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{eqnarray} 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 \\&=&\left(1+\frac{1}{2!}x^2+\frac{1}{4!} x^4+\frac{1}{6!}x^6+\frac{1}{8!}x^8+\cdots\right) +\left(x+\frac{1}{3!}x^3+\frac{1}{5!} x^5+\frac{1}{7!}x^7+\cdots\right) \\&=&\cosh{\left(x\right)}+\sinh{\left(x\right)}\;\ldots\;xが複素数でない=実数の場合 \\&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/coshx.html}{\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} \\&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/sinhx.html}{\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{eqnarray}sinh(x)のマクローリン展開
a点まわりのテイラー展開
\begin{eqnarray} f(x) &=& \sum_{k=0}^{\infty}\frac{f^{(k)}(a)}{x!}(x-a)^k \;\ldots\;a点まわりのテイラー展開 \\&=& \frac{1}{0!}f^{(0)}(a)(x-a)^0+\frac{1}{1!}f^{(1)}(a)(x-a)^1+\frac{1}{2!}f^{(2)}(a)(x-a)^2+\dotsb \\&&\;\ldots\;f^{(n)}(x): f(x)のn階微分 \end{eqnarray}マクローリン展開(0点まわりのテイラー展開)
\begin{eqnarray} f(x) &=& \left.\sum_{k=0}^{\infty}\frac{f^{(k)}(a)}{x!}(x-a)^k\right|_{a=0} \\&=& \sum_{k=0}^{\infty}\frac{f^{(k)}(0)}{x!}(x-0)^k \\&=& \sum_{k=0}^{\infty}\frac{f^{(k)}(0)}{x!}(x)^k \\&=& \frac{1}{0!}f^{(0)}(0)x^0+\frac{1}{1!}f^{(1)}(0)x+\frac{1}{2!}f^{(2)}(0)x^2+\cdots \end{eqnarray}\(f(x)=\sinh{\left(x\right)}\)のマクローリン展開(0点まわりのテイラー展開)
\begin{eqnarray} \sinh{\left(x\right)} &=& \frac{1}{0!}f^{(0)}(0)x^0&+\frac{1}{1!}f^{(1)}(0)x&+\frac{1}{2!}f^{(2)}(0)x^2+\cdots \\&=&\frac{1}{0!}\left\{\sinh{(0)}\right\}x^0&+\frac{1}{1!}\left\{\cosh{(0)}\right\}x &+\frac{1}{2!}\left\{\sinh{(0)}\right\}x^2 \\&&+\frac{1}{3!}\left\{\cosh{(0)}\right\}x^3&+\frac{1}{4!}\left\{\sinh{(0)}\right\}x^4&+\frac{1}{5!}\left\{\cosh{(0)}\right\}x^5 \\&&+\frac{1}{6!}\left\{\sinh{(0)}\right\}x^6&+\frac{1}{7!}\left\{\cosh{(0)}\right\}x^7&+\frac{1}{8!}\left\{\sinh{(0)}\right\}x^8 +\cdots \;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/cosh-sinh.html}{\frac{\mathrm{d}}{\mathrm{d}x}\cosh{\left(x\right)}=\sinh{\left(x\right)},\;\frac{\mathrm{d}}{\mathrm{d}x}\sinh{\left(x\right)}=\cosh{\left(x\right)}} \\&=&\frac{1}{0!}\cdot 0\cdot x^0&+\frac{1}{1!}\cdot 1\cdot x &+\frac{1}{2!}\cdot 0\cdot x^2 \\&&+\frac{1}{3!}\cdot 1\cdot x^3&+\frac{1}{4!}\cdot 0\cdot x^4&+\frac{1}{5!}\cdot 1\cdot x^5 \\&&+\frac{1}{6!}\cdot 0\cdot x^6&+\frac{1}{7!}\cdot 1\cdot x^7&+\frac{1}{8!}\cdot 0\cdot x^8 +\cdots \;\ldots\;\cosh{(0)}=\frac{e^0+e^{-0}}{2}=\frac{1+1}{2}=1,\;\sinh{(0)}=\frac{e^0-e^{-0}}{2}=\frac{1-1}{2}=0 \\&=&\frac{1}{1!}x^1+\frac{1}{3!}x^3&+\frac{1}{5!} x^5+\frac{1}{7!}x^7 +\cdots \\&=&x+\frac{1}{3!}x^3+\frac{1}{5!} x^5&+\frac{1}{7!}x^7 +\cdots \end{eqnarray}cosh(x)のマクローリン展開
a点まわりのテイラー展開
