\(e^{ix}\)のマクローリン展開(0点まわりのテイラー展開)
\begin{eqnarray}
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
\\&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/ex.html}{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^2x^2+\frac{1}{3!}i^3x^3+\frac{1}{4!} i^4x^4+\frac{1}{5!} i^5x^5+\frac{1}{6!}i^6x^6+\frac{1}{7!}i^7x^7+\frac{1}{8!}i^8x^8
+\cdots
\\&=&1+ix+\frac{1}{2!}(-1)x^2+\frac{1}{3!}(-i)x^3+\frac{1}{4!} x^4+\frac{1}{5!} ix^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\left(x-\frac{1}{3!}x^3+\frac{1}{5!} x^5-\frac{1}{7!}x^7+\cdots\right)
\\&=&\cos{\left(x\right)}+i\sin{\left(x\right)}
\\&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/cosx.html}{\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}
\\&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/sinx.html}{\sin{\left(x\right)}=x-\frac{1}{3!}x^3+\frac{1}{5!} x^5-\frac{1}{7!}x^7+\cdots}
\end{eqnarray}
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