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- 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)##.

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)##.