07/15/2014, 09:43 PM

NOTE :

For those who want to conjecture the inverse of conjecture T1 ;

If f_3(z) is stable , then so is f_1(z).

THIS IS FALSE !

For instance if f_2(z) = eps + z^2.

Notice

1 + z + z^2/sqrt(2) + z^3/sqrt(6) + z^4/sqrt(24) = 0

has the zero :

0.2026 + 1.5304 i

and

1 + z + z^2/sqrt(2) + z^3/sqrt(6) + z^4/sqrt(24) + z^5/sqrt(120) = 0

has the zero :

0.545368 + 1.61261 i

Hence disproving the inverse of conjecture T1.

Btw taking more terms to avoid polynomials : g(z) = 1 + z + z^2/sqrt(2) + z^3/sqrt(6) + z^4/sqrt(24) + z^5/sqrt(120) + ... only makes the case more Obvious.

g(z) has zero's with an unbounded large real part.

( or so it appears , anyway the trends seems growing , unbounded is a " mini conjecture / exercise " for those who are intrested ).

regards

tommy1729

For those who want to conjecture the inverse of conjecture T1 ;

If f_3(z) is stable , then so is f_1(z).

THIS IS FALSE !

For instance if f_2(z) = eps + z^2.

Notice

1 + z + z^2/sqrt(2) + z^3/sqrt(6) + z^4/sqrt(24) = 0

has the zero :

0.2026 + 1.5304 i

and

1 + z + z^2/sqrt(2) + z^3/sqrt(6) + z^4/sqrt(24) + z^5/sqrt(120) = 0

has the zero :

0.545368 + 1.61261 i

Hence disproving the inverse of conjecture T1.

Btw taking more terms to avoid polynomials : g(z) = 1 + z + z^2/sqrt(2) + z^3/sqrt(6) + z^4/sqrt(24) + z^5/sqrt(120) + ... only makes the case more Obvious.

g(z) has zero's with an unbounded large real part.

( or so it appears , anyway the trends seems growing , unbounded is a " mini conjecture / exercise " for those who are intrested ).

regards

tommy1729