# Analytic continuation

• Jun 28th 2008, 04:38 AM
Klaus
Analytic continuation
Hello,
I'm trying to understand analytic continuation:
Quote:

Let f and g be two analytic functions.
If f=g on an open subset of $\displaystyle \mathbb{C}$, then f=g on any larger connected subset.
Let f be $\displaystyle 1+x+\frac{x^2}{2!}$ and g be $\displaystyle 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}$ these two analytic functions.

$\displaystyle f=g$ on the open subset of $\displaystyle \mathbb{C}$ $\displaystyle ]-0.1;0.1[$, but $\displaystyle f\not=g$ on the larger connected subset $\displaystyle \mathbb{R}$ !

Where is the problem ?
• Jun 28th 2008, 05:38 AM
mr fantastic
Quote:

Originally Posted by Klaus
Hello,
I'm trying to understand analytic continuation:
Let f be $\displaystyle 1+x+\frac{x^2}{2!}$ and g be $\displaystyle 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}$ these two analytic functions.

$\displaystyle f=g$ on the open subset of $\displaystyle \mathbb{C}$ $\displaystyle ]-0.1;0.1[$, but $\displaystyle f\not=g$ on the larger connected subset $\displaystyle \mathbb{R}$ !

Where is the problem ?

There's no problem. $\displaystyle f \neq g$ on (-0.1, 0.1). In fact, they are only equal at x = 0 .....