You would see this as well with the Fourier Transform if you did the transform of $u(t)cos(\omega t)$ specifically. The calculator will try to find the Inverse Laplace transform of the given function. This explains why the Fourier Transform of a sine wave would appear as two impulses at $\pm \omega_c$ while as you might deduce from the plot above, the Laplace Transform of the sine wave would appear more as tent poles on the $j\omega$ axis. This is due to the use of one side of the Laplace transform online (normal side) and neglect to use the inverse Laplace transform side. The one precaution is that the Fourier Transform is often given as a bilateral function (t extending from $-\infty$ to $\infty$) so to be truly equivalent unless the function is declared to be causal, we must be using the bilateral Laplace Transform for the two to be exactly identical (which is also seldom used). Apart from this, the Laplace transform calculator with solutions can only calculate normal Laplace transform which is a process known as unilateral Laplace transform. In the inverse laplace transform, we have these and then a low class transform. The second shifting there is the 1st one. We are going to use the rules that we had at the very top. You can see this if you compare the two equations, and the small breakout in upper right-hand corner of the plot above is also showing this, which is the Frequency Response specifically. Plus transforms s over s squared plus one plus universal fast transform of one over s squared Theres plus one. However what is very useful is knowing that the Fourier Transform is the Laplace Transform when $s = j\omega$. Hopefully after reading this the OP will no longer feel the need to plot the Laplace Transform, and in practical application a plot of it is never used beyond showing the pole and zero locations. $$X(s) = \int_0^\infty x(t)e^$ continuously grows larger for larger $t$. Is just the Fourier Transform of a causal function with a weighting exponential: Observe that the unilateral Laplace Transform given as: Given the approach started in the OP's Github code I have this suggestion: The inverse laplace transform calculator works online.
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