You are Here: Home >< Maths

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

Announcements Posted on
Why bother with a post grad? Are they even worth it? Have your say! 26-10-2016
1. As far as I'm aware, a point on a function is said to be analytic if it is possible to express 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
2. (Original post by Brian Moser)
As far as I'm aware, a point on a function is said to be analytic if it is possible to express 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, 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.
3. (Original post by Brian Moser)
As far as I'm aware, a point on a function is said to be analytic if it is possible to express as a Taylor expansion about that point. Is this correct?
Yes, a function is analytic at a point i f it can be expanded around in the form with the condition that it has a positive radius of convergence,

i.e: .

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.

Edit: I see Joostan has that covered.
4. (Original post by joostan)
The point itself is not analytic, 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.
Thanks for the response. What exactly do you mean by 'singularity' though?
5. (Original post by Brian Moser)
Thanks for the response. What exactly do you mean by 'singularity' though?
This seems to have it covered.

## Register

Thanks for posting! You just need to create an account in order to submit the post
1. this can't be left blank
2. this can't be left blank
3. this can't be left blank

6 characters or longer with both numbers and letters is safer

4. this can't be left empty
1. Oops, you need to agree to our Ts&Cs to register

Updated: April 12, 2016
TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Today on TSR

### Who is getting a uni offer this half term?

Find out which unis are hot off the mark here

Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read here first

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams