# Laplace transform of and impulse sampled function using “frequency” convolution

This is a long question, but assume we have this:

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:

Now, the book says if you evaluation the path on the left side, then:

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

For the right side, the book says:

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:

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

Lastly, for the second case:

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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