Monday, March 14, 2011

In Honor of Pi Day

In honor of $\pi$ day here are some cool facts about $\pi$.

1. Not only is $\pi$ an irrational number, it's what we call a transcendental number. This means that we can't compute $\pi$ exactly as the root of a polynomial with rational coefficients (we can however get arbitrarily close this way.) It is among the few known numbers which have this property, though we know there are quite a lot of transcendental numbers (uncountably many, in fact), it's a frustratingly difficult thing to prove that a particular number is transcendental.

2.\[ \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi ^2}{6} \]

3. \[ \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} \]

4. If a given complex valued function $f$ defined on an open set $U$ containing $0$ of complex numbers is given by a power series of the form
\[ f(z) = \sum_{n=0}^{\infty} a_n z^n \]
then the coefficients $a_n$ are given by the formula
\[ a_n = \frac{1}{2\pi i} \int_{\gamma} \frac{f(z)}{z^n} dz \]
for any closed curve $\gamma$ bounding $0$ (of course $\gamma$ needs to be in $U$)

Of further interest on #4 is that the condition that a complex valued function $f$ be given by a power series as above in a given set $U$ is in fact equivalent to the fact that $f$ be only differentiable once in that set. So in complex analysis, having one derivative is equivalent to having all derivatives is equivalent to being analytic. This is a huge departure from real analysis and has to do with a slight strengthening of the derivative obtained when moving from the real line to the complex plane.

5. (Stirling's formula)
\[ \lim_{n\to \infty} \frac{n!}{\sqrt{2\pi n} \left(\frac{n}{e} \right)^n}=1 \]
This gives a way to approximate the large number $n!$ for large values of $n.$


$\pi$ appears in a lot of unexpected places in mathematics, but what would be a post about pi without relating it to another (arguably more important) number $e$?


\[ e^{i\pi}+1 = 0 \]