The notion of a fractional derivative or integral is naturally defined in terms of the integration and differentiation theorems for Laplace/Fourier transforms. Let denote the bilateral Laplace transform of :
where is a complex variable, typically denotes time in seconds, ` ' means ``equals by definition,'' and we assume and all of its integrals and derivatives are absolutely integrable and approach zero as . Then the differentiation theorem for bilateral Laplace transforms states that
where denotes the th derivative of with respect to .2 The proof is quickly derived using integration by parts.3
The integration theorem for Laplace transforms follows as a corollary:
The Laplace transform specializes to the Fourier transform along the axis in the complex plane, where is radian frequency (radians per second), while denotes frequency in Hz (cycles per second).