Re: impulsive question-

tommy1729 wrote:
without thinking about it i ask about this very impulsive and intuitive question :

(and apologies if i asked this before )

f1(2x) = 6*f2(x)*f3(x)*f4(x)
f2(2x) = 6*f1(x)*f3(x)*f4(x)
f3(2x) = 6*f1(x)*f2(x)*f4(x)
f4(2x) = 6*f1(x)*f2(x)*f3(x)

its kind a like the addition forms for exp , cosh , sine etc

but i hope this gives a non-elementary function.

it appears simple addition forms lead to simple functions in general...

a big part of that can be explained by differential equations of course.

maybe this one too.

im sorry i did not bother to investigate this myself.

but perhaps you find it fun to analyse.

regards
tommy1729


Perhaps the following theorem of Weierstrass is helpful:

"There are only three kinds of analytic complex functions of a single variable with an algebraic addition theorem (i.e. F(z1 + z2) is an algebraic function of F(z1) and F(z2)):
(1) the rational functions, (2) the simply periodic functions, (3) the doubly periodic functions."

See also http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jmsj/1158241931

and other results from Google > "algebraic addition theorem"

Johan E. Mebius
.