What does analytic mean? How can you find all non-analytic points of an equation?

As far as I'm aware, a point

x=x_{0}

on a function

f(x)

is said to be analytic if it is possible to express

f(x)

as a Taylor expansion about that point. Is this correct?

Secondly, how would you find all the points on a function that are not analytic?

Thanks

Reply 1

joostan

13

Original post by Brian Moser

As far as I'm aware, a point

x=x_{0}

on a function

f(x)

is said to be analytic if it is possible to express

f(x)

as a Taylor expansion about that point. Is this correct?

Secondly, how would you find all the points on a function that are not analytic?

Thanks

The point itself is not analytic,

f

is analytic at the point. Other than this you have the gist of the definition, more precisely, here about means on some open set, but otherwise its fine.
Typically, a function is not analytic at a point if it has some kind of 'bad behaviour' normally some kind of singularity in the function.
Alternatively you can simply take the complement of the set of analytic points.

Edit: It's worth noting that also if any of the function's derivatives have some kind of bad behaviour this is also true, and there may not necessarily be a singularity in the function, but rather in its derivatives.

(edited 8 years ago)

Reply 2

Zacken

22

Original post by Brian Moser

As far as I'm aware, a point

x=x_{0}

on a function

f(x)

is said to be analytic if it is possible to express

f(x)

as a Taylor expansion about that point. Is this correct?

Yes, a function is analytic at a point

x = x_0

i f it can be expanded around

x_0

in the form

\displaystyle f(x) = \sum_{n=0}^{\infty} b_n(x-x_0)^n

with the condition that it has a positive radius of convergence,

i.e:

\displaystyle \lim_{n \to \infty} \left|\frac{b_{n}}{b_{n+1}} \right| \neq 0

.

Secondly, how would you find all the points on a function that are not analytic?

Thanks

For rational functions, they're not analytic when their denominators are zero but I don't know a general method for finding non-analytical points for a general function. I think you can test individual points using differentiability, C-R equations and the likes, but finding non-analyticity evades me right now, so I'll leave you in the hands of somebody who actually knows something about any of this. :yes:

Edit: I see Joostan has that covered. :smile:

(edited 8 years ago)

Reply 3

Brian Moser

OP

12

Original post by joostan

The point itself is not analytic,

f

is analytic at the point. Other than this you have the gist of the definition, more precisely, here about means on some open set, but otherwise its fine.
Typically, a function is not analytic at a point if it has some kind of 'bad behaviour' normally some kind of singularity in the function.
Alternatively you can simply take the complement of the set of analytic points.

Edit: It's worth noting that also if any of the function's derivatives have some kind of bad behaviour this is also true, and there may not necessarily be a singularity in the function, but rather in its derivatives.