27 - Power Series

Real Power Series

Real Power Series is

$$\sum _{k=0}^{\infty }{c}_k{x}^k$$

Where $c_k\in \mathbb{R}$ for all $k\ge 0$ and $x\in \mathbb{R}$

Complex Power Series

Complex Power Series is

$$\sum _{k=0}^{\infty }{c}_k{z}^k$$

Where $c_k\in \mathbb{C}$ for all $k\ge 0$ and $z\in \mathbb{C}$

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Exponential Function

$$\exp :\mathbb{C}\to \mathbb{C}\text{ by }\exp (z)=\sum _{k=0}^{\infty }\frac{z^k}{k!}$$

and we write $e^z$ for $\exp (z)$

Sine Function

$$\sin :\mathbb{C}\to \mathbb{C}\text{ by }\sin (z)=\sum _{k=0}^{\infty }(-1{)}^k\frac{z^{2k+1}}{(2k+1)!}$$

Cosine Function

$$\cos :\mathbb{C}\to \mathbb{C}\text{ by }\cos (z)=\sum _{k=0}^{\infty }(-1{)}^k\frac{z^{2k}}{(2k)!}$$

Hyperbolic Sine Function

$$\sinh :\mathbb{C}\to \mathbb{C}\text{ by }\sinh (z)=\sum _{k=0}^{\infty }\frac{z^{2k+1}}{(2k+1)!}$$

Hyperbolic Cosine Function

$$\cosh :\mathbb{C}\to \mathbb{C}\text{ by }\cosh (z)=\sum _{k=0}^{\infty }\frac{z^{2k}}{(2k)!}$$