# 2変数関数のテイラー（Taylor）の定理の導出

$f\left(a+h,b+k\right)=f\left(a,b\right)$$+\frac{1}{1!}\left(h\frac{\partial }{\partial x}+k\frac{\partial }{\partial y}\right)f\left(a,b\right)$$+\frac{1}{2!}\left(h\frac{\partial }{\partial x}+{k\frac{\partial }{\partial y}\right)}^{2}f\left(a,b\right)$$+\cdots$$+\frac{1}{n!}\left(h\frac{\partial }{\partial x}{+k\frac{\partial }{\partial y}\right)}^{n}f\left(a,b\right)$$+{R}_{n+1}$

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

## ■導出

$f\left(a+h,b+k\right)$  は次のように表すことができる．

$f\left(a+h,b+k\right)=f\left(a,b\right)$$+\underset{a}{\overset{a+h}{\int }}{f}_{x}\left(x,b\right)dx$$+\underset{b}{\overset{b+k}{\int }}{f}_{y}\left(a+h,y\right)dy$

この式を部分積分すると　　　積分はこちら

$=f\left(a,b\right)+h{f}_{x}\left(a,b\right)+\frac{1}{2}{h}^{2}{f}_{xx}\left(a,b\right)$$+\frac{1}{6}{h}^{3}{f}_{xxx}\left(a,b\right)$

$+\cdots +\frac{1}{\left(n-1\right)!}{h}^{n-1}{\left(\frac{\partial }{\partial x}\right)}^{n-1}f\left(a,b\right)$$+\frac{1}{n!}{h}^{n}{\left(\frac{\partial }{\partial x}\right)}^{n}f\left(a,b\right)$

$+\underset{a}{\overset{a+h}{\int }}{\left\{-\frac{1}{\left(n+1\right)!}{\left(a+h-x\right)}^{n+1}\right\}}^{\prime }{\left(\frac{\partial }{\partial x}\right)}^{n+1}f\left(x,b\right)dx$

$+k{f}_{y}\left(a+h,b\right)+\frac{1}{2}{k}^{2}{f}_{yy}\left(a+h,b\right)$$+\frac{1}{6}{k}^{3}{f}_{yyy}\left(a+h,b\right)$

$+\cdots +\frac{1}{\left(n-1\right)!}{k}^{n-1}{\left(\frac{\partial }{\partial y}\right)}^{n-1}f\left(a+h,b\right)$$+\frac{1}{n!}{k}^{n}{\left(\frac{\partial }{\partial y}\right)}^{n}f\left(a+h,b\right)$

$+\underset{b}{\overset{b+k}{\int }}{\left\{-\frac{1}{\left(n+1\right)!}{\left(b+k-y\right)}^{n+1}\right\}}^{\prime }{\left(\frac{\partial }{\partial y}\right)}^{n+1}f\left(a+h,y\right)dy$

となることがわかる．

${f}_{y}\left(a+h,b\right)$ は次のように書き換えることができる．

${f}_{y}\left(a+h,b\right)={f}_{y}\left(a,b\right)+{\int }_{a}^{a+h}\frac{\partial }{\partial x}{f}_{y}\left(x,b\right)dx$

${f}_{yy}\left(a+h,b\right)={f}_{yy}\left(a,b\right)+{\int }_{a}^{a+h}\frac{\partial }{\partial x}{f}_{yy}\left(x,b\right)dx$

${f}_{yyy}\left(a+h,b\right)={f}_{yyy}\left(a,b\right)+{\int }_{a}^{a+h}\frac{\partial }{\partial x}{f}_{yyy}\left(x,b\right)dx$

・・・・・・

${\left(\frac{\partial }{\partial y}\right)}^{n-1}f\left(a+h,b\right)={\left(\frac{\partial }{\partial y}\right)}^{n-1}f\left(a,b\right)$$+\underset{a}{\overset{a+h}{\int }}{\left(\frac{\partial }{\partial y}\right)}^{n-1}f\left(x,b\right)dx$

${\left(\frac{\partial }{\partial y}\right)}^{n}f\left(a+h,b\right)={\left(\frac{\partial }{\partial y}\right)}^{n}f\left(a,b\right)$$+\underset{a}{\overset{a+h}{\int }}{\left(\frac{\partial }{\partial y}\right)}^{n}f\left(x,b\right)dx$

となる．よって

$f\left(a+h,b+k\right)$

$=f\left(a,b\right)+h{f}_{x}\left(a,b\right)+\frac{1}{2}{h}^{2}{f}_{xx}\left(a,b\right)$$+\frac{1}{6}{h}^{3}{f}_{xxx}\left(a,b\right)$

$+\cdots +\frac{1}{\left(n-1\right)!}{h}^{n-1}{\left(\frac{\partial }{\partial x}\right)}^{n-1}f\left(a,b\right)\frac{1}{n!}{h}^{n}{\left(\frac{\partial }{\partial x}\right)}^{n}f\left(a,b\right)$$+\underset{a}{\overset{a+h}{\int }}{\left\{-\frac{1}{\left(n+1\right)!}{\left(a+h-x\right)}^{n+1}\right\}}^{\prime }{\left(\frac{\partial }{\partial x}\right)}^{n+1}f\left(x,b\right)dx$

