Tetration Forum
The super of exp(z)(z^2 + 1) + z. - Printable Version

+- Tetration Forum (https://math.eretrandre.org/tetrationforum)
+-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1)
+--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3)
+--- Thread: The super of exp(z)(z^2 + 1) + z. (/showthread.php?tid=1073)



The super of exp(z)(z^2 + 1) + z. - tommy1729 - 03/14/2016

Im very intrested in The super of exp(z)(z^2 + 1) + z.
Notice it has only 2 fixpoints.

Also The super of exp(z)(z^2 + 1) + z = L.
Does it have solutions z for every L ? Why ?
They determine the branch structure / singularities.

Regards

Tommy1729


RE: The super of exp(z)(z^2 + 1) + z. - tommy1729 - 03/15/2016

So we need to show f = exp(z) (1+x^2) + z is surjective to the complex plane.

F maps [- oo , oo ] to [- oo , oo ].

Also conj(f(z)) = f(conj(z)).

Hence f is surjective on the reals.

By picard if f(z) =\= L then this L is unique.

But by the above f(conj(z)) =\= conj L.
Hence contradicting picard.
Therefore f is surjective to the complex plane.

So the singularities of its super are bounded.

Q.E.D.

Regards

Tommy1729