Transcription
You know functions. You might even know derivatives. But let me ask you something nobody asked you in calculus class. What's between them?
A function sits at one spot, its derivative sits one step over. I want to know what lives in the gap. Can I take a half derivative? And for this, we need half to really earn its name, like a square root. Root nine is the thing you multiply by itself to get nine. So a half derivative is the thing you apply twice to get one whole derivative.
Now that might sound like some nonsense, but it's not, and the question is over 300 years old. In around 1695, right after Leibniz invented his notation, he got a letter from L'Hôpital. Yep, the same one, the rule guy, asking the obvious troublemaker question. What if n isn't a whole number, something like 1/2? Apparently, Leibniz wrote back, and this is basically the real quote, "An apparent paradox from which one day useful consequences will be drawn." And he was right, it just took a while.
So how might we go about building something like this? Take for example x². By the power rule, its derivative is 2x. Multiply by the power, subtract one from the power. So maybe its half derivative should be something like 1.5 x to the 1.5. Well, the issue here is if you apply that same rule again, you get something weirder when we should get 2x. And this gives us our first real insight to how these half derivatives should work.
Differentiating doesn't just lower powers by one, it throws off the coefficients in a very particular way. X to the fourth goes to 4x cubed. X to the seventh goes to 7x to the sixth. So, whatever a half derivative is, it has to know exactly how much coefficient to release on each half step.
So, where do these coefficients really come from? If you take n derivatives of x to the k, you might notice a pattern and get this formula. You can check it against our examples and it works. And so now our plan is pretty obvious. Plug in n = 1/2 and immediately we get an issue because traditionally k factorial only makes sense for whole numbers. And our formula would need fractional factorials.
And so if you're a long time viewer of this channel, you know where we're going. It's the gamma function. The famous gamma function fills in the factorials in between the whole numbers and still acts like factorials at those whole numbers, keeping everything working how it should. So, let's just rewrite our formula with gamma function notation. And now we should be able to plug in pretty much whatever we want.
How about 1/2? We'll just take a simple example. Take our function to be x. What's the half derivative of x? Well, according to this formula, that's gamma of two over gamma of 3/2 x to the 1/2. Now, you do need to know something about the gamma function with these half factorials. That's another video, but gamma of 3/2 is root pi over 2. And so, the half derivative of X is 2 by root pi root X.
Now, that's pretty strange. I don't see any circles lying around, but for our rule to hold, we should be able to take this derivative again and get to the true derivative of X, which is 1. So, throw it into our formula again, take the half derivative. There's another gamma of 3/2. That cancels with our earlier gamma of 3/2. Gamma of 1 is just 1, X to the 0 is 1, and the whole thing collapses to 1. Two half derivatives gave us one full derivative, at least in this case.
How about with our earlier example of X squared? Take the half derivative of X squared. Deal with some gamma function things. Do it again, and hey, there we go. We get the expected result 2 X.
Now, I know what you're saying. That's all well and good for power functions, but what about other types of functions? What about exponentials or trigonometric functions? Well, every time we differentiate e to the ax, you basically just multiply by a factor of a. So, the natural choice here would be that we multiply by a root a instead of a. Do it twice, you get the first derivative.
How about something like sin X? The derivative of sin X is cos X. Which, if you think about it, is basically just a shift, a translation of the sine function. So, a half derivative should just be half of that translation. Translate it by pi over four rather than pi over two. Do it twice, you get this relationship, which is cosine.
So, where are we at with everything? If we treat the function itself like order zero, the derivative is order one, positive one. Going the other way, integration would be like negative one. And they're sort of not really three separate ideas, they're three points on one continuous line. And now, with these fractional ideas, we can sort of stand anywhere along this line.
But, there's another interesting aspect of this. The first derivative of a function has a picture, it's slope. So, does its second one, how the graph curves or bends. What would the half derivative look like or represent? Well, here's the frustrating part, there's not really a clean answer. After these 300 years, there's not really an agreed geometric meaning for a fractional derivative. People have definitely proposed one, but none of them stuck with me in the way that slope did.
There is another kind of tangle to this. Our half derivative is supposed to handle any function, not just powers and exponentials. And when you tie it together with one definition, it comes from rewriting repeated integration, and surprise, that formula also hides a factorial you can swap for the gamma. This is sometimes called the Riemann-Liouville derivative. And it does something that should probably bother you.
Take the half derivative of a constant. Just take the derivative of one. Now, the regular derivative of one is zero. But the half derivative of one is one over root pi x, not zero. A flat function has a half derivative that isn't flat at all. The reason is that a fractional derivative isn't local. The normal derivative only looks at one instant, the slope right where you're standing. A half derivative looks at the function's entire history. It's an integral over everything that came before. It sort of has memory.
Which is probably why no simple picture ever really stuck. A slope is something you can see at a point. A half derivative isn't at a point at all. It's smeared across everything the function has ever done. And this is probably why mathematicians keep more than one definition around here. The version that sends a constant to zero is a different one. And which one you use depends on which situation you're modeling.
And people do model with these. That memory effect is right for materials that are part solid, part fluid, gels, polymers, living tissue. It shows up in diffusion that runs faster or slower than it should, and in control systems that need finer tuning than the standard ones allow. So, Leibniz paradox did pay off, and it took about three centuries.
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We've filled in the gap between a function and its derivative. It's like there's this smooth dial of half steps and even quarter steps. Really, almost anything you like. You can take a derivative of order 1/2 and square it right back into the real thing. What that thing looks like is still sort of up for grabs.
If you like these mind-bending calculus situations, I think you're really also going to enjoy this video on the screen. Click it right now to watch. I'll see you in that one.