円周率を含む数式
π = Γ ( 1 / 4 ) 4 / 3 agm  ( 1 , 2 ) 2 / 3 2 {\displaystyle \pi ={\frac {\Gamma \left({1/4}\right)^{4/3}\operatorname {agm} (1,{\sqrt {2}})^{2/3}}{2}}} (agmは算術幾何平均)
lim n → ∞ 1 n 2 ∑ k = 1 n ( n mod k ) = 1 − π 2 12 {\displaystyle \lim _{n\rightarrow \infty }{\frac {1}{n^{2}}}\sum _{k=1}^{n}(n\;{\bmod {\;}}k)=1-{\frac {\pi ^{2}}{12}}} (mod は剰余演算)
π = lim n → ∞ 4 n 2 ∑ k = 1 n n 2 − k 2 {\displaystyle \pi =\lim _{n\rightarrow \infty }{\frac {4}{n^{2}}}\sum _{k=1}^{n}{\sqrt {n^{2}-k^{2}}}} (単位円の面積とリーマン和を参照)
π = lim n → ∞ 2 4 n n ( 2 n n ) 2 {\displaystyle \pi =\lim _{n\rightarrow \infty }{\frac {2^{4n}}{n{2n \choose n}^{2}}}} (スターリングの近似)
∫ − ∞ ∞ sech  ( x ) d x = π {\displaystyle \int _{-\infty }^{\infty }\operatorname {sech} (x)\,dx=\pi }
∫ − ∞ ∞ ∫ t ∞ e − 1 / 2 t 2 − x 2 + x t d x d t = ∫ − ∞ ∞ ∫ t ∞ e − t 2 − 1 / 2 x 2 + x t d x d t = π {\displaystyle \int _{-\infty }^{\infty }\int _{t}^{\infty }e^{-1/2t^{2}-x^{2}+xt}\,dx\,dt=\int _{-\infty }^{\infty }\int _{t}^{\infty }e^{-t^{2}-1/2x^{2}+xt}\,dx\,dt=\pi }
∫ − 1 1 1 − x 2 d x = π 2 {\displaystyle \int _{-1}^{1}{\sqrt {1-x^{2}}}\,dx={\frac {\pi }{2}}}
∫ 0 1 1 1 + x 2 d x = π 4 {\displaystyle \int _{0}^{1}{1 \over 1+x^{2}}\,dx={\frac {\pi }{4}}}
Peter Borwein, The Amazing Number Pi
Kazuya Kato, Nobushige Kurokawa, Saito Takeshi: Number Theory 1: Fermat's Dream. American Mathematical Society, Providence 1993, ISBN 0-8218-0863-X.