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We have Picard's Great Theorem:

Every non-constant entire function attains every complex value with at most one exception. Furthermore, every analytic function assumes every complex value, with possibly one exception, infinitely often in any neighborhood of an essential singularity.

Assume we have a function $f$ with a singularity at $z$. Has anyone investigated what structures that are possible in a neighborhood of $z$?

For example, assume that some we have some ordered set of values $\{w_i\}$ in a neighborhood of z. Since the function $f$ assumes every complex value infinitely often, we may agrue by the infinite monkey theorem that there exists some set $\{z_i\}$ in this neighborhood such that $f(z_i) =w_i$ for all $i$ in order almost surely. In other words, we know by Picard's Great Theorem that these values exist, but can we also conclude that they exist with a given ordering inside the neighborhood?

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    $\begingroup$ What does that mean with a given ordering? $\endgroup$
    – Conrad
    Commented Apr 2, 2022 at 17:48

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Studying neighborhoods of essential singularities is exactly the same as studying the divergence of analytic functions towards infinite. I find it easier to state versions of Picard's theorem in that direction and will aim the discussion in that direction.

The fact that a transcendental entire function assumes every complex value infinitely often (except for possibly one Lacunary value), says nothing about what the function does on some sequence towards infinite. For example, $\sin(z)$ is bounded on the real line, hence any sequence of real numbers approaching infinite will be bounded.

Lets start a discussion around bounded for a second, to try an understand "how entire functions diverge". OK, we can be bounded on the real line, can we be bounded on the upper half plane? No, that guarantees a constant. OK, how about a sector, like all $z$ with $\theta<\arg(z)<\theta+\epsilon$, can we be bounded there? Turns out, still no, that will also give us a constant. OK, how about a strip, like $|z|<\epsilon$, can we be bounded on that? Yes, in fact, $\sin(z)$ is bounded on any horizontal strip $r_1<\text{Im}(z)<r_2 $.

OK, how about bounding by another function; for example, if $|f(z)|<|p(z)|$, where $p(z)$ is a polynomial, what does that say about $f(z)$. Turns out, it says that $f(z)$ must also be a polynomial. This also applies to sectors (but not strips). We learn then that transcendental entire functions must grow faster than polynomials, essentially in all sectorial directions.

OK, maybe we change the discussion from "bounded" to "zeros". After all, another way to say Picard's theorem is that for any transcendental function, $f(z)$, and number $a$, $f(z)-a$ has infinitely many zeros (except for possibly one Lacunary value). This is essentially an extension of the Fundamental Theorem of Algebra, given any polynomial, $p(z)$ of degree $n$, then $p(z)-a$ has precisely $n$ zeros.

So, what is up with this Lacunary value. We can certainly think of the classic example, $e^z$ which is never zero. But are there other examples? Turns out, not really. The only functions that exhibit Lacunary values are exponential, this is explicitly shown in the Hadamard factorization of an entire function. $$ f(z) = z^m e^{g(z)} \prod E_p(z/a_n) $$ Essentially, every entire function has an "exponential portion" which can have a Lacunary value, and a factorized portion on it's zeros, which does not have a Lacunary value. The Lacunary value makes more sense now with respect to the Hadamard factorization.

OK, back to bounding. We noted that bounding an entire function by a polynomial means the entire function is a polynomial. Considering the factorization of entire functions as Hardamard factorizations. Is there an ordering to entire functions themselves? For example, if $|f(z)|<Ae^{B|z|}$ does that guarantee what the factorization must look like? Yes, it does. What if we have limited growth above and limited our zeros to a sector or a strip; does that exhibit a property? It sure does, it turns out zeros are not allowed to get arbitrarily close together, essentially you gain restrictions, like $\sum \frac{1}{|a_k|^p}$ must converge (dependent on the restrictions growth size and zero locations).

The field starts exploding with concepts like, growth, order, type, restricted zero loci, etc. All of this is really a discussion about what really happens as you approach infinite from different directions with entire functions.

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