Laplace transform

In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually ⁠ t {\displaystyle t} ⁠, in the time domain) to a function of a complex variable s {\displaystyle s} (in the complex-valued frequency domain, also known as s-domain or s-plane). The functions are often denoted using a lowercase symbol for the time-domain function and the corresponding uppercase symbol for the frequency-domain function, e.g. x ( t ) {\displaystyle x(t)} and ⁠ X ( s ) {\displaystyle X(s)} ⁠. The transform is useful for converting differentiation and integration in the time domain into the algebraic operations multiplication and division in the Laplace domain (analogous to how logarithms are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications in science and engineering, mostly as a tool for solving linear differential equations and dynamical systems by replacing ordinary differential equations and integral equations with algebraic polynomial equations, and by replacing convolution with multiplication. For example, through the Laplace transform, the equation of the simple harmonic oscillator (Hooke's law) x ″ ( t ) + k x ( t ) = 0 {\displaystyle x''(t)+kx(t)=0} is converted into the algebraic equation s 2 X ( s ) − s x ( 0 ) − x ′ ( 0 ) + k X ( s ) = 0 , {\displaystyle s^{2}X(s)-sx(0)-x'(0)+kX(s)=0,} which incorporates the initial conditions x ( 0 ) {\displaystyle x(0)} and ⁠ x ′ ( 0 ) {\displaystyle x'(0)} ⁠, and can be solved for the unknown function ⁠ X ( s ) {\displaystyle X(s)} ⁠. Once solved, the inverse Laplace transform can be used to transform it to the original domain. This is often aided by referencing tables such as that given below....