On the directions of complete indeterminacy of an elliptic function

Tiberiu Popoviciu
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T. Popoviciu

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T. Popoviciu, Sur les directions d’indetermination complète d’une fonction elliptique, Bulletin des Sciences Mathématiques, 2e série, année 193, t. LX (1936), pp. 196-198 (in French).

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1936 a -Popoviciu- Bull. Sci. Math. – Sur les directions d’indetermination complete d’une fonction elliptique

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Bulletin des Sciences Mathématiques

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[Zbl 0014.40803, JFM 62.0429.03]

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1936 a -Popoviciu- Bull. Sci. Math. - On the directions of complete indeterminacy of a function e
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ON THE DIRECTIONS OF COMPLETE INDETERMINATION OF AN ELLIPTIC FUNCTION;

By Mr. Tiberius Popoviciu.
A complete direction of indeterminacy of a function f ( z ) f ( z ) f(z)f(z)f(z)of the complex variable z z zzzis a half-line such that if the variable z z zzzdescribes it, the function takes values ​​as close as one wants to any given value. This notion was introduced by Mr. Paul Montel (').
In this small work we propose to examine the directions of complete indeterminacy of an elliptic function. For such a function, it is obviously sufficient to consider only the half-lines passing through the origin. In a course given at the University of Cluj (Romania) Mr. Paul Montel has already demonstrated, and in a very simple way, that in every angle there exists at least one direction of complete indeterminacy. In other words, the set of directions of complete indeterminacy of an elliptic function is everywhere dense in every angle.
In the following, we propose to determine all the directions of complete indeterminacy of an elliptic function.
Let ω 1 , ω 2 ω 1 , ω 2 omega_(1),omega_(2)\omega_{1}, \omega_{2}ω1,ω2two independent periods represented by points A, C and Δ Δ Delta\DeltaΔa half-line passing through the origin O and included, to fix the ideas, in the angle AOC. We consider the network of parallelograms formed from the initial parallelogram OABC constructed on OA, OC. There exists an infinity of parallelograms containing segments of Δ Δ Delta\DeltaΔ.
Let us bring, by translations equal to periods, all these segments into the parallelogram OABC. We will have in OABC
popoviciu.
an insinity of segments ò which have their ends on the sides. There are segments ò which have one end on AB and others which have one end on BC.
The set of values ​​taken by the function on the half-line Δ Δ Delta\DeltaΔcoincides, by virtue of the periodicity, with the set of values ​​taken in the parallelogram OABC on the segments δ δ delta\deltaδ.
The half-right Δ Δ Delta\DeltaΔcut AB at the affix point ω 1 + α ω 2 ω 1 + α ω 2 omega_(1)+alphaomega_(2)\omega_{1}+\alpha \omega_{2}ω1+αω2and BC at the affix point 1 α ω 1 + ω 2 1 α ω 1 + ω 2 (1)/(alpha)omega_(1)+omega_(2)\frac{1}{\alpha} \omega_{1}+\omega_{2}1αω1+ω2Or α α alpha\alphaαis a positive number. We see that the ends on AB of the segments δ δ delta\deltaδare the points ω 1 + ( n α − [ n α ] ) ω 2 ω 1 + ( n α − [ n α ] ) ω 2 omega_(1)+(n alpha-[n alpha])omega_(2)\omega_{1}+(n \alpha-[n \alpha]) \omega_{2}ω1+(nα−[nα])ω2and that the ends on B C B C BCBCBCare the points ω 2 + ( n α − [ n α ] ) ω 1 ω 2 + n α − n α ω 1 omega_(2)+((n)/(alpha)-[(n)/(alpha)])omega_(1)\omega_{2}+\left(\frac{n}{\alpha}-\left[\frac{n}{\alpha}\right]\right) \omega_{1}ω2+(nα−[nα])ω1. n n nnnis here a positive integer and [N] denotes the largest integer contained in N N NNN.
Let us first assume that a is a rational number, then the numbers
