Are there any formulas that include $\pi^n?$

Question

I was thinking about $\pi$ then thought: are there any formulas out there that require $\pi^n$ where $n \in \mathbb{N}$?

For example $\pi^2$ or $\pi^3$, but not just $\pi$? So are there any out there?

Answer 1

Yes, though I'm assuming your asking after interesting formulas.

Some examples are $$\sum_{n=1}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6}$$ As seen from the Basel Problem. Similarly, $$\sum_{n=1}^{\infty} \frac{1}{n^4}=\frac{\pi^4}{90}$$ And so on. As mentioned by @imranfat, generally $$\sum_{k=1}^{\infty} \frac{1}{k^{2n}}=\zeta (2n)=(-1)^{n+1}\frac{B_{2n}(2\pi)^{2n}}{2(2n)!}$$ Where $\zeta(n)$ denotes the Riemann zeta function.

Answer 2

I imagine that you have seen $\pi$ arise while calculating, for example, the area of a circle: $A=\pi r^2$. In fact, if you want to compute the volume of higher dimensional spheres you must calculate higher powers of $\pi$. For example the volume of an 8 dimensional sphere is given by

$$V=\frac{1}{24} \pi^4 r^8.$$

The Wiki page on n-sphere has a nice little section on this subject! In general we have that the volume of an $n$-dimensional sphere is proportional to

$$V(S_n) \propto \pi^{\lfloor n/2\rfloor},$$

so you have formulas involving $\pi$ to any power you desire, provided the dimension is sufficiently large (here $\lfloor \cdot \rfloor$ is the floor function).

• $\frac{1}{2}{\pi}^2r^4\;\;$ volume of the $4$-dimensional sphere
• $\frac{8}{15}{\pi}^2r^5\;\;$ volume of the $5$-dimensional sphere
• $\frac{1}{6}{\pi}^3r^6\;\;$ volume of the $6$-dimensional sphere
• $\frac{16}{105}{\pi}^3r^7\;\;$ volume of the $7$-dimensional sphere
• $\frac{1}{24}{\pi}^4r^8\;\;$ volume of the $8$-dimensional sphere
• $\frac{32}{945}{\pi}^4r^9\;\;$ volume of the $9$-dimensional sphere
• $\frac{1}{120}{\pi}^5r^{10}\;\;$ volume of the $10$-dimensional sphere

Answer 3

For a formula including $\pi^3$: $$ \int_0^{\frac{\pi}{2}}(\ln (\sin x))^2dx = \int_0^{\frac{\pi}{2}}(\ln (\cos x))^2dx = \frac{\pi}{2}(\ln 2)^2 +\frac{\pi^3}{24} $$

Answer 4

The probability distribution function for the normal distribution has $\pi^{-\frac{1}{2}}$ in it: $$f(x) = \frac{1}{\sqrt{2\pi \sigma}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

[I figured I'd answer this under 'community wiki' since there are going to be lots of suggestions.]

Answer 5

The Stirling asymptotic formula for the factorial is

$$n!\approx\sqrt{2\pi n}\left(\frac ne\right)^n.$$

Answer 6

If $\Gamma$ denotes the Gamma function, the volume of the unit ball (open or close) in ${\mathbb{R}}^n$ is $$ m(B_n(0,1)) = \frac{{\pi}^{\frac{n}{2}}}{\Gamma(\frac{n}{2} + 1)}\mbox{,} $$ so, if $n = 2 k$, being $k \in \mathbb{N}$, $$ m(B_n(0,1)) = \frac{{\pi}^k}{\Gamma(k + 1)} = \frac{{\pi}^k}{k !}\mbox{.} $$

Answer 7

You can find $\pi$ in BUCKLING:

https://en.wikipedia.org/wiki/Buckling

Answer 8

Here is a famous (nevertheless surprising!) one for you: $$\int_{\mathbb R} e^{-x^2} dx = \pi^{1/2} $$ More generally: $$\int_{\mathbb R^n} e^{-|x|^2} dx = \pi^{n/2} $$