# Frequency Domain Analysis - the math

I am studying Control Theory from Ogata. My math is a little rusty, so this is a math question about Frequency Domain Analysis.

I get everything upto Equation 8-4

However I don't get the line after that.

The line "where the constant a can be evaluated from Equation (8-2) as follows"

And they write the value of a.

How is
a = G(s) (ωX) /(s^2 + ω^2).

How do they arrive at this value of a?

Can someone help?

And why did they substitute s = -jω after that?

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here is another way

http://img41.imageshack.us/content_round.php?page=done&l=img41/4643/12112011130.jpg [Broken]

i put s = - jw to get value of a, if you put s = jw you will get the value of a bar

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here is another way

http://img41.imageshack.us/content_round.php?page=done&l=img41/4643/12112011130.jpg [Broken]

i put s = - jw to get value of a, if you put s = jw you will get the value of a bar
Thank you. That's a little clearer.

You get values of a & abar by evaluating the limit as s = -jω & s = jω

However, I still don't get why you evaluate these limits?

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Its simply a mathematical way of obtaining the coefficients. You can simply factorize the denominator into complex factors and proceed to obtain the inverse laplace transform.