TITLE: Is there a notion of a complex derivative or complex integral?
QUESTION [8 upvotes]: 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.

REPLY [2 votes]: 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.