02/20/2013, 10:50 PM
However I fear such a function does not exist.
The reason is F(x) must be so different from exp and sinh.
It cannot be of the form exp(x)(x-a)(x-b) and it cannot be of the form exp(x) - exp(ax) - exp(bx). And at the same time F(x) must be entire and have only 2 fixpoints. It seems the Weierstrass factorization theorem implies it must be of a form it cannot be. Also a Dirichlet series for F(x) seems impossible.
The reason is F(x) must be so different from exp and sinh.
It cannot be of the form exp(x)(x-a)(x-b) and it cannot be of the form exp(x) - exp(ax) - exp(bx). And at the same time F(x) must be entire and have only 2 fixpoints. It seems the Weierstrass factorization theorem implies it must be of a form it cannot be. Also a Dirichlet series for F(x) seems impossible.
