Results 1 to 10 of 10

Math Help - cyclic function proof..

  1. #1
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401

    cyclic function proof..

    prove that function f(x)=sinx +sinax
    is cyclic if and only if "a" is rational
    ??
    Follow Math Help Forum on Facebook and Google+

  2. #2
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by transgalactic View Post
    prove that function f(x)=sinx +sinax
    is cyclic if and only if "a" is rational
    ??
    what is your definition of "cyclic"?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401
    cyclic is a function which goes in cycles
    like sinx cosx etc..
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by transgalactic View Post
    prove that function f(x)=sinx +sinax
    is cyclic [the usual word is "periodic"] if and only if "a" is rational
    ??
    If a is rational, a=m/n, then it's easy to see that f is periodic, with period 2nπ.

    It's harder to prove the converse. Suppose that f(x) is periodic. Then f''(x) = -\sin(x) -a^2\sin(ax) will also be periodic, with the same period. But when x=0, f(0)=f''(0)=0. If there is another point, say x=c, at which f(c)=f''(c)=0, and a≠1, then you should be able to show that sin(c) = sin(ac) = 0. With c≠0, this is only possible if a is rational. But if there is no such point c then f cannot be periodic.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401
    you said that if f(0)=f'(0)=f''(0)=0
    but when there is another point which gives the same result
    then its not periodic
    but why should differ +1 and -1

    how to use this definition a=m/n of a rational number in this proof
    ?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641
    Quote Originally Posted by Opalg View Post
    If a is rational, a=m/n, then it's easy to see that f is periodic, with period 2nπ.

    It's harder to prove the converse. Suppose that f(x) is periodic. Then f''(x) = -\sin(x) -a^2\sin(ax) will also be periodic, with the same period. But when x=0, f(0)=f''(0)=0. If there is another point, say x=c, at which f(c)=f''(c)=0, and a≠1, then you should be able to show that sin(c) = sin(ac) = 0. With c≠0, this is only possible if a is rational. But if there is no such point c then f cannot be periodic.
    Question? Maybe I am misremembering but doesnt the period of \sin(ax)+\sin(bx) have something to do with \text{gcf}(a,b) or \text{gcd}(a,b) or something? If so wouldn't that suffice to show that because a is irrational that this doesnt make sense?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401
    i dont know how to prove this thing
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
    From
    Champaign, Illinois
    Posts
    1,163
    Quote Originally Posted by Opalg View Post
    If there is another point, say x=c, at which f(c)=f''(c)=0, and a≠1, then you should be able to show that sin(c) = sin(ac) = 0. With c≠0, this is only possible if a is rational. But if there is no such point c then f cannot be periodic.
    How do you know this?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401
    i know that i need to use
    the definition that if a number is rational
    then it ca be represented by two whole numbers a/b

    what to do here?
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by transgalactic View Post
    i know that i need to use
    the definition that if a number is rational
    then it ca be represented by two whole numbers a/b

    what to do here?
    See my previous comment (#4 above). If f(x)=\sin x +\sin ax is periodic, with period c, then \sin c +\sin ac = f(c) = f(0) = 0 . The second derivative f''(x)=-\sin x - a^2\sin ax will also be periodic with period c, and therefore -\sin c - a^2\sin ac = 0. Add those two equations together, and you see that (1-a^2)\sin ac = 0. Assuming that 1-a^2\ne0, it follows that \sin ac=0. Since \sin c +\sin ac = 0, it also follows that \sin c=0.

    But if \sin x = 0 then x must be a multiple of π. Therefore both c and ac are multiples of π, say c=nπ and ac=mπ. Then a=m/n, which is rational.
    Last edited by Opalg; December 16th 2008 at 11:33 AM. Reason: corrected typos
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Cyclic group, relatively prime numbers proof
    Posted in the Algebra Forum
    Replies: 1
    Last Post: October 11th 2011, 09:45 PM
  2. theoretical question on cyclic function
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: October 22nd 2010, 02:01 AM
  3. Is My Answer Close to Right? Cyclic Group Proof
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: December 10th 2009, 08:22 AM
  4. Prove cyclic subroups => cyclic group
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: October 11th 2009, 08:36 PM
  5. Replies: 1
    Last Post: May 17th 2009, 05:12 AM

Search Tags


/mathhelpforum @mathhelpforum