テイラー展開
sin  x = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) ! x 2 n + 1 for all x {\displaystyle \sin x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\quad {\mbox{ for all }}x}
cos  x = ∑ n = 0 ∞ ( − 1 ) n ( 2 n ) ! x 2 n for all x {\displaystyle \cos x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\quad {\mbox{ for all }}x}
tan  x = ∑ n = 1 ∞ B 2 n ( − 4 ) n ( 1 − 4 n ) ( 2 n ) ! x 2 n − 1 for | x | < π 2 {\displaystyle \tan x=\sum _{n=1}^{\infty }{\frac {B_{2n}(-4)^{n}(1-4^{n})}{(2n)!}}x^{2n-1}\quad {\mbox{ for }}|x|<{\frac {\pi }{2}}}
csc  x = ∑ n = 0 ∞ ( − 1 ) n ( 2 − 2 2 n ) B 2 n ( 2 n ) !
e x = ∑ n = 0 ∞ x n n ! for all x {\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}\quad {\mbox{ for all }}x}