Does Bilateral Laplace Transform Exist for Sin(t) ?

Published 2018-12-02

This had been a question which was bothering me. So here is my question posted on DSP, Stackexchange.

When we were learning Fourier Transforms, we easily derived the fourier transform of sin(t) using it’s Fourier Series coefficients. But it appeared that the Laplace transform (bilateral) didn’t converge. So then I computed the laplace transform for sin(t)*u(t).

(in other words, computing the Unilateral Laplace Transform of sin(t) )

Transform expressions

For the two-sided sine wave:

Its bilateral Laplace transform has no common region of convergence. For the causal signal , the Laplace transform is:

Using the corrected expression in my StackExchange update, the causal signal’s Fourier transform is:

Here, PV denotes the Cauchy principal value. Why doesn’t directly substituting into the Laplace expression recover this Fourier transform?

Mathematica notebook

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How can it happen? Here’s the expert answer..

Figure 1:

References:

Unilateral and bilateral Laplace transforms (Mathematics StackExchange)

Fourier transforms and the Laplace region of convergence (Signal Processing StackExchange)