A thought occurred to me on the way home today. (Do not assume that this kind of thing happens all of the time.)
Suppose I have a function in x. Assume for the moment that the function is well behaved, infinitely differentiable, and is a function of just one variable.
I can take the first derivative of that function.
I can take the second derivative of the function.
And if someone pushed me, I could find the nth derivative of the function for any value of n.
Well, not quite any value of n.
Is there any reason why the domain of n is restricted to natural numbers? (It is for all the calculus I understand, and in most practical situations does not exceed three.) Is it possible to consider “n” in the set of Q or R or even C?
I know that fractional dimensions are possible (fractals) and have practical applications. Is a similar thing possible with differentiation? Has anyone considered this before? does it have any practical uses? Can anyone, for example, tell me the (1+ei)th derivative of f(x)=cosx ?
Well, the derivative of a function is its slope, and the anti-derivative is the area under its curve. I can’t quite conceive of what a fractional derivative would be. That doesn’t mean there couldn’t be one. You say there are fractional dimensions, and I can’t imagine that either.
I suppose you could consider the first anti-derivative of a function to be its -1 derivative (though I’ve never seen anyone do that). The function itself could be the 0 derivative. That means n could be any integer.
I’m sure future posters will expound on fractional derivatives, as well as excoriate my simplistic definitions of derivatives and anti-derivatives.
From friedo’s link: “the foundations of the subject were laid by Liouville in a paper from 1832.” That’s not bad, j_sum1, to be behind the mathematical curve by only 173 years. A lot of people barely catch up to Newton in 1666.
Setting aside the normal warnings about how infinity is not a number, you could theoretically have an infinite derivative in certain situations. If the sequence f, f’, f’’, f’’’, …, converges uniformly on an interval I, then you could define the limit of that sequence to be the infinite derivative of f on I.
Being a limit operator, this preserves addition and multiplication, so it behaves the way a derivative ought to behave.
I know I used the wrong term. But the correct one escaped me. I meant a function that could be differentiated an unlimited number of times. But it is nice to know I inadvertantly stumbled across another interesting question. Thanks for the thought ultrafilter.
I was hoping that either ultrafilter or mathochist would pop in with an answer to my original question. I have not fully read friedo’s link yet, but I note that it only extends to the reals. (Ok, ok, as if that isn’t far enough.) Does anyone care to enlighten me as to the situation with complexes?
In brief, here’s how you get to fractional derivatives: You have a function space F, and you define an operator D: F [symbol]®[/symbol] F as the derivative operator. If you can represent D as a matrix, then you can work with arbitrary real powers of D–so the 1/2th order derivative is a matrix S such that S[sup]2[/sup] = D. Of course, this is highly simplified, so take it with a grain of salt.
Incidentally, if the sequence D, D[sup]2[/sup], D[sup]3[/sup], … converges, you have another notion of the infinite derivative, but I don’t know how well it corresponds to what I proposed earlier.
So what about complex-ordered derivatives? Well, you may run into a problem here because C is not an ordered field, but that’s small potatoes compared to the other problem you might run into: D[sup]x + y[symbol]i[/symbol][/sup] is not necessarily a single-valued function. I don’t know how you might resolve that.
Incidentally, if you have a function that’s infinitely differentiable, you would say it’s an element of C[sup][symbol]¥[/symbol][/sup]. This is an extension of the notion that a continuous function whose nth derivative exists but (n + 1)st does not is an element of C[sup]n[/sup]. The class C[sup]0[/sup] is quite interesting…
That doesn’t save you from having to deal with multiple-valued functions. You’ve heard that exp([symbol]ip[/symbol]) + 1 = 0, right? Well, it’s also equal to 2[symbol]pi[/symbol], 4[symbol]pi[/symbol], -2[symbol]pi[/symbol], and every other integer multiple of 2[symbol]pi[/symbol]. There’s a very good reason to pick 0 as the primary value, but it’s not the only one.
The complex exponential exp() is periodic, not multivalued. exp(i pi)+1 is single-valued. The complex natural logarithm ln, the inverse of exp, is the multivalued one. Over R, ln(1) = 0; but in C, the various branches of ln have ln(1) = 2 pi n i for any integer n.
This doesn’t affect your point above, since the complex power z[sup]w[/sup] is defined as exp(w ln(z)). If z is not restricted to be real then the choice of branch may not be obvious. (Even if z is real we could perversely choose a different branch; so the complex power e[sup]z[/sup] is not strictly equal to the complex exponential exp(z). But the different choices don’t lead to values differing by 2 pi i n.)