2変数関数のテイラー（Taylor）の定理の導出

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

$3$\displaystyle{ \hspace{1}{R}_{ n+1 } }$ $3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2} $ n+1 $ ! \hspace{2}} $ h\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}} { +k\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ }^{ n+1 } f $ a+{\theta}h,b+{\theta}k $ }$ 　　（ただし, $3$\displaystyle{ 0< {\theta}< 1 }$ ）

■導出

$3$\displaystyle{ f $ a+h,b+k $ }$ は次のように表すことができる．

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

この式を部分積分すると　　　⇒積分はこちら

$3$\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 } }$

$3$\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{ +\cdots +\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 } }$

となることがわかる．

次に $3$\displaystyle{ \hspace{1}{f}_{y} $ a+h,b $ }$ ， $3$\displaystyle{ \hspace{1}{f}_{yy} $ a+ h,b $ }$ ， $3$\displaystyle{ \hspace{1}{f}_{yyy} $ a+ h,b $ }$ ，･･･， $3$\displaystyle{{ $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ }^{ n- 1 }f $ a+h,b $ }$ ， $3$\displaystyle{{ $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ }^{n}f $ a+h,b $ }$ について考える．

$3$\displaystyle{ \hspace{1}{f}_{y} $ a+h,b $ }$ は次のように書き換えることができる．

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

同様に

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

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

となる．よって

$3$\displaystyle{f $ a+h,b+k $ }$

$3$\displaystyle{ +\cdots +\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 $ \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 }$

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

$3$\displaystyle{ +k{\displaystyle{\int }}\limits_{a}^{ a+h }\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\hspace{1}{f}_{y} $ x,b $ dx }$ $3$\displaystyle{ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{k}^{2}{\displaystyle{\int }}\limits_{a}^{ a+h }\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\hspace{1}{f}_{ yy } $ x,b $ dx }$ $3$\displaystyle{ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{k}^{3}{\displaystyle{\int }}\limits_{a}^{ a+h }\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}\hspace{1}{f}_{ yyy } $ x,b $ dx }$

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

$3$\displaystyle{ \displaystyle{+ \displaystyle{\int _{b}^{ b+ k } {\left{{ - \frac{\hspace{2}1\hspace{2}}{\hspace{2} \left({ n+ 1 }\right)! \hspace{2}}{\left({ b+ k- y }\right)}^{ n+ 1 } }\right}}^{\prime }{ $\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}}$ }^{ n+ 1 }f $ a+ h,y $ dy }} }$

さらに部分積分をすると　　　⇒積分はこちら

$3$\displaystyle{f $ a+h,b+k $ }$

$3$\displaystyle{ =f $ a,b $ +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}}{h}^{3}\hspace{1}{f}_{ xxx } $ a,b $ }$ $3$\displaystyle{ +\cdots +\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 _{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 } }$

$3$\displaystyle{+k \[ 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 $ +\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{ \left.{ + \displaystyle{\int _{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 } }\right\] }$

$3$\displaystyle{ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{k}^{2} \[ 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{ \left.{ + \displaystyle{\int _{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 } }\right\] }$

$3$\displaystyle{+\frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{k}^{3} \[ 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{\begin{array}{lll}+\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 $ & \vspace{6}\\ \end{array}}$

$3$\displaystyle{ \left.{ + \displaystyle{\int _{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 } }\right\] }$

$3$\displaystyle{ + \cdot \cdot \cdot \cdot \cdot \cdot + }$

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

$3$\displaystyle{ + \frac{\hspace{2}1\hspace{2}}{\hspace{2} n! \hspace{2}}{k}^{n}+ \displaystyle{\int _{a}^{ a+ h } {\left{{ - \left({ a+ h- x }\right) }\right}}^{\prime }$\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}${ $\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}}$ }^{n}f $ x,b $ dx } }$

$3$\displaystyle{ + \displaystyle{\int _{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 } }$

