# Sunday Function

A reader asked me about the hyperbolic trig functions, sinh(x) and cosh(x). What are they for, and do they have an intuitive interpretation in physics? That’s a pretty good question. After all, most of the time you first meet the hyperbolic trig functions in intro calculus, where their rather odd definitions are presented and then…

# Sunday Function

A while back I was driving in my car listening to the radio and was gobsmacked to hear a song (What’s My Name? by Rihanna and Drake) in which the singer’s rap involved accurately estimating a square root. Unfortunately it was in the context of a rather vulgar play on words (“the square root of…

# Sunday Function

Here is a picture of the earth, with latitude and longitude lines in ten-degree increments: Despite the fact that each vaguely rectangular area formed by the intersection of those lines forms a 10 degree by 10 degree region, the actual square-mile area of each rectangle varies. The ones near the poles are smaller than the…

# Sunday Function

In my free time during data acquisition runs and the like, I’ve been paging through Hardy and Wright’s famous textbook An Introduction to the Theory of Numbers. It has something of a legendary reputation among pure math textbooks, and so far as I can tell it is entirely deserved. One of the topics treated in…

# Sunday Function

This Christmas I got a little handheld GPS, which I’ve been using mostly for geocaching. As the device acquires signals from the various satellites dutifully orbiting overhead, it displays your position coordinates and a figure indicating the estimated uncertainty. At the beginning of the acquisition or if the view of the sky is poor, it…

# Sunday Function

In pure mathematics there’s not too many function studied more than the Riemann zeta function. For reasons of historical tradition, the generic variable name that’s usually used is s instead of z. (The function is mostly interesting in terms of complex analysis, so x would be a bit unorthodox too.) It’s defined in the following…

# Sunday Function

Most textbooks, especially ones not aimed at college math majors, give a definition of “function” that seems quite intuitive. They’ll say something along the lines of: a function is a rule that takes an input x and turns it into an output f(x). Formally this isn’t quite right – the essence of a function is…

# Sunday Function

Here’s a straightforward function of two variables, x and y: Its domain is all real x and y, with the single exception of x = y = 0, which would make the denominator 0. But we have experience with functions such as sin(x)/x, where we can find the limiting value as x approaches 0 and…

# Sunday Function

Death and taxes. And dead is dead, but taxes come in a huge panoply of forms. There’s property taxes, excise taxes, sin taxes and income taxes. There’s gas taxes and sales taxes and VAT taxes (yeah, I know) and death taxes. With my politics I’m not a huge fan of any of them, but they…

# Sunday Function

If you go to the bank and open a savings account, the banker might tell you about the virtues of compound interest. He may say something like “Even if you never deposit anything, the rate of change of the money in your account is proportional to the amount of money in your account. The more…