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\begin{document}
 
\noindent
Eric Rasmusen\\
March 14, 2024
\begin{center}
{\large {\bf  Formulas for  $\pi$}}
\end{center}
 
This list is selected from the {\it Wikipedia}  article, \href{https://en.wikipedia.org/w/index.php?title=List_of_formulae_involving_%CF%80&action=edit}{ ``List of Formulas involving $\pi$''. } Some of them I adapted to be $\pi = f(x)$ instead of $x = f(\pi)$  I should really write this all out as eqnarrays. 

\begin{eqnarray} \label{e1}
 \pi  &=  &\frac{C}{d}      \hspace{36pt}   {\rm ( where \  C\ is\  the  \ circumference \  and \ d  \ is  \ the  \ diameter \  of \  a  \ circle.        ) } \\
   \notag &&\\
 \label{e2}
 \pi& =   &\frac{C}{2r}     \hspace{36pt}   {\rm (where \  C \  is  \ the  \ circumference   \ and  \  r   \ is  \  the  \  radius   \ of  \  a   \ circle.         ) } 
\end{eqnarray}


 \begin{equation}\label{e2}
 \pi = \frac{A}{r^2}     \hspace{36pt}   {\rm (    where  \ A  \ is  \ the \  area   \ and  \      r is  \  the  \  radius  \  of   \ a  \ circle.    ) }
\end{equation}


  \begin{equation}\label{e2}
 \pi = \frac{A}{\frac{major}{2} \cdot \frac{minor}{2}},
 \end{equation}
where $A$ is    the area of an ellipse,  $major$ is its major axis, and $minor$ is its minor axis. 

 \begin{equation}\label{e2}
 \pi = \frac{3}{4}  \cdot \frac{V}{r^3}      \hspace{36pt}   {\rm (  where  \ V \  is \  the   \ area  \  of  \  a   \ sphere    \   and   \ r   \ is  \  its  \  radius. 
      ) }
 \end{equation}


 \begin{equation}\label{e2}
 \pi =  \frac{A}{4r^2}    \hspace{36pt}   {\rm (   where \  A  \  is \  a   \ sphere's   \ surface  \  area  \  and  \  r  \   is  \  its   \ radius.       ) }
 \end{equation}


  \begin{equation}\label{e2}
 \pi =    \frac{    \sqrt{2H} }{r^2},    \hspace{36pt}   {\rm (  where\  H\  is\   the\ volume\  and \ r \ is\ the \ radius \ of   \  a   \ 3-hypersphere  \  in   \ 
 4-D    \ space.      ) }
 \end{equation}

   \begin{equation}\label{e2}
 \pi =    \frac{   Sum}{N-2}    \hspace{36pt}   {\rm (  where  \ Sum  \  is  \  the  \  sum  \  of   \   an  \   N-sided  \  regular  \  convex   \ polygon's  \ angles,  \  in  \  radians.        ) }
 \end{equation}


 \begin{equation}\label{e2} 
  \pi =   
 4\left(  
 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots  \right)   \hspace{36pt}  {\rm  (Leibniz's \ formula \ for \ } \pi)
  \end{equation}
    

\begin{equation}\label{e2}
   \pi =  
    2 \sqrt{2}    \left( 1+\frac13-\frac15-\frac17+\frac19+\frac{1}{11}-\cdots  \right)   \hspace{36pt}  {\rm (Newton's  \ formula \ for \ } \pi) 
     \end{equation}

 
 
\begin{equation}\label{e2}
       \pi =    \sum_{k=1}^\infty k\frac{2^kk!^2}{(2k)!} - 3 \hspace{36pt}   {\rm (  Euler's \ series) }
 \end{equation} 
 

 
 
\begin{equation}\label{e2}
 \pi =  
\sqrt[4]{  90\left(    \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4} + \cdots   \right)  }
   \end{equation}
  
