Thread: cyclic function proof..

prove that function f(x)=sinx +sinax
is cyclic if and only if "a" is rational
??

Originally Posted by transgalactic

prove that function f(x)=sinx +sinax
is cyclic if and only if "a" is rational
??

what is your definition of "cyclic"?

cyclic is a function which goes in cycles
like sinx cosx etc..

Originally Posted by transgalactic

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 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.

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
?

Originally Posted by Opalg

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 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 have something to do with or or something? If so wouldn't that suffice to show that because is irrational that this doesnt make sense?

i dont know how to prove this thing

Originally Posted by Opalg

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?

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?

Originally Posted by transgalactic

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 is periodic, with period c, then . The second derivative will also be periodic with period c, and therefore . Add those two equations together, and you see that . Assuming that , it follows that . Since , it also follows that .

But if 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.