Fourier Transform vs Laplace Transform

Say we have a function 𝑓(𝑡). With it let’s create a new function 𝐹(𝑡) such that:

• $F(t)=f(t)e^{-st}H(t)$

where:

• $H(t)=\{\begin{array}{ll}0& t<0\\ 1& \text{otherwise}\end{array}$

Given the above functions: the Laplace transform of 𝑓(𝑡) is the Fourier transform of 𝐹(𝑡)