Let J(x) be the function plotted below.

This is the Bessel function J₁, but we drop the subscript because it’s the only Bessel function we’re interested in for this post. You can think of J as a sort of damped sine.

We can create versions of J with different frequencies by multiplying the argument x by different constants. Suppose we create versions with three different frequencies — J(ax), J(bx), and J(cx) — and integrate their product. This defines a function f of the frequencies.

We can evaluate the integral defining f using Sonine’s formula [1]

where Δ(a, b, c) is the area of a triangle with sides a, b, c, if such a triangle exists, and zero otherwise.

It’s amazing that this formula takes three parameters with no intrinsic geometric meaning and out pops the area of a triangle with such sides.

Numerical (mis)application

It would be ill-advised, but possible, to invert Sonine’s formula and use it to find the area of a triangle; Heron’s formula would be a far better choice. But just for fun, we’ll take the ill-advised route.

from numpy import linspace, pi, sqrt, inf
from scipy.special import j1
from scipy.integrate import quad

def heron(a, b, c):
    s = (a + b + c)/2
    return sqrt(s*(s-a)*(s-b)*(s-c))

def g(a, b, c):
    integrand = lambda x: j1(a*x) * j1(b*x) * j1(c*x)
    i, _ = quad(integrand, 0, inf, limit=500)
    return i

def area(a, b, c):
    return g(a, b, c)*pi*a*b*c/2

print(area(3,4,5), heron(3,4,5))

SciPy’s quad function has difficulty with the integration, and rightfully issues a warning. The code increases the limit parameter from the default value of 50 to 500, improving the accuracy but not eliminating the warning. The area function computes the error of a 3-4-5 triangle to be 5.9984 and the heron function computes the exact value 6.

Update: I tried the numerical integration above in Mathematica and it correctly computed the integral to 10 decimal places with no help. I suspect it is detecting the oscillatory nature of the integral and using Levin’s integration method; when I explicit set the integration to be Levin’s method, I get the same result.

Impossible triangles

You could calculate the area of a triangle from Heron’s theorem or from Sonine’s theorem. The results are identical when the parameters a, b, and c form a valid triangle. But the two approaches diverge when a, b, and c do not form a triangle. If you pass in parameters like (3, 1, 1) then Heron’s theorem gives a complex number and Sonine’s theorem yields zero.

Related posts

• Law of cotangents
• Bessel zero spacing
• Bessell determinants

[1] Discovered by Nikolay Yakovlevich Sonin (1849–1915). The formula is usually referred to as Sonine’s formula rather than Sonin’s formula, as far as I’ve seen. This variation is minor compared to what transliteration does to other Russian names.

Sonine’s formula is more general, extending to Bessel functions J_ν with Re(ν) > ½. I chose ν = 1 for this post because the Sonin’s formula is simplest in this case.

Sometimes it’s useful to apply dimensional analysis where it doesn’t belong, to imagine things having physical dimension when they don’t. This post will look at artificially injecting dimensions into equations involving factorials and related functions.

Factorials

The factorial of n is defined as the product of n terms. If each of these terms had units of length, the factorial would have units of n-dimensional volume. It occurred to me this morning that a lot of identities involving factorials make sense dimensionally if you pretend the terms in the identities have units. The expressions on both sides of an equation have the same dimension, and only terms of the same dimension are added together. This isn’t always the case, so caveat emptor.

We could also think of an nth rising power or nth falling power as an n-dimensional volume. If we do, then the dimensions in a Newton series cancel out, for example.

Dimensional analysis for factorials and related functions could make it easier to remember identities, and easier to spot errors. And it could suggest the correct form of a result before the details of the result are filled in. In other words, artificial dimensional analysis can provide the same benefits of physically meaningful dimensional analysis.

Gamma function

For integers n, Γ(n) = (n − 1)!, and so we could assign dimension n − 1 to Γ(n), and more generally assign dimension z − 1 to Γ(z) for any complex z.

Now for some examples to show this isn’t as crazy as it sounds. For starters, take the identity Γ(z + 1) = z Γ(z). We imagine the left hand side is a z-dimensional volume, and the right hand side is the product of a term with unit length and a term that represents a (z − 1) dimensional volume, so the units check out.

Next let’s take something more complicated, the Legendre duplication formula:

The left hand side has dimension (z − 1) + (z − ½) = 2z − ½. The right hand side has dimension 2z − 1, apparently contradicting our dimensional analysis scheme. But √π = Γ(½), and if we rewrite √π as Γ(½) in the equation above, both sides have dimension 2z − ½. The dimensions in other identities, like the reflection formula, also balance when you replace π with Γ(½)².

Hypergeometric functions

The hypergeometric function F(a, b; c; z) is defined by its power series representation. If we assign dimensions to the coefficients in the series as we’ve done here, then the numerator and denominator of each term have the same dimensions, and so the hypergeometric function should be dimensionless in our sense. This implies we should expect that identities for hypergeometric functions to be dimensionless as well. And indeed they are. I’ll give two examples.

First, let’s look at Gauss’ summation identity

provided the real part of c is greater than the real part of a + b. Notice that the numerator and denominator both have dimension 2c − a − b − 2.

Next, let’s look at Kummer’s identity

Both the numerator and denominator have dimension 3a/2 − b.

Finally, let’s look at a more complicated formula.

In both the terms involving gamma functions, the dimensions of the numerator and denominator cancel out.