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Tag: Math

Everywhere chaotic map on the sphere
Let f be a complex-valued function of a complex variable. The Julia set of f is the set of points where f is chaotic. Julia sets are often complicated fractals, but they can be simple. In this post I want to look at the function f(z) = (z² + 1)² …
Quadruple factorials and Legendre functions
Last week I claimed that double, triple, and quadruple factorials came up in applications. The previous post showed how triple factorials come up in solutions to Airy’s equation. This post will show how quadruple factorials come up in solutions to Legendre’s equation. Legendre’s differential equation is The Legendre functions Pν …
Triple factorials and Airy functions
Last week I wrote in a post on multifactorials in which I said that Double factorials come up fairly often, and sometimes triple, quadruple, or higher multifactorials do too. This post gives a couple examples of triple factorials in practice. One example I wrote about last week. Triple factorial comes …
Gauss’s constant
I hadn’t heard of Gauss’s constant until recently. I imagine I’d seen it before and paid no attention. But once I paid attention, I started seeing it more often. There’s a psychology term for this—reticular activation?—like when you buy a green Toyota and suddenly you see green Toyotas everywhere. Our …
Gamma of integer plus one over integer
The gamma function satisfies Γ(x+1) = x Γ(x) and so in principle you could calculate the gamma function for any positive real number if you can calculate it on the interval (0, 1). For example, So if you’re able to compute Γ(π-3) then you could compute Γ(π). If n is …
Multifactorial
The factorial of an integer n is the product of the positive integers up to n. The double factorial of an even (odd) integer n is the product of the positive even (odd) integers up to n. For example, 8!! = 8 × 6 × 4 × 2 and 9!! …
Discrete sum analog of Gaussian integral
A comment on a recent post lead me to a page of series on Wikipedia. The last series on that page caught my eye: It’s a lot more common to see exp(-πx²) inside an integral than inside a sum. If the summation symbol were replaced with an integration sign, the …
The Calculus of Finite Differences
The Calculus of Finite Differences by L. M. Milne-Thompson is a classic. It covers a great deal of elegant and useful [1] mathematics that isn’t widely known, at least not any more. For a taste of the subject matter of the book, see this post. The book is now in …
Multiple angle identify for cotangent
The previous post discusses generalized replicative functions, functions that satisfy Donald Knuth says in Exercise 40 of Section 1.2.4 of TAOCP that the functions cot πx and csc² πx fall into this class of functions. If this is true of cot πx then it follows by differentiation that it is …
Multiplication theorem rabbit hole
When I started blogging I was reluctant to allow comments. It seems this would open the door to a flood of spam. Indeed it does, but nearly all of it can be filtered out automatically. The comments that aren’t spam have generally been high quality. A comment on my post …
Gamma and the Pi function
The gamma function satisfies Γ(n+1) = n! for all non-negative integers n, and extends to an analytic function in the complex plane with the exception of singularities at the non-positive integers [1]. Incidentally, going back to the previous post, this is an example of a theorem that would have to …
Sawtooth and replicative functions
Here’s something I ran across in Volume 2 of Donald Knuth’s The Art of Computer Programming. Knuth defines the sawtooth function by ((x)) = x – (⌊x⌋ + ⌈x⌉)/2. Here’s a plot. This is an interesting way to write the sawtooth function, one that could make it easier to prove …
Estimating normal tail extreme probabilities
In the previous post I said the probability of a normal distribution being 50 standard deviations from its mean was absurdly small, but I didn’t quantify it. You can’t simply fire up something like R and ask for the probability. The actual value is smaller than can be represented by …
Guide to the recent flurry of posts
I wrote six blog posts this weekend, and they’re all related. Here’s how. Friday evening I wrote a blog post about a strange random number generator based on factorials. The next day my electricity went out, and that led me to think how I would have written the factorial RNG …
Trailing zeros of factorials, revisited
I needed to know the number of trailing zeros in n! for this post, and I showed there that the number is Jonathan left a comment in a follow-up post giving a brilliantly simple approximation to the sum above: … you can do extremely well when calculating the number of …
Filling in gaps in a trig table
The previous post shows how you could use linear interpolation to fill in gaps in a table of logarithms. You could do the same for a table of sines and cosines, but there’s a better way. As before, we’ll assume you’re working by hand with just pencil, paper, and a …
Calculating without electricity
A transformer in my neighborhood blew sometime in the night and my electricity was off this morning. I thought about the post I wrote last night and how I could have written it without electricity. Last night’s post included an example that if n = 8675309, n! has 56424131 digits, …
Floor exercises
The previous post explained why the number of trailing zeros in n! is and that the sum is not really infinite because all the terms with index i larger than log5 n are zero. Here ⌊x⌋ is the floor of x, the largest integer no greater than x. The post …