Chapter 1: Understanding Fractional Derivatives

The concept of derivatives is likely familiar to most high school students. Derivatives measure how a function changes concerning its input, and common notations are widely recognized. We've learned to compute derivatives using the product rule, quotient rule, and chain rule, among other properties.

Have you ever pondered what it means to differentiate a function by half? More specifically, how do we define a derivative of 'order 1/2'? Or even a derivative of 'order 1/n'?

The history of fractional calculus reveals that it was not merely a fleeting idea in modern mathematics; it captivated the minds of many great mathematicians. Imagine Leibniz in a dimly lit library, developing the first ideas of this concept. As time progressed, figures like L'Hôpital, Euler, Lagrange, Laplace, Riemann, Fourier, and Liouville contributed to its evolution, adding complexity to the notion.

Today, we delve into a vast repository of literature dedicated to fractional calculus. Let's start our exploration with the derivative of the exponential function, e^x.

A Bold Approach to Derivatives

Mathematics thrives on patterns and symmetry, forming the foundation of modern mathematical rigor. The derivative of the exponential function is unique in that it remains unchanged when differentiated. Generally, we apply the chain rule for derivatives of the form e^ax.

The first derivative is denoted as D¹, while D² represents the second derivative. This holds true for any integer n, establishing a general form. But what happens when we explore other numbers, like 1/2 or complex values such as 3 + i? Let's take a leap of faith and investigate.

Consider α as a constant, which can be an integer, rational, irrational, or complex. You might wonder what occurs when α = -1 or is any negative number.

Before we dive deeper, let's analyze this. The operator D behaves like a function or operation.

It becomes apparent that D^(-1) corresponds to taking the anti-derivative. We can cautiously conjecture that D^(-n) represents the nth iterated integral, as seen in the context of integrating a function multiple times.

Exploring Sine and Cosine Derivatives

Everyone knows that the derivatives of sine and cosine are closely linked. The derivative of sine is cosine, while the derivative of cosine is negative sine.

Graphing these functions reveals a distinct pattern: each differentiation shifts the graph of sin(x) by π/2 to the left, and the same applies to cos(x).

Here’s a quick exercise: find the derivative of ax. Are you using the chain rule or returning to fundamentals?

From Basics to Gamma Functions

We are familiar with the power rule for differentiation. Can we derive a general expression for D^nx^p?

By multiplying the numerator and denominator by (p-n)!, we can achieve a more streamlined expression.

The natural question is how to substitute the positive integer n with an arbitrary number α. To address this, we can utilize the Gamma function, which Euler introduced in the 18th century to extend the notion of factorials to non-integer values. Its definition is based on a specific integral, which is improper due to its upper limit being infinity.

This property can be deduced by integrating the improper integral when z = z + 1.

With this knowledge, can you see how to reformulate the expression D^nx^p?

Extending Fractional Derivatives

Utilizing the Gamma function, we can extend the definition of fractional derivatives to a broader range of functions. From real analysis, we know any function can be expressed in a Taylor series in terms of powers of x.

Upon term-by-term differentiation, we arrive at a strong candidate for the definition of fractional derivatives.

Addressing Contradictions in Fractional Calculus

However, a contradiction arises. The derivative of the exponential function remains e^x, regardless of how many times it is differentiated.

Comparing this with our earlier expression reveals a discrepancy. The two expressions do not align unless α is a whole number. When α is a whole number, the right-hand side matches the series of e^x. However, when α is a non-integer, we encounter two entirely different functions.

This contradiction has contributed to the scarcity of fractional calculus in elementary textbooks. In classical calculus, where α is an integer, the derivative of an elementary function yields another elementary function. In contrast, fractional calculus deviates from this norm.

Iterated Integrals and Their Definitions

So far, we have focused on repeated derivatives. Similarly, integrals can also be repeated. We define the following with a definite integral.

In multivariable calculus, we can interchange the order of integration. Fubini's Theorem allows us to move functions outside of inner integrals, leading to simplifications.

Try applying the same process for D^-3 and D^-4! You should observe that, in general, substituting -n with an arbitrary α and replacing factorials with the Gamma function leads us closer to our goal.

If α > -1, we encounter an improper integral, as the limit approaches infinity. When -1 < α < 0, the integral converges, validating our expression for negative α values.

Final Thoughts on Fractional Derivatives

We can represent fractional derivatives using the following notation, specifying limits of integration from b to x.

This is the definition of a fractional derivative! Experiment by analyzing various functions, such as e^x and x^p. Share your insights and approaches in the comments!

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Love, Bella ❤️