This week’s resource post: some pages of notes on special functions:
See also blog posts tagged special functions.
I tweet about special functions occasionally on AnalysisFact
Last week: HTML, LaTeX, and Unicode
Next week: Python resources
When I interviewed Daniel Spielman at this year’s Heidelberg Laureate Forum, we began our conversation by looking for common mathematical ground. The first thing that came up was orthogonal polynomials. (If you’re wondering what it means for two polynomials to be orthogonal, see here.)
JC: Orthogonal polynomials are kind of a lost art, a topic that was common knowledge among mathematicians maybe 50 or 100 years ago and now they’re obscure.
DS: The first course I taught I spent a few lectures on orthogonal polynomials because they kept coming up as the solutions to problems in different areas that I cared about. Chebyshev polynomials come up in understanding solving systems of linear equations, such as if you want to understand how the conjugate gradient method behaves. The analysis of error correcting codes and sphere packing has a lot of orthogonal polynomials in it. They came up in a course in multi-linear algebra I had in grad school. And they come up in matching polynomials of graphs, which is something people don’t study much anymore. … They’re coming back. They come up a lot in random matrix theory. … There are certain things that come up again and again and again so you got to know what they are.
More from my interview with Daniel Spielman:
The Riemann zeta function, introduced by Leonard Euler, is defined by
where the sum is over all positive integers n.
Euler also introduced a multivariate generalization of the zeta function
where the sum is over all decreasing k-tuples of positive integers. This generalized zeta function satisfies the following beautiful identity:
The multivariate zeta function and identities such as the one above are important in number theory and are the subject of open conjectures.
The Airy functions Ai(x) and Bi(x) are independent solutions to the differential equation
For negative x they act something like sin(x) and cos(x). For positive x they act something like exp(x) and exp(-x). This isn’t surprising if you look at the differential equation. If you replace x with a negative constant, you get sines and cosines, and if you replace it with a positive constant, you get positive and negative exponentials.
The Airy functions can be related to Bessel functions as follows:
Here J is a “Bessel function of the first kind” and I is a “modified Bessel function of the first kind.” Also
To verify the equations above, and to show how to compute these functions in Python, here’s some code.
The SciPy function
airy computes both functions, and their first derivatives, at once. I assume that’s because it doesn’t take much longer to compute all four functions than to compute one. The code for
Bi2 below uses
np.where instead of
if ... else so that it can operate on NumPy vectors all at once. You can plot
Ai2 and see that the two curves lie on top of each other. The same holds for
from scipy.special import airy, jv, iv from numpy import sqrt, where def Ai(x): (ai, ai_prime, bi, bi_prime) = airy(x) return ai def Bi(x): (ai, ai_prime, bi, bi_prime) = airy(x) return bi def Ai2(x): third = 1.0/3.0 hatx = 2*third*(abs(x))**1.5 return where(x > 0, third*sqrt( x)*(iv(-third, hatx) - iv(third, hatx)), third*sqrt(-x)*(jv(-third, hatx) + jv(third, hatx))) def Bi2(x): third = 1.0/3.0 hatx = 2*third*(abs(x))**1.5 return where(x > 0, sqrt( x/3.0)*(iv(-third, hatx) + iv(third, hatx)), sqrt(-x/3.0)*(jv(-third, hatx) - jv(third, hatx)))
There is a problem with
Bi2: they return
nan at 0. A more careful implementation would avoid this problem, but that’s not necessary since these functions are only for illustration. In practice, you’d simply use
airy and it does the right thing at 0.
The famous Fibonacci numbers are defined by the initial conditions
F0 = 0, F1 = 1
and the recurrence relation
Fn = Fn-1 + Fn-2
for n > 1.
The Fibonacci polynomials are defined similarly. The have the same initial conditions
F0(x) = 0, F1(x) = 1
but a slightly different recurrence relation
Fn(x) = x Fn-1(x) + Fn-2(x)
for n > 1.
Several families of orthogonal polynomials satisfy a similar recurrence relationship
Fn(x) = p(x) Fn-1(x) + c(n) Fn-2(x).
The table below gives the values of p(x) and c(n) for a few families of polynomials.
|Hermite||2x||2 – 2n||1||2x|
There are two common definitions of the Hermite polynomials, sometimes called the physicist convention and the probabilist convention. The table above gives the physicist convention. For the probabilist definition, change the 2’s to 1’s in the last row and leave the 1 alone.
(If you haven’t heard of orthogonal polynomials, you might be wondering “orthogonal to what?”. These polynomials are orthogonal in the sense of an inner product formed by multiplying two polynomials and a weight, then integrating. Orthogonal polynomials differ by the the weight and the limits of integration. For example, the (physicist) Hermite polynomials are orthogonal with weight function exp(-x2) and integrating over the entire real line. The probabilist Hermite polynomials are very similar, but use the standard normal distribution density exp(-x2/2)/√(2π) as the weight instead of exp(-x2).)
