Let \(f(x)\) and \(g(x)\) be two power series . Let \(x_0\) be a point on the common domain of \(f\) and \(g\) with \(g(x_0)\neq 0\). Then
\begin{equation*} h(x)=\frac{f(x)}{g(x)} \end{equation*}is also a power series with the center \(x_0\) (or it is real analytic at \(x_0\)). [1, Proposition 1.1.12]
The same is true for multidimensional power series [1, Proposition 2.2.2].
Remarks
- For an exact formula for the coefficients of \(h\) see [1, Proposition 1.1.12].
- This result implies that the division of two analytic functions is analytic.
References Link to heading
- S. Krantz and H. Parks, A Primer of Real Analytic Functions. Boston, MA: Birkhäuser Boston, 2002. doi:10.1007/978-0-8176-8134-0