[tex]e^z\: =\: 1+z+\frac{z^2}{2!}+\frac{z^3}{3!}+\frac{z^4}{4!}+\: ...[/tex]
[tex]e^{\frac{1}{z}}\: =\: 1+\frac{1}{z}+\frac{1}{z^22!}+\frac{1}{z^33!}+\frac{1}{z^44!}+\: ...[/tex]

[tex]\cos z\: =\: 1-\frac{z^2}{2!}+\frac{z^4}{4!}-\frac{z^6}{6!}+\frac{z^8}{8!}-...[/tex]
[tex]\cos\frac{1}{z}\: =\: 1-\frac{1}{z^22!}+\frac{1}{z^44!}-\frac{1}{z^66!}+\frac{1}{z^88!}-...[/tex]
[tex]\cos\frac{2}{z}\: =\: 1-\frac{2^2}{z^22!}+\frac{2^4}{z^44!}-\frac{2^6}{z^66!}+\frac{2^8}{z^88!}-...[/tex]

[tex]z^3\: e^{\frac{1}{z}}\: \cos\frac{2}{z}\: =\: z^3\: (1+\frac{1}{z}+\frac{1}{z^22!}+\frac{1}{z^33!}+\frac{1}{z^44!}+...)\: (1-\frac{2^2}{z^22!}+\frac{2^4}{z^44!}-\: \frac{2^6}{z^66!}+\: \frac{2^8}{z^88!}-\: ...)[/tex]