The Lambert W-function is the function w(z) implicitly defined by w exp(w) = z. When I first saw this, I thought that I’d seen something like this come up many times and that it would be really handy to know that this function has a name and has many software implementations1. Since then, however, I’ve hardly ever run into the W-function in application. But here’s an application I ran into recently.
A “giant component” in a random network is a component whose size grows in proportion to the size of the network. For a Poisson random network, let c = p(n – 1) be the expected number of vertices connected to any given vertex. If c > 1 then as n goes to infinity, there will be a giant component with probability 1. S, the proportion of nodes inside the a giant component, satisfies the equation
S = 1 – exp(-cS).
S plays a role similar to that of w in the definition of the W-function, which suggests the W-function might come in handy. And in fact, this book gives this solution for S:
S = 1 + w( -c exp(-c) )/c.
There are a couple issues. First, it’s not obvious that solution is correct. Second, we need to be more careful about how we define w(z) when z is negative.
Given some value of c, let S = 1 + w( -c exp(-c) )/c. Then
w( -c exp(-c) ) = -c(1 – S).
Apply the function f(x) = x exp( x ) to both sides of the equation above. By the definition of the W-function, we have
-c exp(-c) = -c(1 – S) exp( -c(1 – S) ).
From there it easily follows that
S = 1 – exp(-cS).
Now let’s look more carefully at the definition of the W-function. For positive real z, w exp(w) = z has a unique real solution defining w. But in the calculations above, we have evaluated w at -c exp(-c), which is negative since c > 1. For negative arguments, we have to pick a branch of w. Which branch guarantees that S = 1 – exp(-cS)? Any branch2. No matter which solution to w exp(w) = z we pick, the resulting S satisfies the giant component equation. However, the wrong branch might result in a meaningless solution. Since S is a proportion, S should be between 0 and 1.
Let’s go back and say that for our purposes, we will take w(z) to mean the real number no less than -1 satisfying w exp(w) = z. Then for c > 1, the solution S = 1 + w( -c exp(-c) )/c is between 0 and 1. You can show that S = 1 – exp(-cS) has a unique solution for 0 < S < 1, so any other branch would not give a solution in this interval.
Related link: Lambert W poster
1 For example, in Python the Lambert W-function is
scipy.special.lambertw. The function takes two arguments: z and an integer denoting the branch. By default, the branch argument is set to 0, giving the branch discussed in this post.
2 For -1/e < z < 0, there are two real branches of w(z), but there are also infinitely many complex branches.
When trying to understand a complex formula, it helps to first ask what is being related before asking how they are related.
This post will look at addition theorems for Bessel functions. They related the values of Bessel functions at two points to the values at a third point.
Let a, b, and c be lengths of the sides of a triangle and let θ be the angle between the sides of length a and b. The triangle could be degenerate, i.e. θ could be π. The addition theorems here give the value of Bessel functions at c in terms of values of Bessel functions evaluated at b and c.
The first addition theorem says that
J0(c) = ∑ Jm(a) Jm(b) exp(imθ)
where the sum is over all integer values of m.
This says that the value of the Bessel function J0 at c is determined by a sort of inner product of values of Bessel functions of all integer orders at a and b. It’s not exactly an inner product because it is not positive definite. (Sometimes you’ll see the formula above written with a factor λ multiplying a, b, and c. This is because you can scale every side of a triangle by the same amount and not change the angles.)
Define the vectors x, y, and z to be the values of all Bessel functions evaluated at a, b, and c respectively. That is, for integer k, xk = Jk(a) and similar for y and z. Also, define the vector w by wk = exp(ikθ). Then the first addition theorem says
z0 = ∑ xm ym wm.
This is a little unsatisfying because it relates the value of one particular Bessel function at c to the values of all Bessel functions at a and b. We’d like to relate all Bessel function values at c to the values at a and b. That is, we’d like to relate the whole vector z to the vectors x and y.
Define the vector v by vk = exp(inψ) where ψ is the angle between the sides of length b and c. Then the formula we’re looking for is
zn = v-n ∑ xn+m ym wm.
Igor Carron commented on his blog that
… the mathematical tools that we will use in the next 20 years are for the most part probably in our hands already.
He compares this to progress in treating leukemia: survival rates increased dramatically over the last 40 years, not primarily by developing new drugs, but by better applying drugs that were available in 1970.
I find that plausible. I’ve gotten a lot of mileage out of math that was well known 100 years ago.
Related post: Doing good work with bad tools
How are castles and quantum mechanics related? One connection is rook polynomials.
The rook is the chess piece that looks like a castle, and used to be called a castle. It can move vertically or horizontally, any number of spaces.
