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}
\(f(x)=\sin{\left(x\right)}\)のマクローリン展開(0点まわりのテイラー展開)
\begin{eqnarray}
\sin{\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\{ \sin{(0)}\right\}x^0&+\frac{1}{1!}\left\{ \cos{(0)}\right\}x &+\frac{1}{2!}\left\{-\sin{(0)}\right\}x^2
\\&&+\frac{1}{3!}\left\{-\cos{(0)}\right\}x^3&+\frac{1}{4!}\left\{ \sin{(0)}\right\}x^4&+\frac{1}{5!}\left\{ \cos{(0)}\right\}x^5
\\&&+\frac{1}{6!}\left\{-\sin{(0)}\right\}x^6&+\frac{1}{7!}\left\{-\cos{(0)}\right\}x^7&+\frac{1}{8!}\left\{ \sin{(0)}\right\}x^8
+\cdots
\\&=&\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
\\&=&\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}
0 件のコメント:
コメントを投稿