# §10.56 Generating Functions

When $2|t|<|z|$,

 10.56.1 $\displaystyle\frac{\mathop{\cos\/}\nolimits\sqrt{z^{2}-2zt}}{z}$ $\displaystyle=\frac{\mathop{\cos\/}\nolimits z}{z}+\sum_{n=1}^{\infty}\frac{t^% {n}}{n!}\mathop{\mathsf{j}_{n-1}\/}\nolimits\!\left(z\right),$ 10.56.2 $\displaystyle\frac{\mathop{\sin\/}\nolimits\sqrt{z^{2}-2zt}}{z}$ $\displaystyle=\frac{\mathop{\sin\/}\nolimits z}{z}+\sum_{n=1}^{\infty}\frac{t^% {n}}{n!}\mathop{\mathsf{y}_{n-1}\/}\nolimits\!\left(z\right).$
 10.56.3 $\displaystyle\frac{\mathop{\cosh\/}\nolimits\sqrt{z^{2}+2izt}}{z}$ $\displaystyle=\frac{\mathop{\cosh\/}\nolimits z}{z}+\sum_{n=1}^{\infty}\frac{(% it)^{n}}{n!}\mathop{{\mathsf{i}^{(1)}_{n-1}}\/}\nolimits\!\left(z\right),$ 10.56.4 $\displaystyle\frac{\mathop{\sinh\/}\nolimits\sqrt{z^{2}+2izt}}{z}$ $\displaystyle=\frac{\mathop{\sinh\/}\nolimits z}{z}+\sum_{n=1}^{\infty}\frac{(% it)^{n}}{n!}\mathop{{\mathsf{i}^{(2)}_{n-1}}\/}\nolimits\!\left(z\right),$
 10.56.5 $\frac{\mathop{\exp\/}\nolimits\!\left(-\sqrt{z^{2}+2izt}\right)}{z}=\frac{e^{-% z}}{z}+\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-it)^{n}}{n!}\mathop{\mathsf{k}_% {n-1}\/}\nolimits\!\left(z\right).$