2変数関数のテイラー（Taylor）の定理の導出

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

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

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

$3$\displaystyle{ = \[ - $ a+h- x $ \hspace{1}{f}_{x} $ x,b $ \] _{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}_{x} $ x,b $ dx }$

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

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

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

$3$\displaystyle{ =h\hspace{1}{f}_{x} $ a,b $ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{h}^{2}\hspace{1}{f}_{ xx } $ 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}_{ xxx } $ x,b $ dx }$

$3$\displaystyle{ =h\hspace{1}{f}_{x} $ a,b $ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{h}^{2}\hspace{1}{f}_{ xx } $ a,b $ }$ $3$\displaystyle{ +{\displaystyle{\int }}\limits_{a}^{ a+h }{ \{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{ $ a+h- x $ }^{3} \} }^{\prime }\hspace{1}{f}_{ xxx } $ x,b $ dx }$

$3$\displaystyle{ =h\hspace{1}{f}_{x} $ a,b $ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{h}^{2}\hspace{1}{f}_{ xx } $ a,b $ }$ $3$\displaystyle{ + \[ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{ $ a+h- x $ }^{3}\hspace{1}{f}_{ xxx } $ x,b $ \] _{a}^{ a+h } }$ $3$\displaystyle{ - {\displaystyle{\int }}\limits_{a}^{ a+h } \{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{ $ a+h- x $ }^{3} \} \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\hspace{1}{f}_{ xxx } $ x,b $ dx }$

$3$\displaystyle{ =h\hspace{1}{f}_{x} $ a,b $ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{h}^{2}\hspace{1}{f}_{ xx } $ a,b $ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{h}^{3}\hspace{1}{f}_{ xxx } $ a,b $ }$ $3$\displaystyle{ +{\displaystyle{\int }}\limits_{a}^{ a+h } \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{ $ a+h- x $ }^{3} \} \hspace{1}{f}_{ xxxx } $ x,b $ dx }$

このことから，積分を続けていくと

$3$\displaystyle{ +\cdots \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 }f $ a,b $ }}$ $3$\displaystyle{ +\frac{\hspace{2}1\hspace{2}}{\hspace{2} n! \hspace{2}}{h}^{n}{ $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}} $ }^{n}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 }f $ x,b $ dx }$

となる．

$3$\displaystyle{\displaystyle{ \int _{b}^{ b+k } \hspace{1}{f}_{y} $ a+h,b $ dy }}$

$3$\displaystyle{=\displaystyle{ \int _{b}^{ b+k } 1\cdot \hspace{1}{f}_{y} $ a+h,y $ dy }}$

$3$\displaystyle{=\int _{b}^{ b+ k }{\left{{ - $ b+ k- y $ }\right}}^{\prime } \hspace{1}{f}_{y} $ a+ h,y $ dy }$

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

$3$\displaystyle{=k\hspace{1}{f}_{y} $ a+h,b $ +\displaystyle{ \int _{b}^{ b+k } $ b+k- y $ \hspace{1}{f}_{ yy } $ a+h,y $ dy }}$

$3$\displaystyle{=k\hspace{1}{f}_{y} $ a+h,b $ +\displaystyle{ \int _{b}^{ b+k } { \{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{ $ b+k- y $ }^{2} \} }^{\prime }\hspace{1}{f}_{ yy } $ a+h,y $ dy }}$

$3$\displaystyle{ =k\hspace{1}{f}_{y} $ a+h,b $ + \[ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{ $ b+k- y $ }^{2}\hspace{1}{f}_{ yy } $ a+h,y $ \] _{b}^{ b+k } }$ $3$\displaystyle{ - {\displaystyle{\int }}\limits_{b}^{ b+k } \{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{ $ b+k- y $ }^{2} \} \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}}\hspace{1}{f}_{ yy } $ a+h,y $ dy }$

$3$\displaystyle{ =k\hspace{1}{f}_{y} $ a+h,b $ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{k}^{2}\hspace{1}{f}_{ yy } $ a+h,b $ }$ $3$\displaystyle{ +{\displaystyle{\int }}\limits_{b}^{ b+k } \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{ $ b+k- y $ }^{2} \} \hspace{1}{f}_{ yyy } $ a+h,y $ dy }$

$3$\displaystyle{ =k\hspace{1}{f}_{y} $ a+h,b $ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{k}^{2}\hspace{1}{f}_{ yy } $ a+h,b $ }$ $3$\displaystyle{ +{\displaystyle{\int }}\limits_{b}^{ b+k }{ \{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{ $ b+k- y $ }^{3} \} }^{\prime }\hspace{1}{f}_{ yyy } $ a+h,y $ dy }$

$3$\displaystyle{ =k\hspace{1}{f}_{y} $ a+h,b $ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{k}^{2}\hspace{1}{f}_{ yy } $ a+h,b $ }$ $3$\displaystyle{ + \[ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{ $ b+k- y $ }^{3}\hspace{1}{f}_{ yyy } $ a+h,y $ \] _{b}^{ b+k } }$ $3$\displaystyle{ - {\displaystyle{\int }}\limits_{b}^{ b+k } \{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{ $ b+k- y $ }^{3} \} \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}}\hspace{1}{f}_{ yyy } $ a+h,y $ dy }$

$3$\displaystyle{ =k\hspace{1}{f}_{y} $ a+h,b $ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{k}^{2}\hspace{1}{f}_{ yy } $ a+h,b $ }$ $3$\displaystyle{ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{k}^{3}\hspace{1}{f}_{ yyy } $ a+h,b $ }$ $3$\displaystyle{ +{\displaystyle{\int }}\limits_{b}^{ b+k } \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{ $ b+k- y $ }^{3} \} \hspace{1}{f}_{ yyyy } $ a+h,y $ dy }$

このことから，積分を続けていくと

$3$\displaystyle{=k\hspace{1}{f}_{y} $ a+h,b $ }$ $3$\displaystyle{+\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{k}^{2}\hspace{1}{f}_{ yy } $ a+h,b $ }$ $3$\displaystyle{+\frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{k}^{3}\hspace{1}{f}_{ yyy } $ a+h,b $ }$

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

となる．

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最終更新日： 2019年9月17日