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

[tex](a+b)^2=a^2+2ab+b^2[/tex]

and

[tex](a+b)(a-b)=a^2-b^2[/tex]

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

 

[CellCoree]

Hello,

I understand your confusion with the equations and how they lead to Equation 3. Let me explain it to you in a simplified manner.

Firstly, a complex function is a function that takes complex numbers as inputs and outputs complex numbers. In simpler terms, it is a function that involves both real and imaginary numbers.

Laplace transforms are mathematical tools used to solve differential equations, particularly in control theory. They involve transforming a function from the time domain to the frequency domain, making it easier to analyze and solve.

Now, in the context of control theory, we are dealing with a specific type of complex function called a transfer function. This function represents the relationship between the input and output of a control system.

In the equations shown in the screenshot, we are dealing with a transfer function of the form H(s), where s is a complex variable. Equation 1 and 2 represent the real and imaginary parts of this transfer function, respectively. These equations are derived using the Cauchy-Riemann equations, which are a set of conditions that must be satisfied for a complex function to be analytic (meaning it can be represented as a power series).

Equation 3 is obtained by combining the real and imaginary parts of the transfer function, which is a common practice in complex analysis. This equation represents the magnitude and phase of the transfer function, which are important in control theory.

As for why the transfer function is not analytic at s = -1, this is because the Cauchy-Riemann equations are not satisfied at that point. This means that the function cannot be represented as a power series at that point, making it non-analytic.

I hope this explanation helps you understand the concepts better. If you have any further questions, please feel free to ask.