# Dictionary:Laplace transform

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The linear transform pair

${\displaystyle F(s)=\int f(t)e^{-st}dt}$

and

${\displaystyle f(t)={\frac {1}{2\pi i}}\int F(s)e^{st}ds}$

s is a complex number and t is a real one. When the limits of integration are ${\displaystyle \pm \infty }$, 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

${\displaystyle F(s)=lim\int f(t)e^{-st}dt}$

and

${\displaystyle f(t)=\int F(s)e^{+st}ds}$

The one-sided transform is often written with limits 0 to ${\displaystyle \infty }$, 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).