On (real) entire functions and the identity theorem

[psie]

TL;DR Summary

In a footnote in Ordinary Differential Equations by Adkins and Davidson, I read about power series of infinite radius of convergence and that they are "determined completely by its values on $[0,\infty)$". This claim confuses me.

In Ordinary Differential Equations by Adkins and Davidson, in a chapter on the Laplace transform (specifically, in a section where they discuss the linear space $\mathcal{E}_{q(s)}$ of input functions that have Laplace transforms that can be expressed as proper rational functions with a fixed polynomial $q(s)$ in the denominator), I read the following two sentences in a footnote:

In fact, any function which has a power series with infinite radius of convergence [...] is completely determined by its values on $[0,\infty)$. This is so since $f(t)=\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}t^n$ and $f^{(n)}(0)$ are computed from $f(t)$ on $[0,\infty)$.

Both of these sentences confuse me, but especially the latter one. $f^{(n)}$ evaluated at $0$ depends on the values of $f^{(n-1)}$ in an arbitrary small neighborhood around $0$. What do they mean by "$f(t)=\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}t^n$ and $f^{(n)}(0)$ are computed from $f(t)$ on $[0,\infty)$"?

For the first sentence, I suspect they are maybe referring to the identity theorem. Suppose $f$ and $g$ are two analytic functions with domain $\mathbb R$ and suppose they equal on some subinterval of $\mathbb R$ with a limit point in $\mathbb R$. Then they equal on $\mathbb R$, so we can say that an analytic function is completely determined by its values on a subinterval with a limit point in $\mathbb R$, e.g. $[0,\infty)$.

 

[pasmith]

A function $f$ having an infinite radius of convergence means that there exists a sequence $(a_n)_{n \geq 0}$ of real numbers such that $$f(t) = \sum_{n=0}^\infty a_n t^n$$ is true for every $t \in \mathbb{R}$. It follows by direct differentiation (which can be done term-by-term within the radius of convergence) that $$f^{(n)}(0) = n!a_n$$ so that $f^{(n)}(0)$ exists.

Since $f^{(n)}(0)$ exists, it must be equal to the one-sided limit $$\lim_{t \to 0^{+}} \frac{f^{(n-1)}(t) - f^{(n-1)}(0)}{t}$$ which depends only on the values of $f^{(n-1)}$, and hence ultimately of $f$, on the interval $[0, \infty)$.

(The converse does not hold: for $f^{(n)}(0)$ to exist we need the existence and value of the limit to be independent of the direction in which we approach the origin.)

 

[WWGD]

TL;DR Summary: In a footnote in Ordinary Differential Equations by Adkins and Davidson, I read about power series of infinite radius of convergence and that they are "determined completely by its values on $[0,\infty)$". This claim confuses me.

In Ordinary Differential Equations by Adkins and Davidson, in a chapter on the Laplace transform (specifically, in a section where they discuss the linear space $\mathcal{E}_{q(s)}$ of input functions that have Laplace transforms that can be expressed as proper rational functions with a fixed polynomial $q(s)$ in the denominator), I read the following two sentences in a footnote:
Both of these sentences confuse me, but especially the latter one. $f^{(n)}$ evaluated at $0$ depends on the values of $f^{(n-1)}$ in an arbitrary small neighborhood around $0$. What do they mean by "$f(t)=\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}t^n$ and $f^{(n)}(0)$ are computed from $f(t)$ on $[0,\infty)$"?

For the first sentence, I suspect they are maybe referring to the identity theorem. Suppose $f$ and $g$ are two analytic functions with domain $\mathbb R$ and suppose they equal on some subinterval of $\mathbb R$ with a limit point in $\mathbb R$. Then they equal on $\mathbb R$, so we can say that an analytic function is completely determined by its values on a subinterval with a limit point in $\mathbb R$, e.g. $[0,\infty)$.

I believe the Identity theorem, that two functions that agree on a subset containing a limit point ( i.e., not just in a discrete set) are equal everywhere, only applies for Complex-Analytic functions, not Real-Analytic ones. Maybe it applies if the latter can be extended into a Complex-Analytic function ( as its Real part).
,