Infinitely differentiable function which is not analytic

The only way I can correct the proof assumes that the potential is real analytic. Sign up to join this community. The best answers are voted up and rise to the top. Does Physics need non-analytic smooth functions? Ask Question. Asked 8 years, 11 months ago. Active 8 years, 1 month ago. Viewed 13k times. If non-analytic smooth functions are irrelevant to Physics, why is it so?

Remark: analogous questions may arise about Fourier series expansions. Expand the coefficients in Taylor series around a point in the scale of physical interest.

Improve this question. Community Bot 1 2 2 silver badges 3 3 bronze badges. Qfwfq Qfwfq So long as you only care about finitely many terms of the series, this usually does the trick. In fact the Klein-Gordon equation will evolve compactly supported data on one time slice to analytic data on all other time slices. However, I'm puzzled about your statement about the Klein-Gordon equation. It surely satisfies finite speed of propagation one can establish this with energy estimates , see e.

This implies that compactly supported solutions stays compactly supported. Maybe I'm misunderstanding the claim. I have to retract my original comment, or at least weaken it to merely dispute the claim that analyticity is incompatible with the impossibility of superluminal signalling. On the grounds that finite propagation is already incompatible with positive energy. Show 8 more comments. Active Oldest Votes.

Improve this answer. Jeff Harvey Jeff Harvey 5, 2 2 gold badges 24 24 silver badges 34 34 bronze badges. For a detailed statement of the comparison between theory and experiment for g-2 see arxiv. I thought it was a typo. Turns out my English is bad. For some field theories constructed rigorously the function is still analytic even though the Taylor series has zero radius of convergence. The issue is the location of the point around which the Taylor expansion is made: in the middle of the domain of analyticity ordinary summation which requires positive radius of convergence versus on the boundary Borel summation.

Add a comment. To conclude, yes, physicists do need to consider smooth, nonanalytic functions. Liviu Nicolaescu Liviu Nicolaescu Nice answer btw. This is one of his first results, in late 30s. As for Hadamard, here are some useful notes math. In thE notes you link there is an explanation of well posedness but not of the statement that elliptic problems are not well posed, or did I oversee that?

Deane Yang Deane Yang The dependence on time is easily seen to be non-smooth as time approaches zero. He points out that there are PDE's that model physical phenomena, where the initial value problem should not be well-posed but where the initial value problem is well-posed in the real analytic category. The key examples are elliptic PDE's modeling time-independent physical states.

Bazin Bazin 13k 19 19 silver badges 54 54 bronze badges. One would give polynomials as inputs to the equation, and look at the solution. This pathology cannot occur with functions of a complex variable rather than of a real variable.

This example teaches us that functions of a real variable are sometimes ill-behaved in ways to which functions of a complex variable are immune. One of the most important applications of this function is the construction of so-called mollifiers , which are important in theories of generalized functions , like e.

This pathology cannot occur with differentiable functions of a complex variable rather than of a real variable. Indeed, all holomorphic functions are analytic, so that the failure of the function f defined in this article to be analytic in spite of its being infinitely differentiable is an indication of one of the most dramatic differences between real-variable and complex-variable analysis.

By the great Picard theorem, it attains every complex value with the exception of zero infinitely many times in every neighbourhood of the origin. Anonymous Not logged in Create account Log in. Hand W iki. From HandWiki. Namespaces Page Discussion. More More Languages. Short description : Mathematical functions which are smooth but not analytic. The non-analytic smooth function f x considered in the article. Stack Overflow for Teams — Collaborate and share knowledge with a private group.

Create a free Team What is Teams? Learn more. The distinction between infinitely differentiable function and real analytic function Ask Question. Asked 9 years, 5 months ago. Active 9 years, 5 months ago. Viewed 5k times. Can anyone explain more clearly about the distinction between the two classes? Juntao Huang Juntao Huang 1 1 silver badge 9 9 bronze badges. See Part 1 and Part 2.