Digital Library of Mathematical Functions
About the Project
NIST
4 Elementary FunctionsTrigonometric Functions

§4.19 Maclaurin Series and Laurent Series

4.19.1 sinz=z-z33!+z55!-z77!+,
4.19.2 cosz=1-z22!+z44!-z66!+.

In (4.19.3)–(4.19.9), Bn are the Bernoulli numbers and En are the Euler numbers (§§24.2(i)24.2(ii)).

4.19.3 tanz=z+z33+215z5+17315z7++(-1)n-122n(22n-1)B2n(2n)!z2n-1+,
|z|<12π,
4.19.4 cscz=1z+z6+7360z3+3115120z5++(-1)n-12(22n-1-1)B2n(2n)!z2n-1+,
0<|z|<π,
4.19.5 secz=1+z22+524z4+61720z6++(-1)nE2n(2n)!z2n+,
|z|<12π,
4.19.6 cotz=1z-z3-z345-2945z5--(-1)n-122nB2n(2n)!z2n-1-,
0<|z|<π,
4.19.7 ln(sinzz)=n=1(-1)n22n-1B2nn(2n)!z2n,
|z|<π,
4.19.8 ln(cosz)=n=1(-1)n22n-1(22n-1)B2nn(2n)!z2n,
|z|<12π,
4.19.9 ln(tanzz)=n=1(-1)n-122n(22n-1-1)B2nn(2n)!z2n,
|z|<12π.