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This is a long question, but assume we have this: enter image description here

The book uses the frequency convolution theorem to solve this problem. To solve the integral, it uses a contour + residue theorem to solve it. The only problem is you can form this path to the left or right: enter image description here

Now, the book says if you evaluation the path on the left side, then: enter image description here

My first question is why does the integral along TL vanish for this side?

For the right side, the book says: enter image description here

My second question is why do we have to consider two cases for this side?

It seems as if we can apply the logic for the first case here and assume the integral along TL vanishes. Anyway, for the first case the book says: enter image description here

My third question is why do they use the initial value theorem to justify the path integral is zero?

Lastly, for the second case: enter image description here

My fourth question is how do they obtain that the path integral along TL for this case is $-1/2*x(0+)$?

My thanks to anyone who took the time to read all this! Furthermore if there seems to be a consistent misunderstanding of some basic concept it would be great to get resources to read on this subject.

EDIT: Btw, this is taken from a text book called Discrete Time Control Systems by Ogata.

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Proof of sampling theorems. No need to use Dirac comb. Page 1

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enter image description here

enter image description here

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This is barely readable. Please type the text and equations out; if you need help with the equations just try your best with MathJax, other members will gladly improve on that. – AlexR Apr 14 '14 at 23:27
Wow... I will need some time to look over this- but thank you so much for posting! – user1346994 Apr 15 '14 at 2:55

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