What Lies Between a Function and Its Derivative? | Fractional Calculus

Source: What Lies Between a Function and Its Derivative? | Fractional Calculus, Morphocular, 25:27, uploaded 2022-08-11, Watch Later position 1133.

Morphocular begins with (f(x)=x^3) and its first three derivatives. Ignoring the coefficients, the powers descend from (x^3) to (x^2), then (x), then (1=x^0). Fractional powers such as (x^{1/2}) already fill the spaces between these polynomial powers. That suggests a question with a precise test attached to it: can a derivative also have a fractional order?

The requirement behind a half-derivative

The first guess is simple. A half-derivative of (x^3) might reduce the exponent by one half and produce (x^{2.5}). The exponent has the right shape, yet the guess fails when applied twice. Two exponent reductions turn (x^3) into (x^2), whilst the ordinary derivative is (3x^2). A fractional derivative needs a coefficient that composes correctly as well as an exponent that sits halfway between the ordinary cases.

The source therefore starts with the property that gives the idea its name. Applying a half-derivative twice should have the same effect as applying one ordinary derivative:

D^{1/2}D^{1/2}f=Df.

Finding the coefficient for (x^{2.5}) is harder than taking the square root of the exponent, because the first half-derivative changes the exponent before the second one acts. Morphocular invites the viewer to try the calculation, then admits that the first attempt led nowhere. The polynomial picture remains suggestive, so the route changes direction.

Repeated integrals and the gamma function

Integrals provide a more workable starting point. Continuous functions can be integrated even when they cannot be differentiated. The video uses the Weierstrass function as an example of a continuous function whose graph has no derivative anywhere. Repeated integrals also have a useful compression property. Cauchy’s formula for repeated integration turns (n) nested integrals into one integral:

I^n_a f(x)=\frac{1}{(n-1)!}\int_a^x (x-t)^{n-1}f(t)\,dt.

For (f(x)=6x), (n=2), and (a=0), this produces (t^3), the second antiderivative of (6x), with (t) taking the place of (x) inside the integral. The factorial is the remaining obstacle to a fractional order. The gamma function extends the factorial relation to non-integer inputs, so replacing ((n-1)!) with (\Gamma(n)) gives

I^p_a f(x)=\frac{1}{\Gamma(p)}\int_a^x (x-t)^{p-1}f(t)\,dt,

for a positive real order (p). At (p=1/2), (\Gamma(1/2)=\sqrt{\pi}), and the half-integral becomes

I^{1/2}_a f(x)=\frac{1}{\sqrt{\pi}}\int_a^x \frac{f(t)}{\sqrt{x-t}}\,dt.

The appearance of (\sqrt{\pi}) looks arbitrary until the composition test. Apply the half-integral to (f(x)=2x), using (a=0), and the result is

I^{1/2}_0(2x)=\frac{8}{3\sqrt{\pi}}t^{3/2}.

Applying the same operation again gives (t^2), which is the ordinary integral of (2t). The strange coefficient has done the work required of it. The same construction gives a one-third integral whose third application equals one ordinary integral. More generally, composing an order-(p) integral with an order-(q) integral gives an order-((p+q)) integral. This is the Riemann–Liouville fractional integral.

The description corrects one detail in the calculation at 7:34. The trigonometric substitution contributes a whole factor of (\pi) in the numerator. That factor cancels the two (\sqrt{\pi}) terms introduced by applying the half-integral twice.

The animations make the interpolation visible. As (p) moves from (0) to (1), the fractional integrals of (2x) gradually become the parabola (x^2). Moving from (p=1) to (p=2) carries the graph towards (x^3/3). Changing the lower limit from (0) to (-1) changes the integration constants and makes the intermediate expressions more involved, while the fractional order still interpolates between the integer cases.

The Riemann–Liouville route to fractional derivatives

It is tempting to define a derivative as an integral of negative order. This runs into two separate problems. The gamma function is undefined at non-positive integers, so (p=-1) cannot produce an ordinary derivative through the fractional-integral formula. For negative non-integer orders such as (-1/2), the integral diverges near its lower endpoint because the factor (x^{p-1}) is not integrable there. The formula therefore works only for strictly positive (p).

The workaround keeps the fractional integral at a positive order and uses ordinary derivatives to reduce the effective order. For a non-integer order (p), choose the least integer (k) greater than (p), take the fractional integral of order (k-p), and then apply (k) ordinary derivatives:

D^p_a f(x)=\frac{d^k}{dx^k}I^{k-p}_a f(x).