この式を整理すると

$3$\displaystyle{ =f $ a,b $ +h\hspace{1}{f}_{x} $ a,b $ +k\hspace{1}{f}_{y} $ a,b $ }$ $3$\displaystyle{ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{h}^{2}\hspace{1}{f}_{ xx } $ a,b $ }$ $3$\displaystyle{ +hk\hspace{1}{f}_{ yx } $ a,b $ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{k}^{2}\hspace{1}{f}_{ yy } $ a,b $ }$

$3$\displaystyle{ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{h}^{3}\hspace{1}{f}_{ xxx } $ a,b $ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{h}^{2}k\hspace{1}{f}_{ yxx } $ a,b $ }$ $3$\displaystyle{ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}h{k}^{2}\hspace{1}{f}_{ yyx } $ a,b $ }$ $3$\displaystyle{ +\frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{k}^{3}\hspace{1}{f}_{ yyy } $ a,b $ +\cdots }$

$3$\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{ +\frac{\hspace{2}1\hspace{2}}{\hspace{2} $ n- 1 $ ! \hspace{2}}{h}^{ n- 1 }k{ $ \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{ +\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2 $ n- 2 $ ! \hspace{2}}{h}^{ n- 2 }{k}^{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{ +\frac{\hspace{2}1\hspace{2}}{\hspace{2} 6 $ n- 3 $ ! \hspace{2}}{h}^{ n- 3 }{k}^{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{ +\cdots }$ $3$\displaystyle{ +\frac{\hspace{2}1\hspace{2}}{\hspace{2} $ n- 1 $ ! \hspace{2}}h{k}^{ n- 1 }\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}{ $ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ }^{ n- 1 } }$ $3$\displaystyle{ +f $ a,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,b $ }$

$3$\displaystyle{+ \displaystyle{\int _{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 }}$

$3$\displaystyle{ + k\displaystyle{\int _{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 } }$

$3$\displaystyle{+ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}{k}^{2}\displaystyle{\int _{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 }}$

$3$\displaystyle{+ \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{k}^{3}\displaystyle{\int _{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 }+ \cdots }$

$3$\displaystyle{+ \frac{\hspace{2}1\hspace{2}}{\hspace{2} \left({ n- 1 }\right)! \hspace{2}}{k}^{ n- 1 }\displaystyle{\int _{a}^{ a+ h } { \{- \frac{\hspace{2}1\hspace{2}}{\hspace{2} 2 ! \hspace{2}}{ $ a+ h- x $ }^{ 2 } \} }^{\prime }{ $\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}$ }^{ 2 }{ $\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}}$ }^{ n- 1 }f $ x,b $ dx }}$

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

$3$\displaystyle{+ \displaystyle{\int _{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 }}$

$3$\displaystyle{ =f $ a,b $ +\frac{\hspace{2}1\hspace{2}}{\hspace{2} 1! \hspace{2}} \{ h\hspace{1}{f}_{x} $ a,b $ +k\hspace{1}{f}_{y} $ a,b $ \} }$ $3$\displaystyle{ +\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2! \hspace{2}} \{ {h}^{2}\hspace{1}{f}_{ xx } $ a,b $ }$ $3$\displaystyle{ +2hk\hspace{1}{f}_{ yx } $ a,b $ +{k}^{2}\hspace{1}{f}_{ yy } $ a,b $ \} }$

$3$\displaystyle{ +\frac{\hspace{2}1\hspace{2}}{\hspace{2} 3! \hspace{2}} \{ {h}^{3}\hspace{1}{f}_{ xxx } $ a,b $ +3{h}^{2}k\hspace{1}{f}_{ yxx } $ a,b $ }$ $3$\displaystyle{ +3h{k}^{2}\hspace{1}{f}_{ yyx } $ a,b $ }$ $3$\displaystyle{ +{k}^{3}\hspace{1}{f}_{ yyy } $ a,b $ \} +\cdots }$

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

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

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

$3$\displaystyle{ + \left({ n+ 1 }\right)k\displaystyle{\int _{a}^{ a+ h } { \{ - { $ 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 } }$

