Section 1.1: Understanding Integration and Differentiation

It's common knowledge that integrating or differentiating a sum of functions equates to summing the integrated or differentiated functions. For instance, d/dx (f(x) + g(x)) = d/dx f(x) + d/dx g(x), and the same principle applies to integration.

This connection arises because an integral is fundamentally a limit of a sequence of sums containing progressively more terms. Hence, the order of summation is inconsequential, allowing us to sum the individual values of f and g before combining the results, instead of merely summing the overall expression at the same points.

To put it simply, we can conceptualize an integral as an infinitely fine sum—an accumulation of countless infinitesimal values, which ultimately converges to a real number.

This leads to an intriguing question: why not consider infinitely fine products? The product operation is inherently natural and complements addition effectively. But how would we define such products? Instead of relying on foundational principles, we can derive a definition based on the expected behavior of a product.

Specifically, we recognize that the logarithm transforms products into sums, implying that it should similarly convert a fine-grained product into a fine-grained sum. Therefore, the logarithm of our multiplicative integral should correspond to the normal integral of the logarithm.

This insight provides us with the definition we seek, denoting the indefinite multiplicative integral with a capital P as follows. It's important to note that the differential dx appears in the exponent—a deliberate choice that will become clear as we progress. The logarithm thus seamlessly transforms P into ∫ with the natural logarithm applied to the integrand, and the differential dx contributes to the multiplication due to a logarithmic property.

We can further refine the multiplicative integral by establishing definite limits of integration, denoted as follows. It is crucial to ensure that the integral on the right side is well-defined and convergent. When it is, we have a reliable operator that adheres to the logarithmic transformation outlined earlier.

Section 1.2: Defining Multiplicative Differentiation

The initial premise of our discussion revolves around the desired multiplicative properties of this operation. Indeed, we find that the operator behaves multiplicatively, a result that aligns beautifully with the exponent in the multiplicative integral.

Before we dive into practical examples, let's define the inverse operation known as multiplicative differentiation. The definition is straightforward:

This operator serves as the inverse to the multiplicative integral, as you can verify.

Section 2.1: The Product Rule and Chain Rule

There exists an equivalent to the product rule in this context. In the additive world, a product corresponds to exponentiation in the multiplicative realm. Thus, when we perform multiplicative integration of a function raised to another function, we derive a rule that we could refer to as the exponent rule. Here, the sum translates into a product, yet we retain one of the functions in each term.

We also possess a chain rule where the outer function undergoes multiplicative differentiation while the inner function is differentiated normally, owing to its position in the exponent. Experimenting with these formulas will soon make them second nature.

Moreover, we have an analog to integration by parts. By applying the exponent rule to (Pf)^g and taking multiplicative integrals of both sides, we arrive at a multiplicative integration by parts formula. Notably, the two operations cancel each other, forming an exact equivalent to the traditional integration by parts formula, where products morph into exponentiations and divisions replace subtractions.

Let's apply this formula to calculate the multiplicative integral of the well-known exponential function. By leveraging the fact that the multiplicative integral of a constant C corresponds to the exponential function D ⋅ C^x (where D is another constant), we find that it is indeed simpler to use the definition directly.

The essence of calculus is beautifully illustrated here, revealing the fundamental principles that underpin these operations.

Section 2.2: The Product Formula

Consider a product across a subset of natural numbers A, which could include prime numbers, even numbers, or the natural numbers themselves. To facilitate our work with this product, we define a counting function for set A as follows:

This function counts how many elements of A are less than x, noting that x may be a real number, not just a natural number. For instance, if x < 1, the counting function is defined as 0.

A well-known example is the prime counting function, denoted π(x). The counting function for natural numbers corresponds to the floor function, denoted [x], which rounds down to the nearest integer less than or equal to x.

Using this framework, we can compute products over arbitrary subsets of natural numbers. It would be advantageous to rewrite the fraction in the product to manipulate the function f differently. Fortunately, akin to the fundamental theorem of calculus, we have a corresponding theorem for multiplicative calculus:

This theorem allows us to express our product in the following manner:

To achieve the correct fraction factor, we must include the -1 in the exponent.

At this stage, something remarkable occurs. It turns out that the multiplicative integral of a function raised to a constant power is equivalent to raising the multiplicative integral of that function to the power of the constant. This establishes a distributive law for multiplicative calculus, similar to the conventional integral.

We can now apply this knowledge since, within the interval from n to n+1, the counting function remains constant.

Consequently, we find:

This leads us to the Abel multiplication formula!

The relationship between this and the additive equivalent is striking—it's a one-to-one correspondence where sums transition to products, products to exponentiations, subtractions to divisions, and differentiation to multiplicative differentiation, while integration evolves into multiplicative integration.

This connection is truly captivating.

We can now utilize this formula to estimate the growth of factorials. Notably, the counting function for natural numbers is the floor function [x], and evaluating it at a natural number gives us the same number, i.e., [m] = m for m ∈ ℕ.

Furthermore, we note that [x] = x - {x}, where {x} represents the fractional part function, yielding the decimals of x (e.g., {3.14} = 0.14). This fractional part function is periodic with a period of 1, leading to a useful Fourier series.

Employing the product formula on the function f(x) = x, we derive the factorial on the left-hand side. Remarkably, we conclude that the factorial grows asymptotically equal to the expression on the right for some constant C. A more thorough analysis reveals that this constant C is the square root of 2π, often referred to as Stirling's approximation.

All in all, this represents a fascinating new dimension in mathematics, one I am eager to continue developing in private. If you wish to engage in a discussion—perhaps a more technical one—or if anything remains unclear, please feel free to reach out via LinkedIn.

Developing a New Type of Calculus explores these innovative concepts further, providing deeper insights into this intriguing area of study.