Let , and define on for with on .
Show that
Remark. In the case of constant function, i.e., on (provided that ) for some ,
we can compute that on for all , which turns out to be a well known result.
That is,
Let , and define on for with on .
Show that
Remark. In the case of constant function, i.e., on (provided that ) for some ,
we can compute that on for all , which turns out to be a well known result.
That is,
Let and . We can show that is analytic on the whole real line (for example by Faa-di-Bruno's formula) and we have according to the chain rule.
We show by induction that . so the result is true for and if it's true for n we have
hence
. We can now compare the two Taylor series.