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The notion of a fractional derivative or integral is naturally defined in terms of the integration and differentiation theorems for Laplace/Fourier transforms. Let $ X(s)$ denote the bilateral Laplace transform of $ x(t)$ :

$\displaystyle X(s) \isdefs {\cal L}_s\{x\} \isdefs \int_{-\infty}^{\infty} x(t) e^{-st}dt,

where $ s=\sigma+j\omega$ is a complex variable, $ t$ typically denotes time in seconds, `$ \isdef$ ' means ``equals by definition,'' and we assume $ x(t)$ and all of its integrals and derivatives are absolutely integrable and approach zero as $ t\to\pm\infty$ . Then the differentiation theorem for bilateral Laplace transforms states that

$\displaystyle {\cal L}_s\left\{x^{(1)}\right\}
\isdefs {\cal L}_s\left\{\frac{d\,x(t)}{dt}\right\}
\eqsp s\,X(s)

where $ x^{(n)}(t)$ denotes the $ n$ th derivative of $ x(t)$ with respect to $ t$ .2 The proof is quickly derived using integration by parts.3

The integration theorem for Laplace transforms follows as a corollary:

$\displaystyle {\cal L}_s\left\{x^{(-1)}\right\} \isdefs {\cal L}_s\left\{\int_{-\infty}^{t} x(\tau)\,d\tau\right\} \eqsp \frac{1}{s} \, X(s)

The Laplace transform specializes to the Fourier transform along the $ s=j\omega$ axis in the complex plane, where $ \omega = 2\pi f$ is radian frequency (radians per second), while $ f$ denotes frequency in Hz (cycles per second).

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``Spectral Tilt Filters'', by Julius O. Smith IIIand Harrison F. Smith, adapted from publication arXiv:1606.06154 [cs.CE].
Copyright © 2021-06-18 by Julius O. Smith IIIand Harrison F. Smith
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University