# Generalisation of kth derivative to real values of k

The answer to this question is most likely no, but I'm asking anyway: Assume that $f\in C^n(\mathbb {R,R})$. Is their any natural generalisation of the map $$\{1,2,\ldots,n\}\to C(\mathbb{R, R})\\k\mapsto f^{(k)}$$ to a map $$[1, n]\to\{g\colon\mathbb{R\to R}\}\\ x\mapsto f^{(x)}?$$ That is, can we generalise "the $k$th derivative" to "the $x$th derivative" for real values of $x$? What I mean by "natural" is: Anything that has desirable properties and that has been explicitly formulated by someone, somewhere.

• Of course. You can try looking for the keyword "pseudodifferential calculus". Commented Mar 2, 2014 at 10:44
• There exists a whole industry of "fractional calculus" with different definitions of what a "fractional derivative" is. For instance, as mentioned in the previous comment, using the Fourier transform there is a natural extension from integer to fractional derivatives. Commented Mar 2, 2014 at 10:47
• @Gaussler try seeing Calculus of Finite differences Commented Mar 2, 2014 at 10:48
• The Wikipedia article for fractional calculus mentions complex order derivatives too. No clue how that works. Commented Jan 5, 2018 at 19:10