Denjoy–Carleman–Ahlfors theorem
The Denjoy–Carleman–Ahlfors theorem states that the number of asymptotic values attained by a non-constant entire function of order ρ on curves going outwards toward infinite absolute value is less than or equal to 2ρ. It was first conjectured by Arnaud Denjoy in 1907.[1] Torsten Carleman showed that the number of asymptotic values was less than or equal to (5/2)ρ in 1921.[2] In 1929 Lars Ahlfors confirmed Denjoy's conjecture of 2ρ.[3] Finally, in 1933, Carleman published a very short proof.[4]
The use of the term "asymptotic value" does not mean that the ratio of that value to the value of the function approaches 1 (as in asymptotic analysis) as one moves along a certain curve, but rather that the function value approaches the asymptotic value along the curve. For example, as one moves along the real axis toward negative infinity, the function [math]\displaystyle{ \exp(z) }[/math] approaches zero, but the quotient [math]\displaystyle{ 0/\exp(z) }[/math] does not go to 1.
Examples
The function [math]\displaystyle{ \exp(z) }[/math] is of order 1 and has only one asymptotic value, namely 0. The same is true of the function [math]\displaystyle{ \sin(z)/z, }[/math] but the asymptote is attained in two opposite directions.
A case where the number of asymptotic values is equal to 2ρ is the sine integral [math]\displaystyle{ \text{Si}(z)=\int_0^z\frac{\sin \zeta}{\zeta}\,d\zeta }[/math], a function of order 1 which goes to −π/2 along the real axis going toward negative infinity, and to +π/2 in the opposite direction.
The integral of the function [math]\displaystyle{ a\sin(z^2)/z+b\sin(z^2)/z^2 }[/math] is an example of a function of order 2 with four asymptotic values (if b is not zero), approached as one goes outward from zero along the real and imaginary axes.
More generally, [math]\displaystyle{ f(z)=\int_0^z\frac{\sin(\zeta^\rho)}{\zeta^\rho}d\zeta, }[/math] with ρ any positive integer, is of order ρ and has 2ρ asymptotic values.
It is clear that the theorem applies to polynomials only if they are not constant. A constant polynomial has 1 asymptotic value, but is of order 0.
References
- ↑ Arnaud Denjoy (July 8, 1907). "Sur les fonctions entiéres de genre fini". Comptes Rendus de l'Académie des Sciences 145: 106–8. http://gallica.bnf.fr/ark:/12148/bpt6k3099v/f106.image.langFR.
- ↑ T. Carleman (1921). "Sur les fonctions inverses des fonctions entières d'ordre fini". Arkiv för Matematik, Astronomi och Fysik 15 (10): 7.
- ↑ L. Ahlfors (1929). "Über die asymptotischen Werte der ganzen Funktionen endlicher Ordnung". Annales Academiae Scientiarum Fennicae 32 (6): 15.
- ↑ T. Carleman (April 3, 1933). "Sur une inégalité différentielle dans la théorie des fonctions analytiques". Comptes Rendus de l'Académie des Sciences 196: 995–7. http://gallica.bnf.fr/ark:/12148/bpt6k3148d/f995.image.langFR.
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