Extensions to common Laplace and Fourier transforms

The extended versions of common Laplace and Fourier transforms are given. This is achieved by defining a new function f/sub e/(p), p/spl epsiv/C related to the function to be transformed f(t), t/spl epsiv/R. Then f/sub e/(p) is transformed by an integral whose path is defined on an inclined line on the complex plane. The slope of the path is the parameter of the extended definitions which reduce to common transforms with zero slope. Inverse transforms of the extended versions are also defined. These proposed definitions, when applied to filtering in complex ordered fractional Fourier stages, significantly reduce the required computation.