Sunday Function
mspringer | August 24, 2008If you're a regular reader of this site, you might remember a post about this fascinating specimen from the collection of unusual functions. I'm only showing it on the interval [-1,1] for reasons that will become apparent, but outside that region the growth tapers off rapidly and the function approaches 1 for large negative and large positive x.
It's continuous once you define the point at the origin, and it has continuous derivatives of all orders. Nevertheless, if you try to find the power series about the origin, you see that every term in the series is 0. You can try offsetting the…
mspringer | August 17, 2008You all know what the natural log function looks like. Take the number 1, divide it by the natural log, and then find the antiderivative of that function. You'll get the logarithmic integral function. It looks like this:
Sometimes the lower limit of the integral is changed to 2 instead of 0, in order to get rid of that singularity at the origin. But we're only interested in the behavior at large x so it doesn't matter either way.
It turns out that if you count all the prime numbers up to one million, the answer will be approximately li(1000000). This is helpful because it's a lot…
mspringer | August 10, 2008I trust you're having a relaxing Sunday? Mathematical physics can be relaxing too, especially when you just look at it. We're just going to look at this one. In fact, this is a literal mathematical instantiation of Sunday relaxation.
If you fix a wire or a rope at two points and let it hang naturally, it forms a shape called a catenary.
It looks a lot like a parabola, and it turns out that in fact as long as the suspension points aren't too close together compared to the length of the rope, it's a very good approximation. For this particular graph the error from a purely parabolic…
mspringer | August 3, 2008The Sine Function. Calm and dignified, it sits among the royal court of the Elementary Functions, presiding with undulating grace over the trigonometric functions, partnered with the Exponential Function, and showing forth his power over the realms of physics and mathematics.
On one of the less subtle TV networks, this installment might be called When Good Functions Go Bad.
The sine function is ubiquitous in physics. Figuring out vector components, solving differential equations in E&M and quantum mechanics, decomposing Fourier series, you name it. It's about as well-behaved as…