2変数関数のテイラー（Taylor）の定理の導出

(1)式

$3$\displaystyle{\displaystyle{ \int _{a}^{ a+h } \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\hspace{1}{f}_{y} $ x,b $ dx }}$

$3$\displaystyle{=\displaystyle{ \int _{a}^{ a+h } 1\cdot \hspace{1}{f}_{ yx } $ x,b $ dx }}$

$3$\displaystyle{=\displaystyle{ \int _{a}^{ a+h } - { $ a+h- x $ }^{\prime }\cdot \hspace{1}{f}_{ yx } $ x,b $ dx }}$

部分積分法を用いると

$3$\displaystyle{ = \[ - $ a+h- x $ \hspace{1}{f}_{ yx } $ x,b $ \] _{a}^{ a+h } }$ $3$\displaystyle{ - {\displaystyle{\int }}\limits_{a}^{ a+h }- $ a+h- x $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\hspace{1}{f}_{ yx } $ x,b $ dx }$

$3$\displaystyle{ =h\hspace{1}{f}_{ yx } $ a,b $ }$ $3$\displaystyle{ +{\displaystyle{\int }}\limits_{a}^{ a+h } $ a+h- x $ \hspace{1}{f}_{ yxx } $ x,b $ dx }$

$3$\displaystyle{ =h\hspace{1}{f}_{ yx } $ a,b $ }$ $3$\displaystyle{ +{\displaystyle{\int }}\limits_{a}^{ a+h }{ \{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{ $ a+h- x $ }^{2} \} }^{\prime }\hspace{1}{f}_{ yxx } $ x,b $ dx }$

$3$\displaystyle{ =h\hspace{1}{f}_{ yx } $ a,b $ }$ $3$\displaystyle{ + \[ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{ $ a+h- x $ }^{2}\hspace{1}{f}_{ yxx } $ x,b $ \] _{a}^{ a+h } }$ $3$\displaystyle{ - {\displaystyle{\int }}\limits_{a}^{ a+h } \{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{ $ a+h- x $ }^{2} \} \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\hspace{1}{f}_{ yxx } $ x,b $ dx }$

$3$\displaystyle{ =h\hspace{1}{f}_{ yx } $ a,b $ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{h}^{2}\hspace{1}{f}_{ yxx } $ a,b $ }$ $3$\displaystyle{ +{\displaystyle{\int }}\limits_{a}^{ a+h } \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{ $ a+h- x $ }^{2} \} \hspace{1}{f}_{ yxxx } $ x,b $ dx }$

積分を繰り返すと

$3$\displaystyle{ =h\hspace{1}{f}_{ yx } $ a,b $ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{h}^{2}\hspace{1}{f}_{ yxx } $ a,b $ }$ $3$\displaystyle{ +\cdots }$ $3$\displaystyle{ +\frac{\hspace{2}1\hspace{2}}{\hspace{2} $ n- 1 $ ! \hspace{2}}{h}^{ n- 1 }{ $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ }^{ n- 1 }\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}}f $ a,b $ }$

$3$\displaystyle{ +{\displaystyle{\int }}\limits_{a}^{ a+h }{ \{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2} n! \hspace{2}}{ $ a+h- x $ }^{n} \} }^{\prime }{ $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ }^{n}\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}}f $ x,b $ dx }$

(2)式

$3$\displaystyle{\displaystyle{ \int _{a}^{ a+h } \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\hspace{1}{f}_{ yy } $ x,b $ dx }}$

$3$\displaystyle{=\displaystyle{ \int _{a}^{ a+h } 1\cdot \hspace{1}{f}_{ yyx } $ x,b $ dx }}$

$3$\displaystyle{=\displaystyle{ \int _{a}^{ a+h } - { $ a+h- x $ }^{\prime }\cdot \hspace{1}{f}_{ yyx } $ x,b $ dx }}$

部分積分法を用いると

$3$\displaystyle{ = \[ - $ a+h- x $ \hspace{1}{f}_{ yyx } $ x,b $ \] _{a}^{ a+h } }$ $3$\displaystyle{ - {\displaystyle{\int }}\limits_{a}^{ a+h }- $ a+h- x $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\hspace{1}{f}_{ yyx } $ x,b $ dx }$

$3$\displaystyle{=h\hspace{1}{f}_{ yyx } $ a,b $ +\displaystyle{ \int _{a}^{ a+h } $ a+h- x $ \hspace{1}{f}_{ yyxx } $ x,b $ dx }}$

$3$\displaystyle{ =h\hspace{1}{f}_{ yyx } $ a,b $ }$ $3$\displaystyle{ +{\displaystyle{\int }}\limits_{a}^{ a+h }{ \{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{ $ a+h- x $ }^{2} \} }^{\prime }\hspace{1}{f}_{ yyxx } $ x,b $ dx }$

