Talk:Entire function

Completeness of examples

Are the only entire analytic functions polynomials in both x and exp(x)? njh 02:06, 23 June 2005 (UTC)

No. Any function of the form

$a_{0}+a_{1}x+a_{2}x^{2}+...$

with an infinite radius of convergence is entire function. Oleg Alexandrov 02:41, 23 June 2005 (UTC)

An example of an entire function that is not a polynomial in $x$ or $\exp(x)$ is $\exp(\exp(x))$. Another is $\exp(x^{2})$. Eric Kvaalen (talk) 11:50, 8 February 2013 (UTC)

Properties

How would you define a function being holomorphic at ∞? —Preceding unsigned comment added by JumpDiscont (talk • contribs) 03:52, 10 November 2009 (UTC)

Holomorphic at infinity means that it's holomorphic in 1/z. For example, the function 1/z itself. A function cannot be holomorphic everywhere including at infinity unless it is constant. Eric Kvaalen (talk) 11:50, 8 February 2013 (UTC)

An entire function can grow faster than any given function

Thanks, user:PMajer, for the correction in this (adding a constant). But I don't think it's really necessary to give the reader such a long explanation. And the way you did it is overkill! For instance, if $g(x)$ is zero for $x\leq 11$ and then jumps to $e$ at 12, then your method gives $f(11)\geq 1.1^{20}\approx e^{2}$.

I would just put the following (I have changed the wording a bit):

Entire functions may grow as fast as any increasing function: for any increasing function g : [0,+∞) → [0,+∞) there exists an entire function f(z) such that f(x) > g( |x| ) for all real x. Such a function may easily be found in the form:

$f(z)=c+\sum _{k=1}^{\infty }\left({\frac {z}{k}}\right)^{n_{k}}$,

for a constant c and a strictly increasing sequence of positive integers n_k. Any such sum converges everywhere and thus defines an entire function, and if the exponents are chosen appropriately, the inequality f(x) > g( |x| ) also holds, for all real x. (The exponent n₁ can be chosen so that this is true in the interval [2, 3], and then n₂ can be chosen so as to satisfy the inequality in the interval [3, 4], and so on.) Furthermore, if g(x) and h(x) are continuous functions with $g(x)<h(x)$ for all real x, one can find an entire function f(x) such that $g(x)<f(x)<h(x)$ for all real x.

(You told me by e-mail that that last statement follows from the preceding, but it's not obvious so it would be good to give a reference -- unless you can give a very brief explanation.)

Eric Kvaalen (talk) 11:50, 8 February 2013 (UTC)

Hi, well, yes, I initially thought that too many details are not needed, but I have recently added them for you, since you seemed to be interested. Anyway, apart the triviality about the constant, I thought the initial explanation after all was clear enough. So maybe I would just go back to the previous version (without forgetting the constant c ). If I remember well The result about the order density of entire functions in the continuous is due to T.Carleman (1927; I will find the exact reference): actually I planned to add this information, as it seems relevant enough to be included. The proof is not difficult but not trivial; a brief explanation may be nice indeed. --pma 14:19, 11 February 2013 (UTC)

Further remarks:

1) overkill: Actually, here the task is just to construct an entire f above the given g, only this. The given form of f allows a quick solution (because it is automatically an entire function, and then one can fix the constant c and the exponents n_k so that the inequality be satisfied). BUT, of course, this form do not allow a close approximation as you are asking : for instance, such an f has all its derivatives positive on the positive half-line, which means it has in any case quite a strong growth. No way to be close to g, if e.g. g is a logarithm. So the objection about the example with $\scriptstyle g(11)=e$ and $\scriptstyle f(11)=e^{2}$ looks quite naive (if you really care about this issue, we could do better using a sharper inequality than $\scriptstyle \log {\bigg (}1+{\frac {1}{k}}{\bigg )}\geq {\frac {1}{2k}},$ like e.g. $\scriptstyle \log {\bigg (}1+{\frac {1}{k}}{\bigg )}\geq {\frac {1}{k+1}},$ that would give a new exponent around half as big).

2) The sentence you suggest:

Furthermore, if g(x) and h(x) are continuous functions with $g(x)<h(x)$ for all real x, one can find an entire function f(x) such that $g(x)<f(x)<h(x)$ for all real x

is ok, but I think the stated property (Carleman's theorem) deserves more emphasis. Maybe a section of its own (e.g. about "density", after the one about "growth").

3) I'll think about how to make shorter the subsequent explanation. At the moment I have cut the last passages, and left the explicit formula for c and n_k. I agree that the latter is not optimal from the point of view of getting f as smaller as possible , but it is (I think) the simpler choice , and, what matters, shows that the whole construction is really constructive. I did not like so much your suggestion

...and then n₂ can be chosen so as to satisfy the inequality in the interval [3, 4], and so on,

because it sounds a bit misleading, in that it suggests that the choice of the n_k has to be done inductively (one after the other), whereas it's somehow simpler, as each one can be fixed directly.

--pma 12:28, 25 February 2013 (UTC)

Nonzero zeros?

Resolved

What does that mean: "z_k are the non-zero [[zero (complex analysis)|roots]] of f"? — Sebastian 16:27, 17 January 2018 (UTC)

It means that the $z_{k}$ are roots of $f$: $f(z_{k})=0,$ but that they are themselves nonzero numbers: $z_{k}\neq 0.$ –Deacon Vorbis (carbon • videos) 18:37, 17 January 2018 (UTC)

Thanks for the good explanation! — Sebastian 00:14, 18 January 2018 (UTC)

But you are right. Indeed, the expression "non-zero zeros" sounds delightfully nonsensical... (That's why we like it ;) ) pma 14:08, 28 March 2020 (UTC)

Missing an important example of order

The order of Riemann Xi function is 1: see here. 129.104.65.7 (talk) 11:26, 20 November 2024 (UTC)