Laplace Transform
The Laplace transform of a function f(t), defined for all real numbers, is the function F(s), defined by:
- $ F(s) = \mathcal{L} \left\{f(t)\right\}=\int_{-\infty}^{\infty} e^{-st} f(t) \,dt. $
Region of convergence
The Laplace transform F(s) typically exists for all complex numbers such that Re{s} > a, where a is a real constant which depends on behavior of f(t), whereas the transform is defined in a range a < Re{s} < b. The subset of values of s for which the Laplace transform exists is called the region of convergence (ROC) or the domain of convergence.
The integral defining the Laplace transform of a function may not exist when the function has infinite discontinuities in the interval of integration, or when it increases so rapidly that exp(-pt) cannot damp it sufficiently for convergence on the interval to take place and many other regions too. There are no specific conditions that one can check a function against to know in all cases if its Laplace transform can be taken, other than to say the defining integral converges. It is however easy to give theorems on cases where it may or may not be taken.
Fourier transform
The continuous Fourier transform is equivalent to evaluating the bilateral Laplace transform with complex argument s = jω: $ F(\omega) = \mathcal{F}\left\{f(t)\right\} = \mathcal{L}\left\{f(t)\right\}|_{s = j \omega} = F(s)|_{s = j \omega} = \int_{-\infty}^{+\infty} e^{-\jmath \omega t} f(t)\,\mathrm{d}t. $
Note that this expression excludes the scaling factor $ \frac{1}{\sqrt{2 \pi}} $, which is often included in definitions of the Fourier transform.
