円周率を含む数式
S = 4 π r 2 {\displaystyle S=4\pi r^{2}}
π = 3 + 1 2 6 + 3 2 6 + 5 2 6 + 7 2 6 + ⋱ {\displaystyle \pi ={3+{\cfrac {1^{2}}{6+{\cfrac {3^{2}}{6+{\cfrac {5^{2}}{6+{\cfrac {7^{2}}{6+\ddots \,}}}}}}}}}}
π = 4 1 + 1 2 3 + 2 2 5 + 3 2 7 + 4 2 9 + ⋱ {\displaystyle \pi ={\cfrac {4}{1+{\cfrac {1^{2}}{3+{\cfrac {2^{2}}{5+{\cfrac {3^{2}}{7+{\cfrac {4^{2}}{9+\ddots }}}}}}}}}}}
π = 4 1 + 1 2 2 + 3 2 2 + 5 2 2 + 7 2 2 + ⋱ {\displaystyle \pi ={\cfrac {4}{1+{\cfrac {1^{2}}{2+{\cfrac {3^{2}}{2+{\cfrac {5^{2}}{2+{\cfrac {7^{2}}{2+\ddots }}}}}}}}}}}
2 π = 6 + 2 2 12 + 6 2 12 + 10 2 12 + 14 2 12 + 18 2 12 + ⋱ {\displaystyle 2\pi ={6+{\cfrac {2^{2}}{12+{\cfrac {6^{2}}{12+{\cfrac {10^{2}}{12+{\cfrac {14^{2}}{12+{\cfrac {18^{2}}{12+\ddots }}}}}}}}}}}}
n ! ∼ 2 π n ( n e ) n {\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}} (スターリングの近似)
e i π + 1 = 0 {\displaystyle e^{i\pi }+1=0} (オイラーの等式)
∑ k = 1 n φ ( k ) ∼ 3 n 2 π 2 {\displaystyle \sum _{k=1}^{n}\varphi (k)\sim {\frac {3n^{2}}{\pi ^{2}}}} (オイラーのトーティエント関数)
∑ k = 1 n φ ( k ) k ∼ 6 n π 2 {\displaystyle \sum _{k=1}^{n}{\frac {\varphi (k)}{k}}\sim {\frac {6n}{\pi ^{2}}}} (オイラーのトーティエント関数)
Γ ( 1 2 ) = π {\displaystyle \Gamma \left({1 \over 2}\right)={\sqrt {\pi }}} (ガンマ関数)