## Leonhard Euler

###### A Basic Introduction to Fractional Calculus

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###### Zanyar Oğurlu

What does it mean to find a derivative of a function at a certain value? It means finding the slope of the line that is tangent to the function, or rate of change, at that point. What does it mean to find the second derivative of a function at a certain value? It means finding the rate of change of its slope, or the curvature, at that point. Here is the tricky question: How do we find the nd derivative of a particular function? What does it even mean to perform such a calculation? Well, it means very little. It means something very rarely, like, maybe if we are dealing with very complicated fractals, it can mean something then. Other than that, it is just fun to think about, which is the motivation behind this article.

Let’s start with some easy stuff. We start by evaluating the m’th derivatives of a simple power function.

Now, let’s formulate the pattern that we observe above.

What happens if we take m to be a non-integer number? In that case, we are faced with a problem with the factorials in the expression that we obtained. Factorials are only defined for integer numbers. We approach this problem simply by replacing the factorial with an equivalent function, which is the Gamma function. But what is the Gamma function? Let’s delve more into that.

The gamma function is defined as follows,

We have the following equivalency between the gamma function and the factorial,

Let's evaluate the gamma function using integration by parts to understand why it shows such an interesting property,

Here, we obtained one of the fundamental definitions of the factorial by deriving that

We also need the initial conditions to be correctly set for the function. By placing z=1, we find that . Using these two results, we also find that . Hence, we see the following relation between the factorial and the gamma function.

Thus, we have proven that

Now, we can go back to our simple example on fractional calculus. We had stated that

Here, if we replace the factorial with the gamma function, we obtain the following result,

Now, let’s move on to an example. Let’s evaluate the nd derivative of x

We are not going to calculate it in this article, but evaluates to . Hence, we have obtained the result,

Now we can get into more complicated fractional calculus. Beyond the simple level, what is interesting about fractional calculus is that there are many approaches to it that actually give different results. But here, I am going to focus on one of the most popular ones, the Riemann-Liouville fractional calculus.

We start from Cauchy’s formula for n-fold integral,

At this point, Riemann realized that this formula could be directly applied to an integral where n is not an integer number by replacing factorial with the gamma function. Hence,

For instance, if our function is defined as f(x) = sin(x), and we are taking its nd integral, we can evaluate this integral as

Now, let’s get to generalized fractional derivatives. Cauchy’s formula for n-fold integral cannot be directly applied as a derivative for negative values of n. We need an extra step to get there. Hence, we use the following rule,

, where n is selected an integer number to make the entire method meaningful, and n > m. Again, if we define our function as f(x)=sin(x), and we are taking its nd derivative, we can evaluate it as

References:

Baliarsingh, P. “On a Fractional Difference Operator.” Alexandria Engineering Journal, vol. 55, no. 2, 2016, pp. 1811–1816., doi:10.1016/j.aej.2016.03.037.

“Gamma Function.” From Wolfram MathWorld, mathworld.wolfram.com/GammaFunction.html.

Munkhammar, Joakim. “Riemann Liouville Fractional Derivatives and the Taylor Riemann Series.” Upsala University Department of Mathematics, 2004.

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