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$$f(x)=\int_1^{x} \frac{1}{\sqrt[m]{P(t)}}\;dt$$

$P(x)$ is polynomial with degree $n$.

$m$ is an positive integer and $m>1$

What is the algoritm to determine $f^{-1}(x)$ is periodic function via using $P(x)$ and $m$ without evaluting the integral ? Is there also a way to find period without evaluting the integral?

Example 1: $P(x)=x^2$ , $m=2$

$$f_1(x)=\int_1^{x} \frac{1}{t}\;dt=\ln x$$


$$f_1^{-1}(x+2k\pi i)=e^{x+2k\pi i}=e^x=f_1^{-1}(x)$$ $k$ is an integer $$f_1^{-1}(x+2k\pi i)=f_1^{-1}(x)$$ $f_1^{-1}(x)$ is a periodic function

Example 2: $P(x)=1-x^2$ , $m=2$ $$f_2(x)=\int_1^{x} \frac{1}{\sqrt{1-t^2}}\;dt=\arcsin x-\frac{\pi}{2}$$

$$f_2^{-1}(x)=\sin {(x+\frac{\pi}{2})}$$

$$f_2^{-1}(x+2k\pi )=\sin{(x+\frac{\pi}{2}+2k\pi )}=\sin{(x +\frac{\pi}{2})}=f_2^{-1}(x)$$ $k$ is an integer $$f_2^{-1}(x+2k\pi )=f_2^{-1}(x)$$ $f_2^{-1}(x)$ is a periodic function

Example 3: $P(x)=(1+x^2)^2=1+2x^2+x^4$ , $m=2$ $$f_3(x)=\int_1^{x} \frac{1}{1+t^2}\;dt=\arctan x -\frac{\pi}{4}$$

$$f_3^{-1}(x)=\tan (x+\frac{\pi}{4})$$

$$f_3^{-1}(x+k\pi )=\tan{(x+\frac{\pi}{4}+k\pi )}=\tan{(x+\frac{\pi}{4} )}=f_3^{-1}(x)$$ $k$ is an integer $$f_3^{-1}(x+k\pi )=f_3^{-1}(x)$$ $f_3^{-1}(x)$ is a periodic function

Example 4: $P(x)=x^4$ , $m=2$

$$f_4(x)=\int_1^{x} \frac{1}{t^2}\;dt=\frac{x-1}{x}$$


$f_4^{-1}(x)$ is not periodic function

Thanks a lot for answers

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You probably have to clarify what you mean by periodicity here. A periodic function can never have an inverse, since it is never one-to-one. What you are really doing here in your examples is calculating a local inverse, and then using analytic continuation. In the case where $m=2$, and where the polynomial $P$ has no repeated roots, you get elliptic integrals (degrees 3 and 4) and hyperelliptic integrals (degree 5 or larger). – Lukas Geyer Oct 21 '12 at 20:52
@Lukas Geyer : Yes,I mention local inverse in my question. I wish to know a way to find which functions can be periodic when we get local inverse of it. Could we determine if the local inverse of elliptical integrals or hyperelliptic integrals are periodic via checking $P(x)$ or not? (for example if ($P(x)=1+x+x^5$) and $m=2$) And also what about case $m=3$ or for bigger m values? Without evulatiıng integral can we understand if local inverse of $f(x)$ is periodic function or not? Thanks a lot for answer. – Mathlover Oct 21 '12 at 21:27
Perhaps you can ask on MO? – awllower Jan 23 '13 at 12:43
Apparently your variables are complex numbers. Hence the double periodicity should be counted in? Also, I think this has something to do with the genus of the algebraic curve defined by the polynomial expression. It is however necessary for me to give it furthre thoughts later, in order to derive a criterion, if there is any... – awllower Jan 23 '13 at 12:49
The reason it might involve the genus comes from the fact that the periodicity occurs when the integral has a kernel defining a complex lattice. Hence its properties depends upon the properties of such curves in the comlex plane. – awllower Jan 23 '13 at 12:51

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