sigma+1))]^2}\\
& = & \frac{1}{(\omega-j(\sigma+1))^2}\\
& = & \frac{1}{(-j)^2(\sigma+1+j\omega)^2}\\
& = & -\frac{1}{\sigma+j\omega+1}
\end{eqnarray*}

The function G is not analytic in -1 since it doesn't exist there. Indeed, G(-1) is undefined and is a pole. (so it's not even a removable singularity)

 

[phiby]

Hi phiby!
(snip solution)

Awesome. Thanks a lot. I got what you did (the simplification of the equation), but didn't get how you knew you had to do that to simplify the original stuff.

I studied a lot of engineering math 20 years ago & I am getting back to it after 20 years (almost did none of this in the 20 years). So it's taking me a little time to get this.

In the original page (my flickr link), I first didn't get how it was separated into Gx & Gy, so I went back & did a review of partial fractions & then it became simple.

So my question is - what part of math should I review to do what you did above?

 

[micromass]

Awesome. Thanks a lot. I got what you did (the simplification of the equation), but didn't get how you knew you had to do that to simplify the original stuff.

I studied a lot of engineering math 20 years ago & I am getting back to it after 20 years (almost did none of this in the 20 years). So it's taking me a little time to get this.

In the original page (my flickr link), I first didn't get how it was separated into Gx & Gy, so I went back & did a review of partial fractions & then it became simple.

So my question is - what part of math should I review to do what you did above?

Well, the things I used where the equations

$$(a+b)^2=a^2+2ab+b^2$$

and

$$(a+b)(a-b)=a^2-b^2$$

If you know these very well, then you can find the above solution.