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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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ON THE DIRECTIONS OF COMPLETE INDETERMINATION OF AN ELLIPTIC FUNCTION;
By Mr. Tiberius Popoviciu.
A complete direction of indeterminacy of a functionf(z)f(z)of the complex variablezzis a half-line such that if the variablezzdescribes 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.
Letomega_(1),omega_(2)\omega_{1}, \omega_{2}two independent periods represented by points A, C andDelta\Deltaa 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 ofDelta\Delta.
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.
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-lineDelta\Deltacoincides, by virtue of the periodicity, with the set of values ​​taken in the parallelogram OABC on the segmentsdelta\delta.
The half-rightDelta\Deltacut AB at the affix pointomega_(1)+alphaomega_(2)\omega_{1}+\alpha \omega_{2}and BC at the affix point(1)/(alpha)omega_(1)+omega_(2)\frac{1}{\alpha} \omega_{1}+\omega_{2}Oralpha\alphais a positive number. We see that the ends on AB of the segmentsdelta\deltaare the pointsomega_(1)+(n alpha-[n alpha])omega_(2)\omega_{1}+(n \alpha-[n \alpha]) \omega_{2}and that the ends onBCBCare the pointsomega_(2)+((n)/(alpha)-[(n)/(alpha)])omega_(1)\omega_{2}+\left(\frac{n}{\alpha}-\left[\frac{n}{\alpha}\right]\right) \omega_{1}.nnis here a positive integer and [N] denotes the largest integer contained inNN.
Let us first assume that a is a rational number, then the numbers
{:(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*}
{:(2)(n)/(alpha)-[(n)/(alpha)]:}\begin{equation*} \frac{n}{\alpha}-\left[\frac{n}{\alpha}\right] \tag{2} \end{equation*}
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 segmentsdelta\delta. It follows that, in this case, the half-lineDelta\Deltais not a direction of complete indeterminacy.
Now suppose thatalpha\alphais 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 segmentsdelta\deltaare 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-lineDelta\Deltais a direction of complete indeterminacy.
of complete indeterminacy or not, depending on whether the number a is irrational or rational.
Lettheta_(1),theta_(2)\theta_{1}, \theta_{2}, the angles of the directionsOA,OC\mathrm{OA}, \mathrm{OC}with the managementDelta\Delta; We have
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}}
which is also equal to the ratio of the distances of the pointsA,CA, Cto the half-rightDelta\DeltaThe rationality of the numberalpha\alphadoes not depend on periodsomega_(1),omega_(2)\omega_{1}, \omega_{2}chosen. Indeed, if we take two other independent periods
{:[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}é
the numberalpha\alphawill become(a alpha+b)/(c alpha+d)\frac{a \alpha+b}{c \alpha+d}. We can therefore state the following theorem:
For a directionDelta\Deltais of complete indeterminacy it is necessary and sufficient that the ratio of the distances toDelta\Deltaof any two independent period points is irrational.
Of course, we only consider periods that are not onDelta\Delta.
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^{\circ}series, vol. LX; July 1936.)
(Extract from the Bulletin des Sciences mathématiques,2^(@)2^{\circ}series, vol. LX; July 1936.)
- (^(1){ }^{1}) Paul Montel, On the series of rational fractions (Mathematical publications of the University of Belgrade, vol. 1, 1932, p. 11).
- Finally, we see that the half-lineDelta\Deltais a direction