What does a "half derivative" mean?

Question

I was looking at fractional calculus on Wikipedia, specifically this section and came across the half derivative of the function $y=x$ which is $\frac{d^{1/2}y}{dx^{1/2}}=\frac{2\sqrt{x}}{\sqrt{\pi}}$ . The derivative tells the slope at any point on the curve, but what does the "half derivative" mean - it's obviously not $\frac{1}{2}$ the derivative of $y=x$ which would be just $\frac{1}{2}$.

I do not have a very deep understanding of calculus - I have just taken Calc 1 & 2 out of a 4 series, but anything helps!

Also, I have checked similar questions, but they did not seem to answer my question that I have bolded.

Answer 1

Short answer: The half-derivative $H$ is some sort of operator (it isn't uniquely defined by this property) such that $H(Hf) = f'$.

Long answer: We can think of the derivative as a linear operator $D:X \to X$, where $X$ is some convenient (say, smooth) space of functions. The $n$th order derivative is then, by definition, the $n$-fold composition $D^n = D\circ \cdots \circ D:X \to X$. Clearly $D^n D^m = D^{n+m}$. Here we've restricted the index $n$ to an integer, but what if we allowed it to be a real number? That is, we want a family of operators $D_t$, with $t\geq 0$ real, such that

• $D_t$ behaves nicely with respect to $t$;
• $D_1$ is just the ordinary derivative $D$;
• $D_t D_s = D_{t + s}$.

(I'm not going to make the first point more precise here, but we ideally want something analogous to continuity or smoothness in $t$. I haven't defined what exactly the space $X$ is or what its geometry looks like, so I'm going to evade the point for now.) Thus, for example, we get an operator $D_{1/2}$ with $D_{1/2} D_{1/2} f = D_1 f = f'$ for suitable $f$.

Where do we get such an operator $D_t$? One place to start is Cauchy's integral formula: \begin{align*} f^{(-n)}(x) = \frac{1}{(n-1)!}\int_0^x (x - \xi)^{n-1} f(\xi)\, d\xi, \end{align*} where $f^{(-n)}(x)$ denotes the antiderivative of $f$, all normalized to have $f^{(-n)}(0) = 0$ for $n > 0$. The factorial above is only defined for positive integers $n$, but we can use the relation $\Gamma(n) = (n - 1)!$ to define something similar for arbitrary $t\geq 0$: \begin{align*} I_t f(x) = \frac{1}{\Gamma(t)}\int_0^x (x - \xi)^{t-1} f(\xi)\, d\xi, \end{align*} Clearly $I_t f(x)$ is just $f^{(-t)}$ if $t$ is a positive integer, and we can show with a bit of work that $I_t(I_s f) = I_{t+s} f$.

Now, that's for an antiderivative. In order to get to the derivative $D_t$ for non-integer $t$, we can use the definition above to get rid of the fractional part. Since $D(If) = f$, we can define \begin{align*} D_t f= \frac{d^n}{dx^n} \left(I_\tau f\right) \end{align*} for $t = n - \tau$ with $n$ an integer and $\tau\in [0, 1)$. This is not the only possible construction of a fractional derivative $D_t$, though.

Answer 2

The half derivative itself doesn't have much physical interpretation (though I believe there is a field called fractional quantum mechanics which may use it)

So why does it exist if its not any real physical thing.

I will explain.

Lets consider the idea of counting children at a school. We use whole numbers (positive integers) to count children. The statements are 5,6...201992 children each are meaningful in the sense that they exist mathematically AND have a physical interpretation.

But the set of Numbers isn't just whole numbers. It includes numbers like $1/2$ and $2^{1/2}$. So we could try to ask well what's half a child or square root of 2 children?

These are meaningless questions in the sense that you can't have half a child (contrary to popular belief disassembly and reassembly of children is not an easy or practical thing to do). Irrational quantities are even harder to produce. Put simply, they just DO NOT appear in that context.

