1変数関数 \(f(x)\) のテイラー展開は以下のようになるのだった。
\begin{eqnarray}
f(x + h) &=& f(x) + f'(x) h + \frac{1}{2!} f^{”}(x) h^2 + \cdots + \frac{1}{n!} f^{({n})}(x) h^n + \cdots\\
&=& f(x) + h \frac{d}{dx} f(x) + \frac{1}{2!} \left(h\frac{d}{dx}\right)^2 f(x) + \cdots + \frac{1}{n!} \left(h\frac{d}{dx}\right)^n f(x) + \cdots
\end{eqnarray}
2変数関数 \(f(x, y)\) のテイラー展開も同様。
\begin{eqnarray}
f(x+h, y+k) = f(x,y) &+& \left(h\frac{\partial}{\partial x} + k\frac{\partial}{\partial y}\right) f(x,y) \\
&+& \frac{1}{2!} \left(h\frac{\partial}{\partial x} + k\frac{\partial}{\partial y}\right)^2 f(x,y) + \cdots \\
&+& \frac{1}{n!} \left(h\frac{\partial}{\partial x} + k\frac{\partial}{\partial y}\right)^n f(x,y) + \cdots
\end{eqnarray}
各項を具体的に計算すると…
\begin{eqnarray}
\left(h\frac{\partial}{\partial x} + k\frac{\partial}{\partial y}\right) f(x,y) &=& h \frac{\partial f}{\partial x} + k \frac{\partial f}{\partial y}\\
\left(h\frac{\partial}{\partial x} + k\frac{\partial}{\partial y}\right)^2 f(x,y) &=&
\left(h\frac{\partial}{\partial x} + k\frac{\partial}{\partial y}\right) \left( h \frac{\partial f}{\partial x} + k \frac{\partial f}{\partial y}\right)\\
&=& h^2 \frac{\partial^2 f}{\partial x^2} + hk \frac{\partial^2 f}{\partial x \partial y} + kh \frac{\partial^2 f}{\partial y \partial x} + k^2 \frac{\partial^2 f}{\partial y^2} \\
&=& h^2 \frac{\partial^2 f}{\partial x^2} + 2 hk \frac{\partial^2 f}{\partial x \partial y}+ k^2 \frac{\partial^2 f}{\partial y^2}\\
\left(h\frac{\partial}{\partial x} + k\frac{\partial}{\partial y}\right)^3 f(x,y) &=& \dots
\end{eqnarray}