e^xとマクローリン展開と双曲線凾数

$e^x$とマクローリン展開と双曲線凾数

$$\begin{array}{rcl}{e}^x& =& 1+x+\frac{1}{2!}{x}^2+\frac{1}{3!}{x}^3+\frac{1}{4!}{x}^4+\frac{1}{5!}{x}^5+\frac{1}{6!}{x}^6+\frac{1}{7!}{x}^7+\frac{1}{8!}{x}^8+\cdots \\ & =& (1+\frac{1}{2!}{x}^2+\frac{1}{4!}{x}^4+\frac{1}{6!}{x}^6+\frac{1}{8!}{x}^8+\cdots )+(x+\frac{1}{3!}{x}^3+\frac{1}{5!}{x}^5+\frac{1}{7!}{x}^7+\cdots )\\ & =& \cosh \left(x\right)+\sinh \left(x\right)\dots x\mathrm{が}\mathrm{複}\mathrm{素}\mathrm{数}\mathrm{で}\mathrm{な}\mathrm{い}=\mathrm{実}\mathrm{数}\mathrm{の}\mathrm{場}\mathrm{合}\\ & & \dots \cosh \left(x\right)=1+\frac{1}{2!}{x}^2+\frac{1}{4!}{x}^4+\frac{1}{6!}{x}^6+\frac{1}{8!}{x}^8+\cdots \\ & & \dots \sinh \left(x\right)=x+\frac{1}{3!}{x}^3+\frac{1}{5!}{x}^5+\frac{1}{7!}{x}^7+\cdots \\ & =& \frac{e^x+e^{-x}}{2}+\frac{e^x-e^{-x}}{2}=\frac{e^x+e^{-x}+e^x-e^{-x}}{2}=\frac{2e^x}{2}\\ & =& e^x\end{array}$$