Introduction

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 $x(t)$ :

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

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

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

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:

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

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).