For (p=1/2), this means taking a half-integral and differentiating the result once. Applied to (x^2), the operation gives

D^{1/2}_0x^2=\frac{8}{3\sqrt{\pi}}t^{3/2}.

A second half-derivative gives (2t), the ordinary derivative of (t^2). The construction passes the original composition test. The source calls this the Riemann–Liouville fractional derivative.

Putting the fractional derivative and fractional integral into one operator produces a piecewise rule: positive orders use the derivative construction and negative orders use the integral construction. Morphocular calls the combined operator a differintegral, then suggests “derivagral” as a better name. The joke is brief, though the unified operator becomes the video’s main way to inspect intermediate orders.

Polynomials, sine, and the lower limit

For (t^2), the polynomial examples show the expected progression between an integral and a derivative. The sine curve gives a more revealing test. With the lower limit set to (a=0), changing the order from (0) to (1) gradually carries (\sin(t)) towards (\cos(t)). The intermediate half-derivative only looks like a shifted sine wave at first glance. Its two visible peaks have different heights, so the curve falls outside sine waves.

The exponential (e^t) exposes the role of the lower limit. Since every ordinary derivative of (e^t) is (e^t), one might expect the fractional versions to remain exponential too. With (a=0), the intermediate curves leave the exponential family. Ordinary integration already explains the source of the trouble:

\int_0^t e^x\,dx=e^t-1,

so a constant enters the result and persists through higher-order integrals. Choosing (a=-\infty) removes that boundary contribution for this function. The fractional derivatives then remain copies of (e^t), including at the intermediate orders.

This dependence on the lower limit belongs to the mathematics. An integer derivative is local: its value at a point depends on the function in an arbitrarily small neighbourhood of that point. A fractional derivative can also depend on what the function did earlier in its domain, and changing the lower limit changes how much of that domain the operator can see. This is why fractional derivatives are called non-local or said to have memory. When the fractional order becomes an integer, the operator becomes local again. Two non-local half-derivatives can therefore compose into one local derivative.

A visualisation without a settled interpretation

The source leaves the geometric meaning of fractional derivatives open. Their dependence on distant parts of the function makes an interpretation comparable to the slope of a tangent line difficult, and Morphocular says that no general interpretation has become widely accepted.

The video offers one way to visualise a fractional integral. The ordinary integral from (0) to (t) is the area under (f(x)). In the half-integral formula, the factor multiplying (f(x)) depends on both (x) and (t). Calling this factor (\mu_t(x)), one can treat it as changing the height of the graph at each point. That preserves the formula, though it makes the original shape hard to see.

A second view leaves the height of each small rectangle unchanged and stretches or squashes its base horizontally according to (\mu_t(x)). The transformed rectangles form an area whose value is the half-integral, whilst the original curve remains visible above it. Varying (t) shows how the half-integral grows in relation to the ordinary integral.

The picture has two limits. The defining composition rule, applying two half-integrals to get one full integral, remains hard to see. The same height or horizontal transformation can also visualise other integral transforms, including the Laplace transform, so the picture does not identify what is specifically fractional about the operation.

Morphocular’s own conclusion stays tentative. The source suspects that fractional calculus may have less to do with ordinary derivatives and integrals than the interpolation story suggests. The relation between a fractional operator and its integer neighbours may turn out to be a useful consequence of a deeper idea, whilst its best interpretation lies elsewhere. The video leaves that idea open and assigns the operators no settled meaning.

The derivative zoo

The Riemann–Liouville derivative depends on the order in which ordinary derivatives and fractional integrals are applied. One can take the fractional integral first and then differentiate, as in the construction above. One can also differentiate first and finish with the appropriate fractional integral. The two procedures both use partial cancellation between derivatives and integrals, yet they can produce different results.

The second construction is the Caputo fractional derivative. It is one of many formulations, so fractional calculus has no single definitive fractional derivative. The video presents the Riemann–Liouville derivative as a well-known choice and notes that it has shortcomings left outside the presentation. The source’s review of alternative definitions treats this lack of uniqueness as part of the subject, with no final formula resolving it.

What Lies Between a Function and Its Derivative? | Fractional Calculus

notes.

What Lies Between a Function and Its Derivative? | Fractional Calculus

The requirement behind a half-derivative

Repeated integrals and the gamma function

The Riemann–Liouville route to fractional derivatives

Polynomials, sine, and the lower limit

A visualisation without a settled interpretation

The derivative zoo

Further reading / references

related notes