$3$\displaystyle{ =h\hspace{1}{f}_{ yyx } $ a,b $ }$ $3$\displaystyle{ + \[ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{ $ a+h- x $ }^{2}\hspace{1}{f}_{ yyxx } $ x,b $ \] _{a}^{ a+h } }$ $3$\displaystyle{ - {\displaystyle{\int }}\limits_{a}^{ a+h } \{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{ $ a+h- x $ }^{2} \} \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\hspace{1}{f}_{ yyxx } $ x,b $ dx }$

$3$\displaystyle{ =h\hspace{1}{f}_{ yyx } $ a,b $ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{h}^{2}\hspace{1}{f}_{ yyxx } $ a,b $ }$ $3$\displaystyle{ +{\displaystyle{\int }}\limits_{a}^{ a+h } \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{ $ a+h- x $ }^{2} \} \hspace{1}{f}_{ yyxxx } $ x,b $ dx }$

(1)式と同様に，積分を繰り返すと

$3$\displaystyle{ =h\hspace{1}{f}_{ yyx } $ a,b $ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{h}^{2}\hspace{1}{f}_{ yyxx } $ a,b $ }$ $3$\displaystyle{ +\cdots }$ $3$\displaystyle{ +\frac{\hspace{2}1\hspace{2}}{\hspace{2} $ n- 2 $ ! \hspace{2}}{h}^{ n- 2 }{ $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ }^{ n- 2 }{ $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ }^{2}f $ a,b $ }$

$3$\displaystyle{ +{\displaystyle{\int }}\limits_{a}^{ a+h }{ \{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2} $ n- 1 $ ! \hspace{2}}{ $ a+h- x $ }^{ n- 1 } \} }^{\prime }{ $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ }^{ n- 1 }{ $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ }^{2}f $ x,b $ dx }$

(3)式

$3$\displaystyle{\displaystyle{ \int _{a}^{ a+h } \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\hspace{1}{f}_{ yyy } $ x,b $ dx }}$

$3$\displaystyle{=\displaystyle{ \int _{a}^{ a+h } 1\cdot \hspace{1}{f}_{ yyyx } $ x,b $ dx }}$

$3$\displaystyle{=\displaystyle{ \int _{a}^{ a+h } - { $ a+h- x $ }^{\prime }\hspace{1}{f}_{ yyyx } $ x,b $ dx }}$

$3$\displaystyle{ = \[ - $ a+h- x $ \hspace{1}{f}_{ yyyx } $ x,b $ \] _{a}^{ a+h } }$ $3$\displaystyle{ - {\displaystyle{\int }}\limits_{a}^{ a+h }- $ a+h- x $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\hspace{1}{f}_{ yyyx } $ x,b $ dx }$

$3$\displaystyle{ =h\hspace{1}{f}_{ yyyx } $ a,b $ }$ $3$\displaystyle{ +{\displaystyle{\int }}\limits_{a}^{ a+h }{ \{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{ $ a+h- x $ }^{2} \} }^{\prime }\hspace{1}{f}_{ yyyxx } $ x,b $ dx }$

$3$\displaystyle{ =h\hspace{1}{f}_{ yyyx } $ a,b $ }$ $3$\displaystyle{ + \[ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{ $ a+h- x $ }^{2}\hspace{1}{f}_{ yyyxx } $ x,b $ \] _{a}^{ a+h } }$ $3$\displaystyle{ - {\displaystyle{\int }}\limits_{a}^{ a+h } \{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{ $ a+h- x $ }^{2} \} \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\hspace{1}{f}_{ yyyxx } $ x,b $ dx }$

$3$\displaystyle{ =h\hspace{1}{f}_{ yyyx } $ a,b $ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{h}^{2}\hspace{1}{f}_{ yyyxx } $ a,b $ }$ $3$\displaystyle{ +{\displaystyle{\int }}\limits_{a}^{ a+h } \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{ $ a+h- x $ }^{2} \} \hspace{1}{f}_{ yyyxxx } $ x,b $ dx }$

積分を繰り返すと

$3$\displaystyle{ =h\hspace{1}{f}_{ yyyx } $ a,b $ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{h}^{2}\hspace{1}{f}_{ yyyxx } $ a,b $ }$ $3$\displaystyle{ +\cdots }$ $3$\displaystyle{ +\frac{\hspace{2}1\hspace{2}}{\hspace{2} $ n- 3 $ ! \hspace{2}}{h}^{ n- 3 }{ $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ }^{ n- 3 }{ $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ }^{3}f $ a,b $ }$

$3$\displaystyle{ +{\displaystyle{\int }}\limits_{a}^{ a+h }{ \{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2} $ n- 2 $ ! \hspace{2}}{ $ a+h- x $ }^{ n- 2 } \} }^{\prime }{ $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ }^{ n- 2 }{ $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ }^{3}f $ x,b $ dx }$

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最終更新日： 2023年1月21日