452 pi

[tommy1729]

in reply to the continuum sum and the continuum product ,

if i take the continuum product of (sin(x) +5/4) i get a period of 452 pi.

i wonder if mike3 gets the same result.

regards

tommy1729

[tommy1729]

the simple but vague idea keeps spinning in my head.

to compute a real period of a real to real function ( that is not double periodic and analytic ) :

f(x) is the continuum product.

compute the superfunction of f(f^-1(x) + i).

use the fixpoint to find the period of that superfunction.

( e.g. regular tetration base e , clearly has the same period as e^Lz.)

use a riemann mapping to get R -> R and reconsider the new period.

rotate the period by a suitable 4th root of unity.

that is the real period.

end.

it needs to be made formal ...

tommy1729

[tommy1729]

mike ?

[mike3]

Continuum product has lowest period \( 2\pi \), so since 452 is a multiple of 2, yes.

Note that

\( \prod_{n=0}^{x-1} sin(n) + 5/4 = \exp\left(\sum_{n=0}^{x-1} \log(\sin(n) + 5/4)\right) \).

Now we consider the Fourier expansion of \( \log(\sin(x) + 5/4) \), from which we find the continuum sum. The zeroth-order Fourier coefficient is 0 (though I don't have a proof, rather was done via numerical integration -- an explicit anti-derivative requires the poly-logarithm and is horrifically complicated, at least according to Wolfram's integrator), so the continuum sum will be periodic and have period equal to the original function, i.e. \( 2 \pi \) (since there is no constant term in the Fourier series and thus the continuum sum reduces to a simple coefficient transformation which does not alter the period), and thus the continuum product will have the same period (if \( g \) is periodic with period \( P \), then \( f \circ g \) is periodic with the same period.).

[tommy1729]

Continuum product has lowest period \( 2\pi \), so since 452 is a multiple of 2, yes.

Note that

\( \prod_{n=0}^{x-1} sin(n) + 5/4 = \exp\left(\sum_{n=0}^{x-1} \log(\sin(n) + 5/4)\right) \).

Now we consider the Fourier expansion of \( \log(\sin(x) + 5/4) \), from which we find the continuum sum. The zeroth-order Fourier coefficient is 0 (though I don't have a proof, rather was done via numerical integration -- an explicit anti-derivative requires the poly-logarithm and is horrifically complicated, at least according to Wolfram's integrator), so the continuum sum will be periodic and have period equal to the original function, i.e. \( 2 \pi \) (since there is no constant term in the Fourier series and thus the continuum sum reduces to a simple coefficient transformation which does not alter the period), and thus the continuum product will have the same period (if \( g \) is periodic with period \( P \), then \( f \circ g \) is periodic with the same period.).

thanks.

i wonder why i got the larger 452 , maybe i need to ajust my precision for estimations. ( i was afraid of conj 2pi since some numerical data indicated otherwise so i took a multiple for certainty that did fit )

at least we are working on the same method.

the idea that periodicity is not altered is intresting. ( in general , maybe in particular for tetration and real iterations of arbitrary functions ? )

im still thinking about an algoritm to find the period of a function ( taylor series )

and the continuum sum might be an important step ...

dont know if you ever considered that ...

btw giving the maximum value of the above continuum product in closed form might also be intresting.

i need more time to study all this ...

[tommy1729]

related is (amongst others)

conjecture

sup continuum product (sin(x) + 5/4) * inf continuum product (sin(x) + 5/4) = 1.

( i havent checked numerically for this specific function , but the general idea is clear )

question

sin(x) + 5/4 is the continuum product of ... ?

in general inverse continuum sum seems an intresting natural question ...

conditional continuum products also come to my mind :

if sin(x) >= 0 f(x) = continuum prod ( sin(x) + 5/4 )
if sin(x) < 0 g(x) = continuum prod ( sin(x) + 5/4 )

and i wonder about things like e.g.

lim f(x) ^ 1/x = sup continuum prod ( sin(x) + 5/4)

anyway you get the idea that this is not finished ...