Can smooth and analytic be used interchangeably?

[dextercioby]

Can "smooth" and "analytic" be used interchangeably?

My guess is 'yes'.

Daniel.

 

[D H]

A real analytic function is smooth, but not necessarily the other way around. For a counter-example, see an infinitely differentiable function that is not analytic.

 

[Hurkyl]

No. Smooth merely means infinitely differentiable, whereas analytic means that it has to be locally equal to its power series!

(But if we're talking complex differentiation, then being once differentiable is sufficient for meing analytic. Complex numbers are magical!)

The classic counterexample is the function:

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

which is infinitely differentiable at x=0: in fact, we have that $f^{(n)}(0) = 0$ for all of its derivatives!

So, this function is clearly not equal to a power series on any neighborhood of zero, and thus is not analytic there.

 

[HallsofIvy]

"smooth" tends to be ambiguous. I have seen smooth used to mean only differentiable and "infinitely smooth" to mean infinitely differentiable. I have even seen reference to "sufficiently smooth" to mean "as differentiable as you need".

Hurkyl is right, though, even an infinitely differentiable function is not necessarily analytic.