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4 Elementary FunctionsTrigonometric Functions

§4.19 Maclaurin Series and Laurent Series

4.19.1 sinz=zz33!+z55!z77!+,
4.19.2 cosz=1z22!+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)n122n(22n1)B2n(2n)!z2n1+,
|z|<12π,
4.19.4 cscz=1z+z6+7360z3+3115120z5++(1)n12(22n11)B2n(2n)!z2n1+,
0<|z|<π,
4.19.5 secz=1+z22+524z4+61720z6++(1)nE2n(2n)!z2n+,
|z|<12π,
4.19.6 cotz=1zz3z3452945z5(1)n122nB2n(2n)!z2n1,
0<|z|<π,
4.19.7 ln(sinzz)=n=1(1)n22n1B2nn(2n)!z2n,
|z|<π,
4.19.8 ln(cosz)=n=1(1)n22n1(22n1)B2nn(2n)!z2n,
|z|<12π,
4.19.9 ln(tanzz)=n=1(1)n122n(22n11)B2nn(2n)!z2n,
|z|<12π.