0
$\begingroup$

Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is an analytic function, and at one point $x_1 \in \mathbb{R}$ we know the Taylor series for $f$ centered at $x_1$. Does this determine $f$ uniquely? If not, is there some number of points such that knowing the Taylor series at each point determines $f$?

$\endgroup$
  • $\begingroup$ It depends on the radius of convergence of the Taylor Series. Such as the series for $\ln(x+1)$ has a radius of convergence of $-1\le x\le 1$. $\endgroup$ – aleden Jan 26 '18 at 2:35
1
$\begingroup$

The question is equivalent to the following: suppose f has an infinite Taylor series expansion around each point in $\mathbb R$ and suppose f is 0 in a neighborhood of some number $a$. Does it follow that $f=0$ everywhere? The answer is YES! If $f(x) \neq 0$ for some $x>a$. Consider $inf \{x>a:f(x) \neq0\}$. If this infimum is $b$ then all the derivatives of f at b are 0 because the functions vanishes in a left-neighborhood of b. The Taylor seried expansion show that f vanishes in a right neighborhood of b too and this contradicts the definition of b. We have proved that $f=0$ on $(a,\infty)$ and a similar argument shows it vanishes in $(-\infty,a)$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.