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 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 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…

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…

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…

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…

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.…

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. Special functions Gamma and related functions Probability distributions Conjugate…

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…

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,…

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 function relationships Gamma function identities Probability distribution relationships…

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…

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…

Bessel functions are sometimes called cylindrical functions because they arise naturally from physical problems stated in cylindrical coordinates. Bessel functions have a long list of special properties that make them convenient to use. But because so much is known about…

This post will present a couple asymptotic series, explain how to use them, and point to some applications.

Last night I produced the plot below and was very surprised at the jagged spike. I knew the curve should be smooth and strictly increasing. My first thought was that there must be a numerical accuracy problem in my code,…