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…
Blog Archives
Castles and quantum mechanics
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.…
Six analysis and probability diagrams
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…
Doubly periodic 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…
College math in a single symbol
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,…
Diagram of special function relationships
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…
Diagram of Bessel function 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…
Diagram of gamma function identities
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…
How to visualize Bessel functions
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…
Two useful asymptotic series
This post will present a couple asymptotic series, explain how to use them, and point to some applications.
Bug in SciPy’s erf function
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,…
Three surprises with bc
bc is a quirky but useful calculator. It is a standard Unix utility and is also available for Windows. One nice feature of bc is that you can set the parameter scale to indicate the desired precision. For example, if…
Math library functions that seem unnecessary
This post will give several examples of functions include in the standard C math library that seem unnecessary at first glance. Function log1p(x) = log(1 + x) The function log1p computes log(1 + x). How hard could this be to…
Stand-alone error function erf(x)
The question came up on StackOverflow this morning how to compute the error function erf(x) in Python. The standard answer for how to compute anything numerical in Python is “Look in SciPy.” However, this person didn’t want to take on…
The gamma function
The gamma function Γ(x) is the most important function not on a calculator. It comes up constantly in math. In some areas, such as probability and statistics, you will see the gamma function more often than other functions that are…
