Built on Facts

You know about the tangent function, tan(x). If you draw a right triangle, the tangent is the ratio of the two sides which comprise the right angle. If the angle between the long sides and each of the short sides is 45 degrees, then clearly the two short sides have the same length. That means the tangent of 45 degrees is 1.

Now the tangent function has an inverse, which we call the arctangent. It’s our Sunday Function, and it looks like this:

In mathematics we never use degrees as a measurement of angle, we use radians. There’s 2π radians per 360 degrees, and so a 45 degree angle is equal to π/4 radians. And since the tangent of 45 degrees is 1, the arctangent of 1 must be 45 degrees. In radians, arctan(1) = π/4.

Like all analytic functions, the arctangent function can be expanded in terms of a power series, which in this case takes the following form:

Plug into that the fact that with x = 1, arctan(1) = π/4 and we get:

So, if you add up the reciprocals of the odd integers with negative signs inserted appropriately and multiply the result by 4, you’ll get pi. Clearly you can’t actually add up an infinite number of fractions, but the more you include the closer you’ll get. It’s an interesting example of pi cropping up some place in mathematics that has no clear relationship to circles at all. It happens a truly surprising number of times.

Incidentally, you wouldn’t want to use this series to compute a large number of digits of pi. It closes in on pi very slowly and so you have to calculate the series out to a tremendous distance in order to get reasonable accuracy. In the real world there’s more complicated but much more quickly converging series that are used to actually find millions of digits worth of pi.

Comments

#1 jr
November 2, 2008

“So, if you add up the reciprocals of the off integers”

What do you get if you take the on integers?
#2 Max Fagin
November 2, 2008

“There’s 2? radians per 260 degrees”

Matt: Your powers of physics are great, but I’m sorry, you are not yet powerful enough to redefine Pi away from it’s usual 2? per 360 degrees

Only the aliens in ‘Contact’ can do that, and I am pretty sure they don’t write this blog.
#3 Odysseus
November 2, 2008

This isn’t exactly meant to be a redefinition of pi, but a new interpretation of ‘circle’, of course.

Actually, for a long time the arctan was used for a very clever approximation of pi, as pi/4 happens to equal 4*arctan(1/5)-arctan(1/239), which gives a fairly quickly converging series.

You omitted one of the arctan’s most interesting properties, which is that its derivative is 1/(1+x^2). This feature is not just beautiful, but also extremely helpful when it comes to integrating some classes of rational functions.
#4 Matt Springer
November 2, 2008

Ok, ok, I admit I wrote this late at night and included a few typos. They’re fixed (well, at least those two are) now!

As for the other very interesting properties of arctan, generally Sunday Function just talks about one interesting property of the function. There’s just not space or time for me to do otherwise. But don’t despair, I plan on revisiting functions on many occasions, as their interesting properties are inexhaustible.
#5 eddie
November 2, 2008

While we’re on the subject of sub-editing, I think your last sentence might have meant to read:

In the real world there’s more complicated but much more quickly converging series…

That’s one that get’s past the spell check.

Back to the science. For large values (+ve or -ve), the graph converges on (+/-)pi/2 as the tangent of that goes to (+/-)infinity. But every cycle the tan function goes to that.

I have been interested in these multi-valued functions since using the intersection of a tangent function with another curve to map energy levels.
#6 Brian X
November 3, 2008

I happen to prefer tan(1/x) myself. What happens to the graph as x approaches zero is a complete and total mindscrew.
#7 BlackGriffen
March 1, 2009

arctan is not analytic, at least not everywhere. It has poles at x = +/- i. I mean, the series you gave has radius of convergence 1, so you need to use some trig identities or some kind of analytic continuation to get the value everywhere.