**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 $n.$

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$\pi$ 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 $e$?

$\mathbf{n+1}$

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