# §4.33 Maclaurin Series and Laurent Series

 4.33.1 $\mathop{\sinh\/}\nolimits z=z+\frac{z^{3}}{3!}+\frac{z^{5}}{5!}+\cdots,$
 4.33.2 $\mathop{\cosh\/}\nolimits z=1+\frac{z^{2}}{2!}+\frac{z^{4}}{4!}+\cdots.$
 4.33.3 $\mathop{\tanh\/}\nolimits z=z-\frac{z^{3}}{3}+\frac{2}{15}z^{5}-\frac{17}{315}% z^{7}+\cdots+\frac{2^{2n}(2^{2n}-1)\mathop{B_{2n}\/}\nolimits}{(2n)!}z^{2n-1}+\cdots,$ $|z|<\frac{1}{2}\pi$.

For $\mathop{B_{2n}\/}\nolimits$ see §24.2(i). For expansions that correspond to (4.19.4)–(4.19.9), change $z$ to $iz$ and use (4.28.8)–(4.28.13).