So why am I telling you this? Here's why, lets ask the question not what 1/2 means in therms of children but how it came along. It came along because we wanted to generalize the set of numbers to include stuff in between the integers. It came along for applications besides counting children and is in fact most specifically an "accidental-byproduct" of the existence of division.

So what's a fractional derivative? We can easily answer the question that the nth derivative is the "rate of change of the rate of change ... (Repeat n times) of the rate of change of the function". This like children is a discrete structure. Only whole numbers (and if you include integrals then negative integers as well (like a backwards derivative)) work.

The fractional derivative is a consequence of the question "what is the function whom I apply twice to get a first derivative". Rather than "what is the rate of... Rate of change of the function"

So in short. It's an interesting question where we extend our level of control and understanding of calculus but it shares little similarity with the more physical forms that calculus originally had.

Answer 3

Often with new mathematical objects, it's not that the object has no meaning, but that we're asking the wrong question, or from the wrong context. It makes sense to double my apples, but it makes no sense to multiply my apples by i. That's asking the wrong question. One way of understanding multiplication by a complex number is that it scales for real values and rotates for imaginary values, so if we want a physical example that makes sense for multiplying imaginary numbers, allowing for rotation in the context would enable us to use this fact. Numbers of apples doesn't work, but perhaps modifying the velocity vector of my car would. We aren't accustomed to saying, "Double our velocity" or "i-ble" our velocity (turn 90 degrees), but it would make some sense, if we're asking the right question. We definitely could talk about multiplying by imaginary numbers whenever we rotate ourselves, even though we commonly don't.

What question, what context, could make sense for a half derivative? We have one famous context that I know of, namely the Riemann Hypothesis and the functions that surround it. There are probably other contexts.

In the Riemann Hypothesis, we have essentially a Fourier analysis of prime numbers, and the F(s) frequency space function is related somehow to the Riemann zeta function. (Question for the audience for my benefit: IS the frequency space function equivalent to the Re(z)=1/2 slice?). Hopefully my ignorance won't hinder the discussion here too much.

The Riemann Zeta function is an interesting one, which for $Re(z) > 1$, can be simply defined like this:

$$ \zeta(z) = \sum_{k=1}^\infty \frac{1}{k^z} $$

Since this combines both exponentiation and addition, this makes things tricky for moving around the graph of $\zeta(z)$. How would we consider shifting by $z+2$, for example? For integer amounts of shift, we might be able to rely on binomial expansion or something similar, but the number of terms becomes unwieldy quickly.

Differentiation and integration of a polynomial provides a convenient alternative, since they are linear operators, and modify the exponents. If we could differentiate $\zeta(z)$ in respect to k, the exponent would change, providing every term with a multiplicative factor but we can deal with that.

Confirmation of differentiation being somehow related to the values of $\zeta(z)$ can be found in the equations that link the Riemann zeta function and the polygamma function $\psi_n(z)$, where different values of $z$ for the Riemann zeta function correspond to different amounts of derivative in the polygamma function:

$$ \psi_n(1) = (-1)^{n+1} n! \zeta(n+1) $$

where $\psi_n(z)$ is the nth derivative of the digamma function. But for fractional derivatives, and using the gamma function instead of the factorial, this can be made valid for non-integer n.

But the points in question for the Riemann zeta function are well defined for Re(z) for integer $z$, whereas the points in question are all on the Re(z) = 1/2 line. We could probably use half derivatives of $\psi_n(z)$ to get there, and mayble we could use imaginary derivatives to traverse imaginary-wise along the critical line. I would wager that this impetus is what led Riemann and Cauchy to developing the beginnings of fractional calculus in the first place.

Other contexts might deal with exponential functions, which are eigenfunctions for derivatives and calculus. Repeated differentiation will lead to different eigenvalues. If the eigenvalues can be real valued instead of just integer valued, and if those eigenvalues have physical meaning, then fractional calculus will also find meaning there.