It is well known that for analytic function defined on interval $I$ we have $$ f(x)=\sum^{\infty}_{k=0}f^{k}(0)\frac{x^{k}}{k!} $$ and for function defined on $I^{n}\rightarrow \mathbb{R}$ we have $$ f(x)=\sum^{\infty}_{|\alpha|=0}\frac{1}{|\alpha|! }\partial^{\alpha}f(0)x^{\alpha} $$ where $\alpha$ is the multi-index. However, the notation becomes difficult when I work with $$ f:I^{n}\rightarrow \mathbb{R}^{m} $$ because even if I can write out the function in coordinates, I lose some of the global picture how they united in $\mathbb{R}^{m}$. Clearly, the best way to work with this is to introduce the derivative as a matrix valued function in $\mathbb{R}^{mn}$. However, this brings the inconvenience of introducing the Frechet derivative and as an example the total second derivative would be in space $\mathbb{R}^{mn^{2}}$. The Taylor formula now becomes $$ f(x)=f(0)+f'(0)\cdot x+\frac{1}{2}(f''(0)(x)\cdot x)\cdots $$ I am sure this must be well known in literature. But I am having trouble to find a nice readable form that I can manipulate the indices to show some basic properties like when is $f$ analytic on $I^{n}$. I found the high dimensional derivative very difficult to manipulate algebraically. Can someone give me a hint how this operates in real life?
For example, for now the "working definition" I used for $F$ being analytic is
$$ |\partial^{\alpha}F_i(0)|\le \frac{|\alpha|!K^{|\alpha|+1}}{C(d,|\alpha|)},\forall i,\forall |\alpha|>0 $$ where $C(d,|\alpha|)$ is a constant independent of $F(x)$ denoting how many terms we have with degree $|\alpha|$. This should be equivalent to the real one using Frechet derivative using triangular inequality. But I feel this is not the right definition and it is also cumbersome to use.
(This question came from my attempt to prove Cauchy-Kowalevski theorem)
Reference:
http://en.wikipedia.org/wiki/Fr%C3%A9chet_derivative http://en.wikipedia.org/wiki/Derivative#Higher_derivatives
Try Apostol, "Mathematical Analysis." In the edition I have (third printing, Dec. 1960) it appears in section 6-11 of chapter 6. There are some notational conventions established which facilitate the writing of Taylor's formula. You might find them useful for what you want to do.