\begin{eqnarray} f(x) &=& \sum_{k=0}^{\infty}\frac{f^{(k)}(a)}{x!}(x-a)^k \;\ldots\;a点まわりのテイラー展開 \\&=& \frac{1}{0!}f^{(0)}(a)(x-a)^0+\frac{1}{1!}f^{(1)}(a)(x-a)^1+\frac{1}{2!}f^{(2)}(a)(x-a)^2+\dotsb \\&&\;\ldots\;f^{(n)}(x): f(x)のn階微分 \end{eqnarray}マクローリン展開(0点まわりのテイラー展開)
\begin{eqnarray} f(x) &=& \left.\sum_{k=0}^{\infty}\frac{f^{(k)}(a)}{x!}(x-a)^k\right|_{a=0} \\&=& \sum_{k=0}^{\infty}\frac{f^{(k)}(0)}{x!}(x-0)^k \\&=& \sum_{k=0}^{\infty}\frac{f^{(k)}(0)}{x!}(x)^k \\&=& \frac{1}{0!}f^{(0)}(0)x^0+\frac{1}{1!}f^{(1)}(0)x+\frac{1}{2!}f^{(2)}(0)x^2+\cdots \end{eqnarray}\(f(x)=\cosh{\left(x\right)}\)のマクローリン展開(0点まわりのテイラー展開)
\begin{eqnarray} \cosh{\left(x\right)} &=& \frac{1}{0!}f^{(0)}(0)x^0&+\frac{1}{1!}f^{(1)}(0)x&+\frac{1}{2!}f^{(2)}(0)x^2+\cdots \\&=&\frac{1}{0!}\left\{\cosh{(0)}\right\}x^0&+\frac{1}{1!}\left\{\sinh{(0)}\right\}x &+\frac{1}{2!}\left\{\cosh{(0)}\right\}x^2 \\&&+\frac{1}{3!}\left\{\sinh{(0)}\right\}x^3&+\frac{1}{4!}\left\{\cosh{(0)}\right\}x^4&+\frac{1}{5!}\left\{\sinh{(0)}\right\}x^5 \\&&+\frac{1}{6!}\left\{\cosh{(0)}\right\}x^6&+\frac{1}{7!}\left\{\sinh{(0)}\right\}x^7&+\frac{1}{8!}\left\{\cosh{(0)}\right\}x^8 +\cdots \;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/cosh-sinh.html}{\frac{\mathrm{d}}{\mathrm{d}x}\cosh{\left(x\right)}=\sinh{\left(x\right)},\;\frac{\mathrm{d}}{\mathrm{d}x}\sinh{\left(x\right)}=\cosh{\left(x\right)}} \\&=&\frac{1}{0!}\cdot 1\cdot x^0&+\frac{1}{1!}\cdot 0\cdot x &+\frac{1}{2!}\cdot 1\cdot x^2 \\&&+\frac{1}{3!}\cdot 0\cdot x^3&+\frac{1}{4!}\cdot 1\cdot x^4&+\frac{1}{5!}\cdot 0\cdot x^5 \\&&+\frac{1}{6!}\cdot 1\cdot x^6&+\frac{1}{7!}\cdot 0\cdot x^7&+\frac{1}{8!}\cdot 1\cdot x^8 +\cdots \;\ldots\;\cosh{(0)}=\frac{e^0+e^{-0}}{2}=\frac{1+1}{2}=1,\;\sinh{(0)}=\frac{e^0-e^{-0}}{2}=\frac{1-1}{2}=0 \\&=&\frac{1}{0!}x^0+\frac{1}{2!}x^2&+\frac{1}{4!} x^4+\frac{1}{6!}x^6&+\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 \end{eqnarray}cosh, sinhの微分
\(\cosh{\left(x\right)}\)の微分
\begin{eqnarray} \frac{\mathrm{d}}{\mathrm{d}x}\cosh{\left(x\right)} &=& \frac{\mathrm{d}}{\mathrm{d}x}\frac{e^x+e^{-x}}{2} \\&=& \frac{1}{2}\left(\frac{\mathrm{d}}{\mathrm{d}x}e^x+\frac{\mathrm{d}}{\mathrm{d}x}e^{-x}\right) \\&&\;\ldots\;\frac{\mathrm{d}}{\mathrm{d}x}Cf(x)=C\frac{\mathrm{d}}{\mathrm{d}x}f(x) \\&&\;\ldots\;\frac{\mathrm{d}}{\mathrm{d}x}\left(f(x)+g(x)\right)=\frac{\mathrm{d}}{\mathrm{d}x}f(x)+\frac{\mathrm{d}}{\mathrm{d}x}g(x) \\&=& \frac{1}{2}\left(\frac{\mathrm{d}}{\mathrm{d}x}e^x+\frac{\mathrm{d}}{\mathrm{d}x}e^{-x}\right) \\&=& \frac{1}{2}\left\{e^x+(-1)e^{-x}\right\} \\&=& \frac{1}{2}\left(e^x-e^{-x}\right) \\&=& \sinh{\left(x\right)} \end{eqnarray}\(\sinh{\left(x\right)}\)の微分