 

 \begin{equation}\label{e2}
 \pi =  \sqrt{ 12\left(\frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + \cdots    \right)}
 \end{equation} 
 
  \begin{equation}\label{e2}
 \pi =  \sqrt{ 24 \left( \frac{1}{2^2} + \frac{1}{4^2} + \frac{1}{6^2} + \frac{1}{8^2} + \cdots   \right)}
 \end{equation} 
 
  \begin{equation}\label{e2}
 \pi =  \sqrt{  8 \left(  \frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + \frac{1}{7^2} + \cdots   \right)}
 \end{equation} 

 \begin{equation}\label{e2}
 \pi =  \sqrt[3]{  32 \left(  \frac{1}{1^3} - \frac{1}{3^3} + \frac{1}{5^3} - \frac{1}{7^3} + \cdots  \right) }
 \end{equation} 

 
 
 \begin{equation}\label{e2}
 \pi =  \sum_{n=-\infty}^\infty \frac{(-1)^n}{n+1/2}
 \end{equation} 
 
 
   \begin{equation}\label{e2}
 \pi =  \sqrt{\sum_{n=-\infty}^\infty \frac{1}{(n+1/2)^2}}
 \end{equation} 
 
 
 
\begin{equation}\label{e2}
 \pi =  
 \sqrt{  6\left( \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots  \right)  }  \;\;\;\;\;({\rm from \ Euler})
 \end{equation}
 
  \begin{equation}\label{e2}
 \pi = 4\left( \frac{3}{4} \cdot \frac{5}{4} \cdot \frac{7}{8} \cdot \frac{9}{8} \cdot \frac{11}{12} \cdot \frac{13}{12} \cdots\right),
 \end{equation} 
 from Euler, where the numerators are the odd primes; each denominator is the multiple of four nearest to the numerator.
 
 \begin{equation}\label{e2}
 \pi =  2 \int_{-1}^1 \sqrt{1-x^2}\,dx    \hspace{36pt}   {\rm (which \  is  \ to  \  integrate  \  two  \  halves  \  of  \  a  \  unit  \  circle     ) }
 \end{equation}
 and, similarly, 
 \begin{equation}\label{e2}
 \pi =    \int_{-1}^1   \frac{1}{ \sqrt{1-x^2}}\,dx    \hspace{36pt}   {\rm (which \  is  \ to  \  integrate   \ two  \  halves  \  of  \  a  \ unit   \ circle     ) }
 \end{equation}
 
 
 
 

 \begin{equation}\label{e2}
 \pi =  \int_{-\infty}^\infty\frac{1}{1+x^2} dx      \hspace{36pt}   {\rm ( from \  the  \ Cauchy  \  density  \  and   \  \href{https://en.wikipedia.org/wiki/Witch_of_Agnesi}{ the  \ Witch  \  of   \ Agnesi. }        ) }
 \end{equation}

 
  \begin{eqnarray}\label{e2}
 & {\rm Since \ }  f(x) =\frac {1} { \sigma \sqrt{2\pi} }  e ^{-\frac{1}{2}  \left(\frac{x-\mu}{ \sigma} \right)^2} \hspace{36pt} {\rm (the\ normal \  density}),\\
  1  =& \int_{-\infty}^\infty  \frac{1}{ \sqrt{2\pi} }    e ^{-x^2 /2} dx \hspace{36pt}
 {\rm (integrating \ and \  setting  \ } \sigma=1, \mu=0),  {\rm so } \\
 \pi  = &  \frac{1}{ 2}  \left( \int_{-\infty}^\infty      e^{-x^2/2}\,dx \right)^2 \\
  \pi   \approx &   \left( \sum_{x=0}^N  \frac{1}{N}    e^{-x^2/2}  \right)^2    \hspace{36pt} {\rm (doubling \   the  \  sum  \  from  \  0  \  to  \  N)}\\
 \end{eqnarray}
 
  
  \begin{equation}\label{e2}
 \pi =  \int_{-\infty}^\infty  \frac{sin (x)}{x} dx ,
 \end{equation}
 
 Add the fluid dynamics equation. 
 
 Add the Euler identity. 
 
 
 
   
  
 
  Peter Borwein, \href{http://www.pi314.net/eng/borwein.php}{The World of Pi} website. 
  
  Peter Borwein,    \href{http://www.pi314.net/ref/P159.pdf} {``The Amazing Number $\Pi$''} (2000). 
 
  \end{document}