(1) n α − [ n α ] , n = 1 , 2 , … , (1) n α − [ n α ] , n = 1 , 2 , … , {:(1)n alpha-[n alpha]","quad n=1","2","dots",":}\begin{equation*} n \alpha-[n \alpha], \quad n=1,2, \ldots, \tag{1} \end{equation*}(1)nα−[nα],n=1,2,…,
(2) n α − [ n α ] (2) n α − n α {:(2)(n)/(alpha)-[(n)/(alpha)]:}\begin{equation*} \frac{n}{\alpha}-\left[\frac{n}{\alpha}\right] \tag{2} \end{equation*}(2)nα−[nα]
only take a finite number of distinct values. In this case, there is only a finite number of distinct segments ò. Now, the set of values ​​taken by the function on a segment ò is obviously everywhere non-dense in the plane of the function and this will also be the case for the set of values ​​taken on a finite number of segments δ δ delta\deltaδ. It follows that, in this case, the half-line Δ Δ Delta\DeltaΔis not a direction of complete indeterminacy.
Now suppose that α α alpha\alphaαis an irrational number. In this case the numbers (I) are everywhere dense in the interval (0,1) and the same is true of the numbers (2). It follows that the segments δ δ delta\deltaδare everywhere dense in the parallelogram OABC . Now, any given value is taken at least once in the parallelogram OABC. We can therefore affirm that, by virtue of the continuity of the function, the set of values ​​taken on the segments o is everywhere dense in the plane of the function. It follows that, in this case, the half-line Δ Δ Delta\DeltaΔis a direction of complete indeterminacy.
of complete indeterminacy or not, depending on whether the number a is irrational or rational.
Let θ 1 , θ 2 θ 1 , θ 2 theta_(1),theta_(2)\theta_{1}, \theta_{2}θ1,θ2, the angles of the directions OA , OC OA , OC OA,OC\mathrm{OA}, \mathrm{OC}OA,OCwith the management Δ Δ Delta\DeltaΔ; We have
α = | ω 1 ω 2 | sin ⁡ 0 1 sin ⁡ 0 2 α = ω 1 ω 2 sin ⁡ 0 1 sin ⁡ 0 2 alpha=|(omega_(1))/(omega_(2))|(sin 0_(1))/(sin 0_(2))\alpha=\left|\frac{\omega_{1}}{\omega_{2}}\right| \frac{\sin 0_{1}}{\sin 0_{2}}α=|ω1ω2|sin⁡01sin⁡02
which is also equal to the ratio of the distances of the points HAS , C HAS , C A,CA, CHAS,Cto the half-right Δ Δ Delta\DeltaΔThe rationality of the number α α alpha\alphaαdoes not depend on periods ω 1 , ω 2 ω 1 , ω 2 omega_(1),omega_(2)\omega_{1}, \omega_{2}ω1,ω2chosen. Indeed, if we take two other independent periods
ω 1 ′ = has ω 1 + b ω 2 , ω 2 ′ = c ω 1 + d ω 2 ( has , b , c , d real and integer ; has d − b c ≠ 0 ) , ω 1 ′ = has ω 1 + b ω 2 , ω 2 ′ = c ω 1 + d ω 2 ( has , b , c , d  real and integer  ; has d − b c ≠ 0 ) , {:[omega_(1)^(')=aomega_(1)+bomega_(2)","quadomega_(2)^(')=comega_(1)+domega_(2)],[(a","b","c","d" réels et entiers ";ad-bc!=0)","]:}\begin{gathered} \omega_{1}^{\prime}=a \omega_{1}+b \omega_{2}, \quad \omega_{2}^{\prime}=c \omega_{1}+d \omega_{2} \\ (a, b, c, d \text { réels et entiers } ; a d-b c \neq 0), \end{gathered}ω1′=hasω1+bω2,ω2′=cω1+dω2(has,b,c,d real and integer ;hasd−bc≠0),
the number α α alpha\alphaαwill become a α + b c α + d a α + b c α + d (a alpha+b)/(c alpha+d)\frac{a \alpha+b}{c \alpha+d}hasα+bcα+d. We can therefore state the following theorem:
For a direction Δ Δ Delta\DeltaΔis of complete indeterminacy it is necessary and sufficient that the ratio of the distances to Δ Δ Delta\DeltaΔof any two independent period points is irrational.
Of course, we only consider periods that are not on Δ Δ Delta\DeltaΔ.
We see, therefore, that almost all directions passing through the origin are of complete indeterminacy. We have only a countable set of half-lines which are not of complete indeterminacy. It should be noted that these latter directions are everywhere dense in every angle.
(Extract from the Bulletin des Sciences mathématiques, 2 ∘ 2 ∘ 2^(@)2^{\circ}2∘series, vol. LX; July 1936.)
  1. ( 1 1 ^(1){ }^{1}1) Paul Montel, On the series of rational fractions (Mathematical publications of the University of Belgrade, vol. 1, 1932, p. 11).
  2. Finally, we see that the half-line Δ Δ Delta\DeltaΔis a direction
1936

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