$+k{f}_{y}\left(a,b\right)+\frac{1}{2}{k}^{2}{f}_{yy}\left(a,b\right)+\frac{1}{6}{k}^{3}{f}_{yyy}\left(a,b\right)$$+\cdots$$+\frac{1}{\left(n-1\right)!}{k}^{n-1}{\left(\frac{\partial }{\partial y}\right)}^{n-1}f\left(a,b\right)$$+\frac{1}{n!}{k}^{n}{\left(\frac{\partial }{\partial y}\right)}^{n}f\left(a,b\right)$

$+k\underset{a}{\overset{a+h}{\int }}\frac{\partial }{\partial x}{f}_{y}\left(x,b\right)dx$$+\frac{1}{2}{k}^{2}\underset{a}{\overset{a+h}{\int }}\frac{\partial }{\partial x}{f}_{yy}\left(x,b\right)dx$$+\frac{1}{6}{k}^{3}\underset{a}{\overset{a+h}{\int }}\frac{\partial }{\partial x}{f}_{yyy}\left(x,b\right)dx$

$+\cdots$$+\frac{1}{\left(n-1\right)!}{k}^{n-1}\underset{a}{\overset{a+h}{\int }}{\left(\frac{\partial }{\partial y}\right)}^{n-1}f\left(x,b\right)dx$$+\frac{1}{n!}{k}^{n}\underset{a}{\overset{a+h}{\int }}{\left(\frac{\partial }{\partial y}\right)}^{n}f\left(x,b\right)dx$

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

さらに部分積分をすると　　　積分はこちら

$f\left(a+h,b+k\right)$

$=f\left(a,b\right)+h{f}_{x}\left(a,b\right)+\frac{1}{2}{h}^{2}{f}_{xx}\left(a,b\right)$$+\frac{1}{6}{h}^{3}{f}_{xxx}\left(a,b\right)$$+\cdots +\frac{1}{\left(n-1\right)!}{h}^{n-1}{\left(\frac{\partial }{\partial x}\right)}^{n-1}f\left(a,b\right)$$+\frac{1}{n!}{h}^{n}{\left(\frac{\partial }{\partial x}\right)}^{n}f\left(a,b\right)$

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

$+k{f}_{y}\left(a,b\right)+\frac{1}{2}{k}^{2}{f}_{yy}\left(a,b\right)+\frac{1}{6}{k}^{3}{f}_{yyy}\left(a,b\right)$$+\cdots$$+\frac{1}{\left(n-1\right)!}{k}^{n-1}{\left(\frac{\partial }{\partial y}\right)}^{n-1}f\left(a,b\right)$$+\frac{1}{n!}{k}^{n}{\left(\frac{\partial }{\partial y}\right)}^{n}f\left(a,b\right)$

$+k\left[h{f}_{yx}\left(a,b\right)+\frac{1}{2}{h}^{2}{f}_{yxx}\left(a,b\right)+\cdots$$+\frac{1}{\left(n-1\right)!}{h}^{n-1}{\left(\frac{\partial }{\partial x}\right)}^{n-1}\frac{\partial }{\partial y}f\left(a,b\right)$

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

$+\frac{1}{2}{k}^{2}\left[h{f}_{yyx}\left(a,b\right)+\frac{1}{2}{h}^{2}{f}_{yyxx}\left(a,b\right)$$+\cdots$$+\frac{1}{\left(n-2\right)!}{h}^{n-2}{\left(\frac{\partial }{\partial x}\right)}^{n-2}{\left(\frac{\partial }{\partial y}\right)}^{2}f\left(a,b\right)$

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

$+\frac{1}{6}{k}^{3}\left[h{f}_{yyyx}\left(a,b\right)+\frac{1}{2}{h}^{2}{f}_{yyyxx}\left(a,b\right)$$+\cdots$$\begin{array}{l}+\frac{1}{\left(n-3\right)!}{h}^{n-3}{\left(\frac{\partial }{\partial x}\right)}^{n-3}{\left(\frac{\partial }{\partial y}\right)}^{3}f\left(a,b\right)\end{array}$

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

$+\cdot \cdot \cdot \cdot \cdot \cdot +$

$+\frac{1}{\left(n-1\right)!}{k}^{n-1}\left[h\frac{\partial }{\partial x}{\left(\frac{\partial }{\partial y}\right)}^{n-1}f\left(a,b\right)$$+\underset{a}{\overset{a+h}{\int }}{\left\{-\frac{1}{2!}{\left(a+h-x\right)}^{2}\right\}}^{\prime }{\left(\frac{\partial }{\partial x}\right)}^{2}{\left(\frac{\partial }{\partial y}\right)}^{n-1}f\left(x,b\right)dx\right]$

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

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

この式を整理すると

$=f\left(a,b\right)+h{f}_{x}\left(a,b\right)+k{f}_{y}\left(a,b\right)$$+\frac{1}{2}{h}^{2}{f}_{xx}\left(a,b\right)$$+hk{f}_{yx}\left(a,b\right)+\frac{1}{2}{k}^{2}{f}_{yy}\left(a,b\right)$

