Weak theorem about an almost identical function: PROOF

[bo198214]

First it can be shown that \( J \) (as an entire function) does not even omit one value from \( \mathbb{C} \). (I give the proof in a next post, though this assertion is not really necessary for the following conclusions.)

Ok, here the proof. We first show that:
Each 1-periodic entire function \( p \) has a fixed point.

Proof. Assume that \( p \) has no fixed point i.e. that there is no \( z \) such that \( p(z)=z \). In this case the entire function \( q(z):=p(z)-z \) omits 0. But then there is also no \( z \) such that \( p(z+1)=z+1 \), i.e. \( p(z)-z=1 \) hence \( q \) omits 1. But then \( q \) is an entire function that omits 2 values and must be a constant (little Picard): \( p(z)=z+c \). But this function is not 1-periodic for any \( c \). Contradiction.\( \boxdot \)

Now lets look whether there is a \( z \) for each \( w \) such that \( w=J(z)=p(z)+z \). We see that \( w-p(z) \) is also 1-periodic entire and hence has a fixed point.\( \boxdot \)