How to make semi-periodic funtions complete periodic?

[MWresearch]

Suppose I have a function that has some periodic properties related to discontinuities to the left of x=0 and is mostly monotonic or just a continuous to the right of x=0, like for instance y=gamma(x). How do I make such a function and other special functions always periodic over all the domain without reflection symmetry? If I do gamma(-x^2) the function is periodic over the entire domain, but instead of continuing the periodic property from x=-infinity to x=infinity, the function is reflected over the y axis. I want it to be more rotationally symmetric like tan(x).

[studiot]

Gamma (-x²) doesn't look periodic to me.

 

 

 

Edited March 13, 2015 by studiot

[MWresearch]

Use wolframalpha, type gamma(-x^2) and don't restrict the domain. You'll get something that looks kind of like cosecant.

[studiot]

Well post the result then.

[MWresearch]

This site is so glitched I can barely even post. In fact, mostly I can't post, I have to refresh the page 3 times to make a single post. I definitely can't post that link. You'll just have to do the exhausting work of typing in a url and equation to see for yourself.

Edited March 13, 2015 by MWresearch

[John]

I haven't looked into it terribly much, but one way we can generate something periodic is by using modular arithmetic, e.g. check out

[math]y = \Gamma (x \textnormal{ mod } 2)[/math]: http://www.wolframalpha.com/input/?i=y+%3D+gamma(x+mod+2)

and

[math]y = \Gamma \left(-(x \textnormal{ mod } 2)^{2}\right)[/math]: http://www.wolframalpha.com/input/?i=y+%3D+gamma(-(x+mod+2)^2)

Edit: I'm assuming that instead of "rotational symmetry," you meant "translational symmetry," since the latter is closer to what periodicity implies, but if you do want rotational symmetry, more elaborate constructions may be required. For example, going back to the gamma function, we can do something like [math]y = \frac{x}{|x|} \Gamma(|x|)[/math]: http://www.wolframalpha.com/input/?i=y+%3D+%28x%2F%7Cx%7C%29gamma%28%7Cx%7C%29

Edited March 13, 2015 by John

[imatfaal]

This site is so glitched I can barely even post. In fact, mostly I can't post, I have to refresh the page 3 times to make a single post. I definitely can't post that link. You'll just have to do the exhausting work of typing in a url and equation to see for yourself.

 

Really? What browser do you use - and are you free of malware. I regularly switch between all major browsers to check we are ok - I haven't had that sort of problem. This is a pretty vanilla implementation of forum software - that has glitches but not so major that we cannot post; although we are getting a major version update soon which may well take a fair while getting used to.

 

And on your second comment - please bear in mind this is a forum where people try to chat and help out others with questions; try being a little less confrontational and a bit more laid back.

 

On the major question in the OP - gamma (-x^2) is not what I would think of as periodic. It merely switches from positive to negative every x. If you think of the break down of what gamma (-x^2)

 

gamma (-x^2) = - (-x^2)! / x^2

 

the denominator will always be positive - the numerator is basically the product of a list of negative numbers x in length which will be positive for even x and negative for odd x. Thus the answer will flip from positive to negative - this makes it look periodic - but it is not what I would call periodic

[MWresearch]

John: That modular notation does look like an interesting tool that I never considered using before, but I don't think modular-forced equations will differentiate and integrate nicely nor have much use in identities, neither will absolute values. I was hoping for something more traditional and continuous, more like sqrt(x^2) instead of |x|.

 

Imaatfal: Using internet explorer and okay.

Yeah I don't think it's purely periodic either, hence the phrase "semi-periodic." Only the discontinuities on the negative side of gamma(x) are periodic and I don't know what that's called, but that phenomena occurs in other equations involving gamma, exponents and trigonometry.

 

Look at tan(x). The same translational pattern occurs to the left and to the right. I want to do that with gamma(x), I want the pattern on the negative part to continue to the positive side of the domain unreflected.

[imatfaal]

multiply by x/[sqrt(x^2)] - which basically gives you minus one for x<0 and plus one for x>0. That flips everything on the x<0 side

Browsing with IE running under Win 7 Pro N - no problems. Will see if I hear of any other people having trouble.

[John]

Well, [math]|x| = \sqrt{x^{2}}[/math] in any case, so changing from one to the other won't affect much.

As for the rest, at a quick glance, I don't think modular arithmetic makes things terribly messy. For differentiation and integration, taking the derivative (or integral) of the starting function and then replacing instances of x with x mod n would probably work, and I'm not sure where the problem would lie with identities.

Edited March 13, 2015 by John

[MWresearch]

Mutiplying by what imaatfal said almost did it in a way, but I ended up only getting the positive side and it was still reflected.

 

And with john, something doesn't add up. I remember from statistics that we used something like sqrt(x^2) for equations dealing with variance and standard deviation specifically to avoid the discontinuity caused by the vertex of |x| at x=0 or wherever the vertex happens to be. Maybe it's something else I'm thinking of?