The linear transform pair
s is a complex number and t is a real one. When the limits of integration are , the transform is two-sided. The two-sided Laplace transform becomes identical with the Fourier transform when s is purely imaginary. More often the one-sided transform is used, especially in the study of transient waveforms. In this case, where f(t) is causal, the integral is
The one-sided transform is often written with limits 0 to , the limit being implied. Laplace transforms may not exist for all values of s and hence many Laplace transforms are limited to strips of convergence, the ranges of values for the real part of s for which the above intearals are finite. The Laplace transform domain is often called the s-plane. See Sheriff and Geldart (1995, 545–546).