$+\frac{1}{6}{h}^{3}{f}_{xxx}\left(a,b\right)+\frac{1}{2}{h}^{2}k{f}_{yxx}\left(a,b\right)$$+\frac{1}{2}h{k}^{2}{f}_{yyx}\left(a,b\right)$$+\frac{1}{6}{k}^{3}{f}_{yyy}\left(a,b\right)+\cdots$

$+\frac{1}{n!}{h}^{n}{\left(\frac{\partial }{\partial x}\right)}^{n}f\left(a,b\right)$$+\frac{1}{\left(n-1\right)!}{h}^{n-1}k{\left(\frac{\partial }{\partial x}\right)}^{n-1}\frac{\partial }{\partial y}f\left(a,b\right)$$+\frac{1}{2\left(n-2\right)!}{h}^{n-2}{k}^{2}{\left(\frac{\partial }{\partial x}\right)}^{n-2}{\left(\frac{\partial }{\partial y}\right)}^{2}f\left(a,b\right)$

$+\frac{1}{6\left(n-3\right)!}{h}^{n-3}{k}^{3}{\left(\frac{\partial }{\partial x}\right)}^{n-3}{\left(\frac{\partial }{\partial y}\right)}^{3}f\left(a,b\right)$$+\cdots$$+\frac{1}{\left(n-1\right)!}h{k}^{n-1}\frac{\partial }{\partial x}{\left(\frac{\partial }{\partial y}\right)}^{n-1}$$+f\left(a,b\right)$$+\frac{1}{n!}{k}^{n}{\left(\frac{\partial }{\partial y}\right)}^{n}f\left(a,b\right)$

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

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

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

$+\frac{1}{6}{k}^{3}{\int }_{a}^{a+h}{\left\{-\frac{1}{\left(n-2\right)!}{\left(a+h-x\right)}^{n-2}\right\}}^{\prime }{\left(\frac{\partial }{\partial x}\right)}^{n-2}{\left(\frac{\partial }{\partial y}\right)}^{3}f\left(x,b\right)dx+\cdots$

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

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

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

$=f\left(a,b\right)+\frac{1}{1!}\left\{h{f}_{x}\left(a,b\right)+k{f}_{y}\left(a,b\right)\right\}$$+\frac{1}{2!}\left\{{h}^{2}{f}_{xx}\left(a,b\right)$$+2hk{f}_{yx}\left(a,b\right)+{k}^{2}{f}_{yy}\left(a,b\right)\right\}$

$+\frac{1}{3!}\left\{{h}^{3}{f}_{xxx}\left(a,b\right)+3{h}^{2}k{f}_{yxx}\left(a,b\right)$$+3h{k}^{2}{f}_{yyx}\left(a,b\right)$$+{k}^{3}{f}_{yyy}\left(a,b\right)\right\}+\cdots$

$+\frac{1}{n!}\left\{{\left(h\frac{\partial }{\partial x}\right)}^{n}f\left(a,b\right)+n{\left(h\frac{\partial }{\partial x}\right)}^{n-1}k\frac{\partial }{\partial y}f\left(a,b\right)$$+\frac{n\left(n-1\right)}{2!}{\left(h\frac{\partial }{\partial x}\right)}^{n-2}{\left(k\frac{\partial }{\partial y}\right)}^{2}f\left(a,b\right)$

$+\frac{n\left(n-1\right)\left(n-2\right)}{3!}{\left(h\frac{\partial }{\partial x}\right)}^{n-3}{\left(k\frac{\partial }{\partial y}\right)}^{3}f\left(a,b\right)$$+\cdots +{\left(k\frac{\partial }{\partial y}\right)}^{n}f\left(a,b\right)\right\}$

$+\frac{1}{\left(n+1\right)!}\left[{\int }_{a}^{a+h}{\left\{-{\left(a+h-x\right)}^{n+1}\right\}}^{\prime }{\left(\frac{\partial }{\partial x}\right)}^{n+1}f\left(x,b\right)dx\right$

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

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

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

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

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

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

$=f\left(a,b\right)+\frac{1}{1!}\left(h\frac{\partial }{\partial x}+k\frac{\partial }{\partial y}\right)f\left(a,b\right)$$+\frac{1}{2!}\left(h\frac{\partial }{\partial x}$${+k\frac{\partial }{\partial y}\right)}^{2}f\left(a,b\right)$

$+\frac{1}{3!}{\left(h\frac{\partial }{\partial x}+k\frac{\partial }{\partial y}\right)}^{3}f\left(a,b\right)$$+\cdots$$+\frac{1}{n!}{\left(h\frac{\partial }{\partial x}+k\frac{\partial }{\partial y}\right)}^{n}f\left(a,b\right)$

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

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

ここで，

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

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

${R}_{n+1}$に置き換えると

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

となる．しかし ${R}_{n+1}$

${R}_{n+1}=\frac{1}{\left(n+1\right)!}{\left(h\frac{\partial }{\partial x}+k\frac{\partial }{\partial y}\right)}^{n+1}f\left(a+\theta h,b+\theta k\right)$

である．

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