Series for ArcTan Using Integrals

I'm getting ready to present my superconductor research[1] at the Texas Academy of Science meeting this weekend, so today's post is just a quick note on an easy way to find a series expansion for the arctangent.

Arctan(x) looked a little daunting to get into a series at first. The key to get started is to remember what the derivative of arctan is (picture 1)

At least the derivative is something that looks like it could be pretty easily worked into a series. With a little more thought, it turns out that the anti-derivative, (the integral), of the result above is arctan. So, the process will be to turn the derivative into a series first, and then take the integral of that series to wind up with the series for arctan.

The result of the derivative shown above is a prime candidate for a geometric series expansion(picture 2)

Now that we have the series, all that remains is to integrate and we get (picture 3):

So (picture 4),

References:
1. http://arxiv.org/abs/1208.1870

Picture of the Day (picture 5):

From 2/27/13

Comments

Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

Now available as a Kindle ebook for 99 cents ! Get a spiffy ebook, and fund more physics The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems , there seem to be fewer explanations of the procedure for deriving the conversion, so here goes! What do we actually want? To convert the Cartesian nabla to the nabla for another coordinate system, say… cylindrical coordinates. What we’ll need: 1. The Cartesian Nabla: 2. A set of equations relating the Cartesian coordinates to cylindrical coordinates: 3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system: How to do it: Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables. The chain ...

The Alcubierre Warp Drive Tophat Function and Open Science with Sage

I transferred yesterday's Mathematica file with the Alcubierre warp drive[2] line element and space curvature calculations to the +Sage Mathematical Software System today, (the files been added to the public repository [3]). If you haven't used Sage before, it's a Python based software package that's similar in functionality to Mathematica. Oh, and it' free. I also worked a little more on understanding the theory, but frankly, I made far more progress with the software than the theory. What follows will be a little more of the Alcubierre theory, plus, a cool Sage interactive demo of one of the Alcubierre functions[1], as well as a bit about my first experience with using Sage. Theory The theory is fun, but it's moving slowly. Here's the chalk board from this morning's discussion Alcubierre setup the derivation using something called the 3+1 formalism which means we consider space to be flat, (in this case), slices that are labelled ...

The Valentine's Day Magnetic Monopole

There's an assymetry to the form of the two Maxwell's equations shown in picture 1. While the divergence of the electric field is proportional to the electric charge density at a given point, the divergence of the magnetic field is equal to zero. This is typically explained in the following way. While we know that electrons, the fundamental electric charge carriers exist, evidence seems to indicate that magnetic monopoles, the particles that would carry magnetic 'charge', either don't exist, or, the energies required to create them are so high that they are exceedingly rare. That doesn't stop us from looking for them though! Keeping with the theme of Fairbank[1] and his academic progeny over the semester break, today's post is about the discovery of a magnetic monopole candidate event by one of the Fairbank's graduate students, Blas Cabrera[2]. Cabrera was utilizing a loop type of magnetic monopole detector. Its operation is in...

Copasetic Flow

Search This Blog