8

While reading about fractional calculus in http://arxiv.org/pdf/math/0110241.pdf , I came across the following quote:

Fractional integration and fractional differentiation are generalisations of notions of integer-order integration and differentiation, and include n-th derivatives and n-folded integrals (n denotes an integer number) as particular cases.

Let $D = \frac{d}{dx}$ .

We have found meaningful notions of $D^2$ and $D^{-1}$ (derivative and antiderivative, respectively, of integer-order) and $D^{\frac{1}{2}}$ (derivative of fractional-order) , and we can say that integer differentiation (where given $D^n , n \in \mathbb{Z}$) is a special case of more general fractional differentiation (where given $D^n , n \in \mathbb{R}$).

I'm wondering if there's some meaningful notion of "complex differentiation", say something like $D^i$, where fractional differentiation is a special case (note that anti differentiation is a special case of differentiation; namely, given $D^n$, anti differentiation occurs when $n$ is real and negative). Is this conceivable? If so, are there any apparent applications of this?

Sorry if this is a dumb question (by dumb, I mean something that I could've found elsewhere online). I've searched around and haven't found anything on this yet.

adamcatto
  • 311

2 Answers2

7

Yes. One could perhaps start with Fractional derivatives of imaginary order by E.R. Love (from 1971). This is still an active area of research. (Contrast "fractional calculus", which frequently starts with the Riemann-Liouville differintegral, neither of whom are particularly recent publishers.) See, for instance, Fractional Integrals of Imaginary Order in the Space of Hölder Functions with Polynomial Weight on an Interval or Definitions of Complex Order Integrals and Complex Order Derivatives Using Operator Approach. (The latter essentially extends the R-L differintegral slightly.)

Part of the difficulty is identifying functions that can be differentiated imaginarily- (or complexly-) many times. It would be nice to just have a supply of spaces of functions that can be differintegrated in such a way, and whose set relationships with each other and other well studied spaces of functions were known, but we're not quite there yet.

Eric Towers
  • 67,037
2

I can present an example problem with a complex fractional derivative. Consider the Cauchy pulse (Ref: S.L. Hahn, Hilbert Transforms in Signal Processing, Artech House, 1996)

$$\psi=\frac{1}{1-i\tau}$$

Differentiating $n$ times we obtain

$$\psi(\tau,n)=\psi^{(n)}(\tau)=\frac{d^n\psi}{d\tau^n}=\frac{i^nn!}{(1-i\tau)^n}=\frac{i^n\Gamma(n+1)}{(1-i\tau)^n}$$

And, of course, the Gamma function is our pathway to any fractional derivative. Now, insofar as $\tau \in (-\infty,\infty)$ we can make a change of variables so that $\tau=\text{tan}\theta,\ \theta\in [-\pi/2,\pi/2]$. Then

$$\frac{1}{(1-i\tau)^n}\to \frac{1}{(1-i\text{tan}\theta)^n}=\text{cos}^{n+1}\theta \ e^{i(n+1)\theta}$$

As an example, we plot the ${i^{3/2}}^{th}$ derivative of the Cauchy pulse [note that $i^{3/2}=\frac{\sqrt{2}}{2}(-1+i)$]. Clearly, we can create a infinite number of new plane curves in this manner.

For a more detailed description of the Cauchy pulse and related functions see The Apple of My i. This was my first foray into recreational mathematics five years ago.

The Apple of My *i*

Cye Waldman
  • 7,524