Laplace transform

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Laplace transform, named after <a href="/facts/Pierre-Simon_Laplace/6ktp0txb">Pierre-Simon Laplace</a> , is an <a href="/facts/Integral_transform/AB2yjHJJ">integral transform</a> that converts a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> of a <a href="/facts/Real_number/R02gw5Pb">real</a> <a href="/facts/Variable_(mathematics)/VbQlrvKz">variable</a> (usually 
 
 
 
 t
 
 
 {\displaystyle t}
 
, in the <a href="/facts/Time_domain/1w6mP2MU">time domain</a>) to a function of a <a href="/facts/Complex_number/2w7mqI7e">complex</a> variable 
 
 
 
 s
 
 
 {\displaystyle s}
 
 (in the complex-valued <a href="/facts/Frequency_domain/pyTVrUSq">frequency domain</a>, also known as s-domain, or s-plane).
The transform is useful for converting <a href="/facts/Derivative/7Ito70Dh">differentiation</a> and <a href="/facts/Integral/V7lyV997">integration</a> in the time domain into much easier <a href="/facts/Multiplication/VtcUjpNv">multiplication</a> and <a href="/facts/Division_(mathematics)/FqLKNkmq">division</a> in the Laplace domain (analogous to how <a href="/facts/Logarithm/QeXaV97q">logarithms</a> are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications in <a href="/facts/Science/IJoaD7rs">science</a> and <a href="/facts/Engineering/RFVg8H3I">engineering</a>, mostly as a tool for solving linear <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a> and <a href="/facts/Dynamical_system/782gREl5">dynamical systems</a> by simplifying <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ordinary differential equations</a> and <a href="/facts/Integral_equation/LCu4Imfq">integral equations</a> into <a href="/facts/Algebraic_equation/1QauIGxs">algebraic polynomial equations</a>, and by simplifying <a href="/facts/Convolution/PVPrdz9J">convolution</a> into <a href="/facts/Multiplication/VtcUjpNv">multiplication</a>. Once solved, the inverse Laplace transform reverts to the original domain.
The Laplace transform is defined (for suitable functions 
 
 
 
 f
 
 
 {\displaystyle f}
 
) by the <a href="/facts/Integral/V7lyV997">integral</a>

L
          
        
        {
        f
        }
        (
        s
        )
        =
        
          ∫
          
            0
          
          
            ∞
          
        
        f
        (
        t
        )
        
          e
          
            −
            s
            t
          
        
        
        d
        t
        ,
      
    
    {\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt,}

where s is a <a href="/facts/Complex_number/2w7mqI7e">complex number</a>. It is related to many other transforms, most notably the <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> and the <a href="/facts/Mellin_transform/M0087pMe">Mellin transform</a>. <a href="/facts/Formal_calculation/7VL2szyd">Formally</a>, the Laplace transform is converted into a Fourier transform by the substitution 
 
 
 
 s
 =
 i
 ω
 
 
 {\displaystyle s=i\omega }
 
 where 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
 is real. However, unlike the Fourier transform, which gives the decomposition of a function into its components in each frequency, the Laplace transform of a function with suitable decay is an <a href="/facts/Analytic_function/JhL3z8FP">analytic function</a>, and so has a convergent <a href="/facts/Power_series/vYh6sTps">power series</a>, the coefficients of which give the decomposition of a function into its <a href="/facts/Moment_(mathematics)/GL7vJ1nb">moments</a>. Also unlike the Fourier transform, when regarded in this way as an analytic function, the techniques of <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a>, and especially <a href="/facts/Contour_integral/4ILprcRr">contour integrals</a>, can be used for calculations.

Laplace transform open-in-new

Laplace transform