# Analytic Functions

• Jean-Pierre Serre
Part of the Lecture Notes in Mathematics book series (LNM, volume 1500)

## Abstract

We first fix some notation:
1. 1.

k: field, complete with respect to a non-trivial absolute value. k[[X1,...,X n ]]: formal power series in n variables X1,...,X n .

2. 2.
We use:
1. a.

Greek letters α, β for n-tuples as α = (α1,...,αn), αi ≥ 0, ∈ Z.

2. b.

Latin letters r, s for n-tuples as r = (r1,...,r n ), r i > 0, ∈ R.

3. c.

Latin letters x, y for n-tuples as x = (x1,..., x n ), x i k.

3. 3.
We set:
$$\begin{array}{*{20}{c}} {{{r}^{\alpha }}=r_{1}^{{{{\alpha }_{1}}}}\cdots r_{n}^{{{{\alpha }_{n}}}}} \\ {{{x}^{\alpha }}=x_{1}^{{{{\alpha }_{1}}}}\cdots x_{n}^{{{{\alpha }_{n}}}}} \\ {{{X}^{\alpha }}=X_{1}^{{{{\alpha }_{1}}}}\cdots X_{n}^{{{{\alpha }_{n}}}}} \\ {\left| \alpha \right|=\sum {{{\alpha }_{i}}} } \\ {\alpha !=\prod {{{\alpha }_{i}}!} } \\ {\left( {\begin{array}{*{20}{c}} \alpha \\ \beta \\ \end{array}} \right)=\frac{{\alpha !}}{{\beta !\left( {\alpha -\beta } \right)!}}} \\ \end{array}$$

4. 4.

We define: |x| ≤ r (resp. |x| < r) ⇔ |x i | ≤ r i (resp. |x i | < r i ), 1 ≤ in. We define similarly r′ ≤ r, r′ < r, α′ ≤ α, and α′ < α.

5. 5.

We set:

P(r)(x) = { y : |y - x| ≤ r } = Polydisk of radius r about x

P0(r)(x) = { y : |y - x| < r } = Strict polydisk of radiues r about x.

P(r) = P(r)(0)

P0(r) = P0(r)(0).