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In mathematics, the Laplace transform, named after Pierre-Simon Laplace (/ l ə ˈ p l ɑː s /), is an integral transform that converts a function of a real variable (usually , in the time domain) to a function of a complex variable (in the complex-valued frequency domain, also known as s-domain, or s-plane).
Two-sided Laplace transforms are closely related to the Fourier transform, the Mellin transform, the Z-transform and the ordinary or one-sided Laplace transform. If f ( t ) is a real- or complex-valued function of the real variable t defined for all real numbers, then the two-sided Laplace transform is defined by the integral
The unilateral Laplace transform takes as input a function whose time domain is the non-negative reals, which is why all of the time domain functions in the table below are multiples of the Heaviside step function, u(t). The entries of the table that involve a time delay τ are required to be causal (meaning that τ > 0).
A Laplace transform can be defined for functions on time scales, which uses the same table of transforms for any arbitrary time scale. This transform can be used to solve dynamic equations on time scales. If the time scale is the non-negative integers then the transform is equal [2] to a modified Z-transform: ′ {[]} = {[+]} +
In mathematics, the Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain. The Laplace transform can be used in some cases to solve linear differential equations with given initial conditions. First consider the following property of the Laplace transform:
The state-transition equation is defined as the solution of the linear homogeneous state equation. The linear time-invariant state equation given by = + + (), with state vector x, control vector u, vector w of additive disturbances, and fixed matrices A, B, E can be solved by using either the classical method of solving linear differential equations or the Laplace transform method.
Once () is known, we can compute its Laplace transform, calculate the Laplace transform of / and then take the inverse transform (or try to) to find /. For the tautochrone problem, T ( y 0 ) = T 0 {\displaystyle T(y_{0})=T_{0}\,} is constant.
In the case of a boundary put at infinity with the boundary condition setting the solution to zero at infinity, then one has an infinite-extent Green's function. For the three-variable Laplace operator, one can for instance expand it in the rotationally invariant coordinate systems which allow separation of variables.