Generalize the Factorial to Real Numbers (from Scratch)

The video begins by reviewing factorials for natural numbers (e.g., 3! = 3*2*1 = 6). It introduces the problem of calculating the factorial of a non-natural number like 1/2, explaining that the gamma function can generalize the factorial to include such values. The gamma function is initially presented as a special definite integral, with a note that it's offset by one compared to the traditional factorial (Γ(x) = (x-1)!).

The core problem is defined: to generalize the factorial from natural numbers to real numbers. This means finding a function g(x) that passes through the discrete points of natural number factorials and extends smoothly. The key property of factorials, n! = n * (n-1)!, is highlighted as the starting point for finding such a function g that satisfies g(n) = n * g(n-1).

The video draws a parallel between the factorial property g(n) = n * g(n-1) and the differentiation power rule (d/dx x^n = n * x^(n-1)). It identifies the need to balance the 'n' and 'n-1' terms in a differentiation context. The product rule is then introduced as a way to create this balance, particularly when one term is differentiated.

To address the imbalance in differentiation, the exponential function e^(-x) is introduced due to its property of keeping its form when differentiated (with a sign change). By setting v = e^(-x) in the product rule, and u = x^n, a new equation emerges that treats x^n and x^(n-1) more equally within the differentiated context. This leads to defining a new function gn(x) = x^n * e^(-x).

The next step involves using definite integrals to eliminate the variable 'x' and any extra terms. By integrating both sides of the derived equation for gn(x) from 0 to infinity, the extra term becomes zero, thanks to the behavior of gn(x) at x=0 and as x approaches infinity. This crucial step results in a new relationship: G(n) = n * G(n-1), where G(n) is the integral of gn(x) from 0 to infinity.

By defining G(n) = ∫[0 to ∞] x^n * e^(-x) dx, the video shows that this function satisfies the factorial property G(n) = n * G(n-1) and G(0) = 1. This integral definition is then generalized by replacing 'n' with a real number 'x', leading directly to the gamma function, Γ(x+1) = ∫[0 to ∞] t^x * e^(-t) dt. The video highlights that this function extends the factorial for x > -1 and is offset by one compared to the usual factorial notation (Γ(x) = (x-1)!).

The process followed in the video is presented as a 'thought experiment' to understand how one might derive the gamma function from first principles, rather than its historical discovery. It emphasizes that while integration by parts can quickly verify the relationship, the derivation from scratch helps interpret the meaning of the integral. The video concludes by acknowledging that the gamma function is often introduced as a definition and this exploration provides a valuable interpretive journey.