\begin{eqnarray} \frac{\mathrm{d}}{\mathrm{d}x}\sinh{\left(x\right)} &=& \frac{\mathrm{d}}{\mathrm{d}x}\frac{e^x-e^{-x}}{2} \\&=& \frac{1}{2}\left(\frac{\mathrm{d}}{\mathrm{d}x}e^x-\frac{\mathrm{d}}{\mathrm{d}x}e^{-x}\right) \\&&\;\ldots\;\frac{\mathrm{d}}{\mathrm{d}x}Cf(x)=C\frac{\mathrm{d}}{\mathrm{d}x}f(x) \\&&\;\ldots\;\frac{\mathrm{d}}{\mathrm{d}x}\left(f(x)+g(x)\right)=\frac{\mathrm{d}}{\mathrm{d}x}f(x)+\frac{\mathrm{d}}{\mathrm{d}x}g(x) \\&=& \frac{1}{2}\left(\frac{\mathrm{d}}{\mathrm{d}x}e^x-\frac{\mathrm{d}}{\mathrm{d}x}e^{-x}\right) \\&=& \frac{1}{2}\left\{e^x-(-1)e^{-x}\right\} \\&=& \frac{1}{2}\left(e^x+e^{-x}\right) \\&=& \cosh{\left(x\right)} \end{eqnarray}cosh(i x), sinh(i x) (純虚数に対するcosh, sinh)
\(\cosh{\left(ix\right)}\)
$$\begin{eqnarray} \cosh{\left(x\right)}&=&\frac{e^x+e^{-x}}{2} \\\cosh{\left(ix\right)}&=&\frac{e^{ix}+e^{-ix}}{2} \\&=&\frac{\left\{\cos{\left(x\right)}+i\sin{\left(x\right)}\right\}+\left\{\cos{\left(-x\right)}+i\sin{\left(-x\right)}\right\}}{2} \\&&\;\ldots\;e^{ix}=\cos{\left(x\right)}+i\sin{\left(x\right)} \\&=&\frac{\left\{\cos{\left(x\right)}+i\sin{\left(x\right)}\right\}+\left\{\cos{\left(x\right)}-i\sin{\left(x\right)}\right\}}{2} \\&&\;\ldots\;\cos{\left(-x\right)}=\cos{\left(x\right)},\;\sin{\left(-x\right)}=-\sin{\left(-x\right)} \\&=&\frac{\cos{\left(x\right)}\color{red}{+i\sin{\left(x\right)}}\color{black}{+\cos{\left(x\right)}}\color{red}{-i\sin{\left(x\right)}}}{2} \\&=&\frac{\cos{\left(x\right)}+\cos{\left(x\right)}}{2} \\&=&\frac{2\cos{\left(x\right)}}{2} \\&=&\cos{\left(x\right)} \end{eqnarray}$$\(\sinh{\left(ix\right)}\)
$$\begin{eqnarray} \sinh{\left(x\right)}&=&\frac{e^x-e^{-x}}{2} \\\sinh{\left(ix\right)}&=&\frac{e^{ix}-e^{-ix}}{2} \\&=&\frac{\left\{\cos{\left(x\right)}+i\sin{\left(x\right)}\right\}-\left\{\cos{\left(-x\right)}+i\sin{\left(-x\right)}\right\}}{2} \\&&\;\ldots\;e^{ix}=\cos{\left(x\right)}+i\sin{\left(x\right)} \\&=&\frac{\left\{\cos{\left(x\right)}+i\sin{\left(x\right)}\right\}-\left\{\cos{\left(x\right)}-i\sin{\left(x\right)}\right\}}{2} \\&&\;\ldots\;\cos{\left(-x\right)}=\cos{\left(x\right)},\;\sin{\left(-x\right)}=-\sin{\left(-x\right)} \\&=&\frac{\color{red}{\cos{\left(x\right)}}\color{black}{+i\sin{\left(x\right)}}\color{red}{-\cos{\left(x\right)}}\color{black}{+i\sin{\left(x\right)}}}{2} \\&=&\frac{i\sin{\left(x\right)}+i\sin{\left(x\right)}}{2} \\&=&\frac{2i\sin{\left(x\right)}}{2} \\&=&i\sin{\left(x\right)} \end{eqnarray}$$\(cos{(i x)}, sin{(i x)}\) (純虚数に対する\(\cos, \sin\))
\(cos{(i x)}, sin{(i x)}\) (純虚数に対する\(\cos, \sin\))
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