Laplace Transform

  The Laplace transform is an integral transform which has the property of translating certain complicated operations (e.g. the differentiation of a function or the convolution of two functions), into simple algebraic operations in the image (Laplace) space. It can therefore be used to transform certain types of functional equations into algebraic equations. A special case of the Laplace transform is the Fourier transform.

The one-sided Laplace transform of a function F is defined by

where s is a complex parameter. If the integral converges for s = a, a real, the Laplace transform f(s) exists for all s with .

The two-sided Laplace transform is defined by the same formula, with the integral extending from to .

Under appropriate assumptions, the original function F is obtained from the ``image'' function f by the inversion formula

where x is any real number, with a for and for .

For practical use, refer to modern packages (e.g Mathematica, [Wolfram91]).