# Dictionary:Laplace transform

The linear transform pair

and

*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

and

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).