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The Laplace transform

The Laplace transform converts a function of time tt into a function of the frequency variable ss, defined by an integral. Standard transforms of common functions are found in tables.

L{f(t)}=∫0∞e−stf(t) dt\mathcal L\{f(t)\} = \int_0^\infty e^{-st}f(t)\,dtdefinition
L{tn}=n!sn+1\mathcal L\{t^n\} = \frac{n!}{s^{n+1}}powers of t
L{sin⁡ωt}=ωs2+ω2\mathcal L\{\sin\omega t\} = \frac{\omega}{s^2+\omega^2}sine

Symbols

sscomplex frequency variable
ω\omegaangular frequencyrad/s

Example

L{t}=1s2\mathcal L\{t\} = \dfrac{1}{s^2}, and L{sin⁡(2t)}=2s2+4\mathcal L\{\sin(2t)\} = \dfrac{2}{s^2+4}.

The Laplace transform exists only when ss is large enough for the integral to converge.
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Part of Laplace, Fourier and PDEs: The Laplace transform.