e^(ix)のマクローリン展開

$e^{ix}$のマクローリン展開(0点まわりのテイラー展開)

$$\begin{array}{rcl}{e}^{ix}& =& 1+(ix)+\frac{1}{2!}(ix{)}^2+\frac{1}{3!}(ix{)}^3+\frac{1}{4!}(ix{)}^4+\frac{1}{5!}(ix{)}^5+\frac{1}{6!}(ix{)}^6+\frac{1}{7!}(ix{)}^7+\frac{1}{8!}(ix{)}^8+\cdots \\ & & \dots e^x=1+x+\frac{1}{2!}{x}^2+\frac{1}{3!}{x}^3+\frac{1}{4!}{x}^4+\frac{1}{5!}{x}^5+\frac{1}{6!}{x}^6+\frac{1}{7!}{x}^7+\frac{1}{8!}{x}^8+\cdots \\ & =& 1+ix+\frac{1}{2!}{i}^2{x}^2+\frac{1}{3!}{i}^3{x}^3+\frac{1}{4!}{i}^4{x}^4+\frac{1}{5!}{i}^5{x}^5+\frac{1}{6!}{i}^6{x}^6+\frac{1}{7!}{i}^7{x}^7+\frac{1}{8!}{i}^8{x}^8+\cdots \\ & =& 1+ix+\frac{1}{2!}(-1)x^2+\frac{1}{3!}(-i)x^3+\frac{1}{4!}{x}^4+\frac{1}{5!}i{x}^5+\frac{1}{6!}(-1)x^6+\frac{1}{7!}(-i)x^7+\frac{1}{8!}{x}^8+\cdots \\ & =& 1+ix-\frac{1}{2!}{x}^2-i\frac{1}{3!}{x}^3+\frac{1}{4!}{x}^4+i\frac{1}{5!}{x}^5-\frac{1}{6!}{x}^6-i\frac{1}{7!}{x}^7+\frac{1}{8!}{x}^8+\cdots \\ & =& 1-\frac{1}{2!}{x}^2+\frac{1}{4!}{x}^4-\frac{1}{6!}{x}^6+\frac{1}{8!}{x}^8+\cdots +i(x-\frac{1}{3!}{x}^3+\frac{1}{5!}{x}^5-\frac{1}{7!}{x}^7+\cdots )\\ & =& \cos \left(x\right)+i\sin \left(x\right)\\ & & \dots \cos \left(x\right)=1-\frac{1}{2!}{x}^2+\frac{1}{4!}{x}^4-\frac{1}{6!}{x}^6+\frac{1}{8!}{x}^8+\cdots \\ & & \dots \sin \left(x\right)=x-\frac{1}{3!}{x}^3+\frac{1}{5!}{x}^5-\frac{1}{7!}{x}^7+\cdots \end{array}$$