# Is it impossible to prove that some functions are periodic?

• B
Heres an example.

Let G(s) be the moving average of all previous values of f(s).

G(s) and F(s) intersect at multiple points. Is it possible to prove that the intersections happen periodically?

pwsnafu
Could you explain what the 1 over infinity is supposed to be?

mustang19
That's the summation term from 1 to infinity. The software garbled it.

mfb
Mentor
Like this?$$F(s) = \sum_{n=1}^{\infty} \frac{1}{n^{s i +2}}$$
What do you mean by "moving average over all previous values", and does f(s) mean F(s)?
Do you have a proof that they intersect at all? With complex numbers this is not trivial.

mustang19
Like this?$$F(s) = \sum_{n=1}^{\infty} \frac{1}{n^{s i +2}}$$
What do you mean by "moving average over all previous values", and does f(s) mean F(s)?
Do you have a proof that they intersect at all? With complex numbers this is not trivial.
I just graphed the functions. If you dont know what a moving average is then look it up.

As I suspected this problem seems very difficult or impossible to solve.

mfb
Mentor
If you dont know what a moving average is then look it up.
I know what a moving average is, but the definition is not unique (average over which range?), and your description doesn't make sense.

Did you plot both the real and imaginary part?

Currently the problem is not even well-defined.

mustang19
I know what a moving average is, but the definition is not unique (average over which range?), and your description doesn't make sense.

Did you plot both the real and imaginary part?

Currently the problem is not even well-defined.
I plotted both the real and imaginary parts.

The range is over the entire function, or if your software does not support that, then try from the origin to S.

mfb
Mentor
The range is over the entire function
That doesn't make sense either.
How did you define G(s), can you show the formula?
I plotted both the real and imaginary parts.
And they had intersections at the same point?

mustang19
That doesn't make sense either.
How did you define G(s), can you show the formula?And they had intersections at the same point?
I don't have the formula handy. You just average all the previous points.

They intersected at different points. I want to know if either or both of the graphs are periodic.

Im not too concerned about the specifics. If you can answer anything related to this question then that would be interesting. I am just asking if its always possible to prove whether a function is periodic.

mfb
Mentor
You just average all the previous points.
That would work as description if there would be a finite set of those points, but you don't have that. Average from 0 to s? Is your function defined for negative s? Make some limit procedure to "average" from -infinity to s? Something else?
They intersected at different points.
Which means the functions do not intersect. Only their real and imaginary parts.
I am just asking if its always possible to prove whether a function is periodic.
A clear no. You can construct functions that are periodic if and only if some turing machine halts, and there is no algorithm that can determine this for all turing machines (halting problem). Those function definitions will look weird, however (like "f(x)=sin(x) if this turing machine halts, f(x)=x if it does not").

mustang19
That would work as description if there would be a finite set of those points, but you don't have that. Average from 0 to s?
Yes

Is your function defined for negative s?
Yes

Which means the functions do not intersect. Only their real and imaginary parts.A clear no. You can construct functions that are periodic if and only if some turing machine halts, and there is no algorithm that can determine this for all turing machines (halting problem). Those function definitions will look weird, however (like "f(x)=sin(x) if this turing machine halts, f(x)=x if it does not").
Is that a proof? Are you sure they never intersect?

You seem to be claiming I claimed they always intersect at different points. No, I said they intersected at some points and the parts intersected at others.

Anyway you basically answered my questions to a good enough level with your last paragraph.

Last edited:
mfb
Mentor
Are you sure they never intersect?
No, but you claimed they would intersect, and I don't see evidence for that so far.
G(s) and F(s) intersect at multiple points.
An intersection of two functions requires that the function values are identical - for complex numbers, this means both real and imaginary part have to be identical.

No, but you claimed they would intersect, and I don't see evidence for that so far.An intersection of two functions requires that the function values are identical - for complex numbers, this means both real and imaginary part have to be identical.
I really don't see how that helps. If you have any more questions about the function let me know, but it seems pretty well defined for me. Cant you just use the moving average feature in matlab? I really don't see where the ambiguity was, I am able to graph the function fine.

mfb
Mentor
It is well-defined now, after 10 posts with multiple rounds of questions how it is defined.
I really don't see where the ambiguity was, I am able to graph the function fine.
The range of definition and the range you average over.

In addition, the first post had the (probably wrong, but certainly not backed by evidence) claim that the functions intersect.

mustang19
So basically, the example problem is well defined and unanswerable.

jbriggs444
Homework Helper
2019 Award
So basically, the example problem is well defined and unanswerable.
Nothing in this thread has indicated that the example problem (as clarified in #11) is unanswerable. Only that it is unanswered and that it might be unanswerable.

mustang19
Nothing in this thread has indicated that the example problem (as clarified in #11) is unanswerable. Only that it is unanswered and that it might be unanswerable.
So its unanswerable, as far as anyone knows.

jbriggs444
Homework Helper
2019 Award
So its unanswerable, as far as anyone knows.
And it's answerable, as far as anyone knows. Might be answerable, might not be. We do not know. Trying to spin it one way or the other does not change that.

Last edited:
Mark44
Mentor
If you dont know what a moving average is then look it up.