$3$\displaystyle{ + \frac{\hspace{2} $ n+ 1 $n \hspace{2}}{\hspace{2}2\hspace{2}}{k}^{2}\displaystyle{\int _{a}^{ a+ h } { \{ - { $ 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 } }$

$3$\displaystyle{ + \frac{\hspace{2} $ n+ 1 $n $ n- 1 $ \hspace{2}}{\hspace{2} 3! \hspace{2}}{k}^{3}\displaystyle{\int _{a}^{ a+ h } { \{ - { $ 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 }+ \cdots }$

$3$\displaystyle{ + \frac{\hspace{2} $ n+ 1 $n $ n- 1 $ \cdots 4\cdot 3 \hspace{2}}{\hspace{2} \left({ n- 1 }\right)! \hspace{2}}{k}^{ n- 1 }\displaystyle{\int _{a}^{ a+ h } { \{ - { $ a+ h- x $ }^{2} \} }^{\prime }{ $\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}$ }^{2}{ $\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}}$ }^{ n- 1 }f $ x,b $ dx } }$

$3$\displaystyle{ \left.{ + \frac{\hspace{2} $ n+ 1 $n $ n- 1 $ \cdots 3\cdot 2 \hspace{2}}{\hspace{2} n! \hspace{2}}{k}^{n}\displaystyle{\int _{a}^{ a+ h } { \{- \left({ a+ h- x }\right)\} }^{\prime }{ \frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}$\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}}$ }^{n}f $ x,b $ dx } }\right\] }$

$3$\displaystyle{ + \displaystyle{\int _{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 } }$

$3$\displaystyle{ =f $ a,b $ +\frac{\hspace{2}1\hspace{2}}{\hspace{2} 1! \hspace{2}} $ h\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}+k\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ f $ a,b $ }$ $3$\displaystyle{ +\frac{\hspace{2}1\hspace{2}}{\hspace{2} 2! \hspace{2}} $ h\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }$ $3$\displaystyle{ { +k\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ }^{2}f $ a,b $ }$

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

$3$\displaystyle{ + \frac{\hspace{2}1\hspace{2}}{\hspace{2} $ n+ 1 $ ! \hspace{2}}\left[{\displaystyle{\left{{ {\sum }\limits_{ r=0 }^{n}\hspace{1}{}_{ n+1 }\hspace{1}{C}_{r} {k}^{r}\displaystyle{\int _{a}^{ a+ h } { \{ - { $ a+ h- x $ }^{ n- r+ 1 } \} }^{\prime }{ $\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}$ }^{ n- r+ 1 }{ $\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}}$ }^{r}f $ x,b $ dx } }\right}} }$

$3$\displaystyle{ \left.{ + \displaystyle{\int _{b}^{ b+ k } { \{ - { $ 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 } }\right\] }$

ここで，

$3$\displaystyle{ \frac{\hspace{2}1\hspace{2}}{\hspace{2} $ n+ 1 $ ! \hspace{2}}\left[{\displaystyle{\left{{ {\sum }\limits_{ r=0 }^{n}\hspace{1}{}_{ n+1 }\hspace{1}{C}_{r} {k}^{r}\displaystyle{\int _{a}^{ a+ h } { \{ - { $ a+ h- x $ }^{ n- r+ 1 } \} }^{\prime }{ $\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}$ }^{ n- r+ 1 }{ $\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}}$ }^{r}f $ x,b $ dx } }\right}} }$

を $3$\displaystyle{\hspace{1}{R}_{ n+1 }}$ に置き換えると

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

となる．しかし $3$\displaystyle{\hspace{1}{R}_{ n+1 }}$

$3$\displaystyle{\hspace{1}{R}_{ n+1 }=\frac{\hspace{2}1\hspace{2}}{\hspace{2} $ n+1 $ ! \hspace{2}}{ $ h\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}x \hspace{2}}+k\frac{\hspace{2}{\partial}\hspace{2}}{\hspace{2} {\partial}y \hspace{2}} $ }^{ n+1 }f $ a+{\theta}h,b+{\theta}k $ }$

である．

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最終更新日： 2023年10月16日