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

where denotes the th derivative of with respect to .

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

- Fractional Derivatives and Integrals
- Filter Interpretation
- Exponentially Distributed Real Pole-Zero Pairs
- Importance in Audio Signal Processing
- Summary of Results
- Outline of the Remainder

Download spectilt.pdf

[Comment on this page via email]

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University