Exploring The Fascinating World Of Factorials

Factorials are a cornerstone in mathematics, playing a pivotal role in areas ranging from combinatorics to probability theory and statistical analysis. In this post, we delve into the classical definition of the factorial, introduce the concept of the left factorial as proposed by Serbian mathematician Djuro Kurepa, and explore how factorials can be extended to non-integer values using an integral definition.

The Classical Definition of Factorial

It is customary to define the factorial as:

n! = ∏[k = 1 to n] k, n ∈ N

This product of natural numbers forms the backbone of many mathematical formulas and concepts. The factorial function grows extremely fast and is instrumental in counting problems, series expansions, and probability distributions.

Introducing the Left Factorial

The Serbian mathematician Djuro Kurepa introduced the concept of the left factorial, defined as:

!n = 0! + 1! + 2! + 3! + ⋯ + (n – 1)!

Although 0 is not included in the set of natural numbers N, we define 0! as 1. This might appear to be an axiom at first glance, but its justification comes from a lesser-known definition involving an integral representation of the factorial.

An Alternative Integral Definition of Factorial

Beyond the familiar product formula, the factorial can also be defined through an integral:

n! = ∫[0 to ∞] (x^n) [e^(-x)] dx

Accordingly, for n = 0 we have:

0! = ∫[0 to ∞] (x^0) [e^(-x)] dx = ∫[0 to ∞] [e^(-x)] dx

The solution to this integral is -e^(-x) in the range from 0 to ∞ (or, which is the same, e^(-x) in the range from ∞ to 0). It is easy to conclude that this value is 1.

Factorials Beyond the Natural Numbers

Because the integral definition is not restricted to natural numbers, the factorial can, in theory, be extended to any real (or even complex) number. For example, using a scientific calculator, it can be seen that:

0.5! ≈ 0.8862 (which is (√π) /2)

and

(−0.5)! ≈ 1.7724 (which is √π)

These results imply that the factorial function is a much richer concept than simply the product of a sequence of integers.

Evaluating (1/2)! Through Integration

For n = 1/2​, the factorial is defined by the integral:

(1/2)! = ∫[0 to ∞] [x^(1/2)] [e^(-x)] dx

The Method of Integration by Parts

To evaluate this integral, we use the integration by parts. Recall the derivative of a product of two functions u and v:

(uv)´ = u´v + uv´

Thus, the total differential is:

d(uv) = vdu + udv

Integrating both sides gives:

∫d(uv) = ∫vdu + ∫udv

or equivalently,

uv = ∫vdu + ∫udv

which leads to the well-known formula:

∫udv = uv – ∫vdu

Implementing the Substitution

Introduce the substitution:

• Let u = x^(1/2) so that du = (1/2) x^(-1/2) dx
• Let e^(-x) dx = dv so that v = -e^(-x)

Then the integral becomes:

∫[x^(1/2)] [e^(-x) dx = [-x^(1/2)] [e^(-x)] + (1/2) ∫[x^(-1/2)] [e^(-x)] dx

The first term vanishes in the limits from 0 to ∞, and the remaining integral can be simplified further by substituting t = x^(1/2) (so dt = (1/2 x)^(-1/2) dx), which means that t^2 = x. This substitution transforms the integral into:

∫[0 to ∞] [e^(-t^2)] dt

Connecting to the Gaussian Integral

The integral:

∫[0 to ∞] [e^(-t^2)] dt

is a classic example known as the Gaussian integral. There are two insightful ways to evaluate it:

Approach 1: Squaring the Integral and Solving Using the Polar Coordinate System

Consider:

I^2 = ∫[0 to ∞] [e^(-x^2)] dx ∫[0 to ∞] [e^(-x^2)] dx = ∫[0 to ∞] [e^(-x^2)] dx ∫[0 to ∞] [e^(-y^2)] dy = ∬[0 to ∞] e^[-(x^2 + y^2)] dxdy

The square of the distance of any point from the coordinate origin is r^2 = x^2 + y^2. The product dxdy represents the infinitesimal area of ​​the rectangle with sides dx and dy. By introducing polar coordinates, one side of this rectangle is dr, and the other is the length of the arc of the central angle dθ (θ is the angle that the positive direction of the x-axis subtends with the line connecting the coordinate origin and the observed point). This means that the length of the second side is:

(2rπ / 2π) dθ = rdθ

With this in mind, we get:

I^2 = ∫[0 to ∞] r e^(-r^2) dr ∫[0 to (π/2)] dθ

By substituting r^2 = p (which gives 2rdr = dp or rdr = (1/2) dp), we find:

∫[0 to ∞] r e^(-r^2) dr = ∫[0 to ∞] [e^(-p)] dp = 1/2

It is clear that:

∫[0 to (π/2)] dθ = π/2

so:

I^2 = π/4

or:

I = ∫[0 to ∞] [e^(-t^2)] dt = (√π) / 2

Approach 2: Using the Gaussian Probability Density Function

The Gaussian probability density function is given by:

f(x) = [1/(σ√2)π] e^[(-x^2) / (2σ^2)]

with the normalization condition:

∫[-∞ to ∞] f(x) dx = 1

This implies:

∫[-∞ to ∞] e^[(-x^2) / (2σ^2)] dx = (σ√2)π = (√π)σ√2

So, the solution to the integral is √π multiplied by the square root of the denominator of the exponent of the integrand. Based on analogy, and taking into account that the Gaussian probability density function is an even function (it is symmetric with respect to the ordinate y-axis which means that f(x) = f(-x)), we again conclude that the solution ∫[0 to ∞] [e^(-t^2)] dt is (√π) / 2.

This means that, definitely:

(1/2)! = (√π) / 2

Linking Factorials and Their Properties

One of the well-known properties of the factorial is:

(n + 1)! = (n + 1) n!

For n = -1/2​, this property implies:

(1/2)! = (1/2) (-1/2)!

so:

(-1/2)! = 2 (1/2)!

which means:

(-1/2)! = 2 (√π) / 2 = √π

This result is a beautiful illustration of how the factorial function, when extended via the integral definition, maintains its consistency and opens up connections to other areas of mathematics.

Applications in Probability and Statistics

Calculating the factorial is far from being an abstract exercise – it is essential in studying mathematical probability and statistics. These concepts are crucial when analyzing and processing random signals such as speech, music, images, or noise.

Conclusion

The exploration from the traditional definition of factorial to the concept of left factorial – and further into the realm of non-integer factorials – reveals the depth and versatility of this fundamental mathematical function. Whether defined by a product or an integral, the factorial function interconnects with various branches of mathematics, offering profound insights into probability, statistics, and beyond.

By understanding methods such as integration by parts and appreciating the elegance of the Gaussian integral, we not only expand our mathematical toolkit but also appreciate the continuous evolution and interconnectivity of mathematical concepts.