en>Headbomb: /* References */Various citation cleanup, replaced: | url=http://dx.doi.org/ → | doi=, {{MathSciNet | id = 1501618}} → {{MR|1501618}}, | id={{MR|1501618}} → | mr=1501618 using AWB

2011-08-21T18:39:58Z

References: Various citation cleanup, replaced: | url=http://dx.doi.org/ → | doi=, {{MathSciNet | id = 1501618}} → {{MR|1501618}}, | id={{MR|1501618}} → | mr=1501618 using AWB

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[[File:FBN exp(-1x2).jpeg|thumb|The function ''y'' = ''e''−1/''x''2 is flat at ''x'' = 0.]]
In '''[[mathematics]]''', especially [[real analysis]], a '''flat function''' is a [[smooth function]] ƒ : ℝ → ℝ all of whose [[derivative]]s vanish at a given point ''x''0 ∈ ℝ. The flat functions are, in some sense, the [[antitheses]] of the [[analytic function]]s. An analytic function ƒ : ℝ → ℝ is given by a [[convergent series|convergent]] [[power series]] close to some point ''x''0 ∈ ℝ:
:<math> f(x) \sim \lim_{n\to\infty}\sum_{k=0}^n\frac{f^{(k)}(x_0)}{k!}(x-x_0)^k . </math>
In the case of a flat function we see that all derivatives vanish at ''x''0 ∈ ℝ, i.e. ƒ(''k'')(''x''0) = 0 for all ''k'' ∈ ℕ. This means that a meaningful [[Taylor series]] expansion in a neighbourhood of ''x''0 is impossible. In the language of [[Taylor's_theorem#Statement|Taylor's theorem]], the non-constant part of the function always lies in the remainder ''Rn''(''x'') for all ''n'' ∈ ℕ.

Notice that the function need not be flat everywhere. The [[constant function]]s on ℝ are flat functions at all of their points. But there are other, non-trivial, examples.

==Example==

The function defined by

: <math>f(x) = \begin{cases} e^{-1/x^2} & \text{if }x\neq 0 \\
0 & \text{if }x = 0 \end{cases} </math>

is flat at ''x'' = 0.

==References==

* {{Citation|first=P.|last=Glaister|title=A Flat Function with Some Interesting Properties and an Application|publisher=The Mathematical Gazette, Vol. 75, No. 474, pp. 438–440 |date=December 1991|jstor=3618627}}

[[Category:Real analysis]]
[[Category:Algebraic geometry]]
[[Category:Differential calculus]]
[[Category:Smooth functions]]
[[Category:Differential structures]]

Bateman function - Revision history

en>Headbomb: /* References */Various citation cleanup, replaced: | url=http://dx.doi.org/ → | doi=, {{MathSciNet | id = 1501618}} → {{MR|1501618}}, | id={{MR|1501618}} → | mr=1501618 using AWB