Could someone show me explicitly what is going in the first line? I am confused as I have learnt Laplace transformations but I cannot figure out which Laplace transformation this is.. and the (a) in the first line, is that variable of the Laplace?

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1 Answer 1


The Laplace transform of a function $f(t)$ is defined as

$$\mathcal{L}\{f\}(a)=\int_0^\infty f(t)e^{-at}\,dt$$

where $a$ denotes the parameter of the integral (i.e. $a$ appears in the evaluation).

In your case, for the first line, they set $f(t)=\sin(bt)/t$.

For the second equality, the frequency domain integration property was used.

  • $\begingroup$ what is the frequency domain integrration ppty? $\endgroup$ Commented Sep 24, 2020 at 1:59
  • $\begingroup$ You can find a proof here and is listed as a property here. Usually, to compute laplace transforms, lookup tables are used. See this. $\endgroup$
    – kiyomi
    Commented Sep 24, 2020 at 2:01

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