Mr. Bessel’s eponymous functions

Yesterday I wrote a post showing that the trapezoid rule evaluates the integral very efficiently. But how do we know what the exact integral is for comparison? If you ask Mathematica, it will tell you the integral equals −2π J1(1) where J1 is a Bessel function. This may seem like rabbit out of a hat, but it’s actually a simple calculation given the integral definition of Bessel functions: Since cosine is even, we can write our integral over [−π, π] as twice the integral over [0, π]. Then a change of variables turns this into the definition of Jn(z) with n = 1 and z = 1. A deeper question is what have we accomplished by just giving a new name to essentially the same problem we started with. Another question is why in the world are Bessel functions defined as above. As for what we’ve accomplished, we’ve related out integration problem to a very well-studied function. Bessel functions have been studied for two centuries and it’s easy to find software to evaluate them. Even the usually…