To determine if there exists a holomorphic function ( f ) on ( \mathbb{C} ) such that ( f(\mathbb{Z}) = {0,1,2,3} ) and ( f(z) \in {0,1,2,3} \implies z \in \mathbb{Z} ), we can draw inspiration from the known solution for the set ( {0,1,2} ).

Key Insight:

The function ( g(z) = 1 - \cos\left(\frac{\pi z}{2}\right) ) works for ( {0,1,2} ) because:

• It is entire (holomorp...

Yes. One can construct such an entire function. Below is a concrete (but not “closed-form elementary”) construction that uses standard approximation/interpolation tools from complex analysis and a simple Rouché-argument to control level sets.

Outline of the construction

• We want an entire f with
  • f(n) ∈ {0,1,2,3} for every integer n (indeed we will force f(n) = n mod 4),
  • and, conversely, ...

Yes, such an entire function exists, although it cannot be written with a short “elementary’’ formula.
One can obtain it by a completely explicit (but admittedly rather long) construction that uses nothing deeper than

• the Weierstrass product theorem (to produce entire functions with prescribed zeros), and
• an elementary inductive perturbation argument (to make sure that the four values 0,1...

<think> We are asked for a holomorphic function on the complex plane that sends integers exactly to the set {0,1,2,3} and only integers are mapped to this set. That is:

1. For every integer n, f(n) ∈ {0,1,2,3}.
2. If f(z) is in {0,1,2,3} for some complex number z, then z must be an integer.

Additionally, note that we have a known example for the set {0,1,2}: g(z) = 1 - cos(πz/2). This function i...

An excellent and deep question that connects several areas of complex analysis. The short answer is no, such a function does not exist.

Here is a step-by-step explanation of why this is the case.

1. Initial Analysis and Key Properties

Let f be the hypothetical function.

1. f is entire: Holomorphic on all of .
2. f is not a polynomial: A non-constant polynomia...

Yes – such an entire function exists.
We cannot write it in a short closed form (one cannot expect that: the conditions single out a function with infinitely many prescribed zeros and with four different “exceptional’’ values), but we can construct it explicitly by standard approximation theorems. One may even choose the function to be a (very large-degree) polynomial.

Below is a complete con...

This is a fascinating question about constructing a holomorphic function with very specific properties. Let me analyze this problem systematically.

Analysis of the Problem

We need a holomorphic function such that:

1. (the image of integers is exactly )
2. (only integers map to ${0,1,2...

Here is a short “no-go’’ proof. Assume, to the contrary, that there existed an entire non-constant
(f\colon\C\to\C) with

(1) (f(n)\in{0,1,2,3}) for every integer (n),
(2) (f(z)\in{0,1,2,3}\implies z\in\Z).

Define the monic quartic
[ P(w);=;w,(w-1),(w-2),(w-3);=;w^4-6w^3+11w^2-6w,. ] Then by (1)–(2) the entire function [ F(z);=;\frac{P\bigl(f(z)\bigr)}{\si...

A Holomorphic Function Mapping Integers (and Only Integers) to {0,1,2,3}

This is an interesting question about constructing a holomorphic function with specific value restrictions.

Understanding the Problem

We need to find a holomorphic function such that:

1. maps integers to the set
2. If for any $z \in \mathbb{...