A rook polynomial is a polynomial whose coefficients give the number of ways rooks can be arranged on a chess board without attacking each other. The coefficient of xk in the polynomial Rm,n(x) is the number of ways you can arrange k rooks on an m by n chessboard such that no two rooks are in the same row or column.
The rook polynomials are related to the Laguerre polynomials by
Rm,n(x) = n! xn Lnm-n(-1/x)
where Lnk(x) is an “associated Laguerre polynomial.” These polynomials satisfy Laguerre’s differential equation
x y” + (n+1-x) y‘ + k y = 0,
an equation that comes up in numerous contexts in physics. In quantum mechanics, these polynomials arise in the solution of the Schrödinger equation for the hydrogen atom.
Related: Relations between special functions
Here are a few diagrams I’ve created that summarize relationships in analysis and probability. Click on a thumbnail image to go to a page with the full image and explanatory text.
Gamma and related functions
Functions like sine and cosine are periodic. For example, sin(x + 2πn) = sin(x) for all x and any integer n, and so the period of sine is 2π. But what happens if you look at sine or cosine as functions of a complex variable? They’re still periodic if you shift left or right, but not if you shift up or down. If you move up or down, i.e. in a pure imaginary direction, sines and cosines become unbounded.
Doubly periodic functions are periodic in two directions. Formally, a function f(z) of complex variable is doubly periodic if there exist two constants ω1 and ω2 such that
f(z) = f(z + ω1) = f(z + ω2)
for all z. The ratio ω1 / ω2 cannot be real; otherwise the two periods point in the same direction. For the rest of this post, I’ll assume ω1 = 1 and ω2 = i. Such functions are periodic in the horizontal (real-axis) and vertical (imaginary-axis) directions. They repeat everywhere their behavior on the unit square.
What do doubly periodic functions look like? It depends on what restrictions we place on the functions. When we’re working with complex functions, we’re typically interested in functions that are analytic, i.e. differentiable as complex functions.
Only constant functions can be doubly periodic and analytic everywhere. Why? Our functions take on over and over the values they take on over the unit square. If a function is analytic over the (closed) unit square then it’s bounded over that square, and since it’s doubly periodic, it’s bounded everywhere. By Liouville’s theorem, the only bounded analytic functions are constants.
This says that to find interesting doubly periodic functions, we’ll have to relax our requirements. Instead of requiring functions to be analytic everywhere, we will require them to be analytic except at isolated singularities. That is, the functions are allowed to blow up at a finite number of points. There’s a rich set of such functions, known as elliptic functions.
There are two well-known families of elliptic functions. One is the Weierstrass ℘ function (TeX symbol
wp, Unicode U+2118) and its derivatives. The other is the Jacobi functions sn, cn, and dn. These functions have names resembling familiar trig functions because the Jacobi functions are in some ways analogous to trig functions.
It turns out that all elliptic functions can be written as combinations either of the Weierstrass function and its derivative or combinations of Jacobi functions. Roughly speaking, Weierstrass functions are easier to work with theoretically and Jacobi functions are easier to work with numerically.
From Prelude to Mathematics by W. W. Sawyer (1955):
There must be many universities today where 95 percent, if not 100 percent, of the functions studied by physics, engineering, and even mathematics students, are covered by the single symbol F(a, b; c; x).
The symbol Sawyer refers to is the hypergeometric function. (There are hypergeometric functions with any number of parameters, but the case with three parameters is so common that it is often called “the” hypergeometric function.) The most commonly used functions in application — trig functions, exp, log, the error function, Bessel functions, etc. — are either hypergeometric functions or closely related to hypergeometric functions. Sawyer continues:
I do not wish to imply that the hypergeometric function is the only function about which mathematics knows anything. That is far from being true. … but the valley inhabited by schoolboys, by engineers, by physicists, and by students of elementary mathematics, is the valley of the Hypergeometric Function, and its boundaries are (but for one or two small clefts explored by pioneers) virgin rock.
Here’s a first pass at a diagram showing the relationships between special functions that commonly come up in applications.
These notes explain what everything on the diagram means.
Other math diagrams
Bessel functions are interesting and useful, but they’re surrounded by arcane terminology. It may seem that there are endless varieties of Bessel functions, but there are not that many variations and they are simply related to each other.
Each letter in the diagram is taken from the traditional notation for a class of Bessel functions. Functions in the same row satisfy the same differential equation. Functions in the same column have something in common in their naming.
The lines represent simple relations between functions. There’s one line conspicuously missing from the diagram, and that’s no accident.
These notes give the details behind the diagram.
Related post: Visualizing Bessel functions
The gamma function has a large number of identities relating its values at one point to values at other points. By focusing on just the function arguments and not the details of the relationships, a simple pattern emerges. Most of the identities can be derived from just four fundamental identities:
The same holds for functions derived from the gamma function: log gamma, digamma, trigramma, etc.
For the details of the relationships, see Identities for gamma and related functions.