Editing Integral transform (section)

==Usage example==

As an example of an application of integral transforms, consider the [[Laplace transform]]. This is a technique that maps [[Differential equation|differential]] or [[integro-differential equation]]s in the [[time domain|"time" domain]] into polynomial equations in what is termed the [[frequency domain|"complex frequency" domain]]. (Complex frequency is similar to actual, physical frequency but rather more general. Specifically, the imaginary component ''ω'' of the complex frequency ''s'' = −''σ'' + ''iω'' corresponds to the usual concept of frequency, ''viz.'', the rate at which a sinusoid cycles, whereas the real component ''σ'' of the complex frequency corresponds to the degree of "damping", i.e. an exponential decrease of the amplitude.) The equation cast in terms of complex frequency is readily solved in the complex frequency domain (roots of the polynomial equations in the complex frequency domain correspond to [[eigenvalues]] in the time domain), leading to a "solution" formulated in the frequency domain. Employing the [[Inverse Laplace transform|inverse transform]], ''i.e.'', the inverse procedure of the original Laplace transform, one obtains a time-domain solution. In this example, polynomials in the complex frequency domain (typically occurring in the denominator) correspond to [[power series]] in the time domain, while axial shifts in the complex frequency domain correspond to damping by decaying exponentials in the time domain.

The Laplace transform finds wide application in physics and particularly in electrical engineering, where the [[Characteristic equation (calculus)|characteristic equations]] that describe the behavior of an electric circuit in the complex frequency domain correspond to linear combinations of exponentially scaled and time-shifted [[damped sinusoid]]s in the time domain. Other integral transforms find special applicability within other scientific and mathematical disciplines.

Another usage example is the kernel in the [[Path integral formulation#Path integral in quantum mechanics|path integral]]:

:<math>\psi(x,t) = \int_{-\infty}^\infty \psi(x',t') K(x,t; x', t') dx'.</math>

This states that the total amplitude <math>\psi(x,t)</math> to arrive at <math>(x,t)</math> is the sum (the integral) over all possible values <math>x'</math> of the total amplitude <math>\psi(x',t')</math> to arrive at the point <math>(x',t')</math> multiplied by the amplitude to go from <math>x'</math> to <math>x</math> {{large|[}}i.e. <math>K(x,t;x',t')</math>{{large|]}}.<ref>Eq 3.42 in Feynman and Hibbs, Quantum Mechanics and Path Integrals, emended edition:</ref>  It is often referred to as the [[propagator]] for a given system. This (physics) kernel is the kernel of the integral transform. However, for each quantum system, there is a different kernel.<ref>[http://physics.stackexchange.com/questions/156273/mathematically-what-is-the-kernel-in-path-integral Mathematically, what is the kernel in path integral?]</ref>