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 ${\displaystyle \exp(z)}$ approaches zero, but the quotient ${\displaystyle 0/\exp(z)}$ does not go to 1.

Examples

The function ${\displaystyle \exp(z)}$ is of order 1 and has only one asymptotic value, namely 0. The same is true of the function ${\displaystyle \sin(z)/z,}$ 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 ${\displaystyle {\text{Si}}(z)=\int _{0}^{z}{\frac {\sin \zeta }{\zeta }}\,d\zeta }$, 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 ${\displaystyle a\sin(z^{2})/z+b\sin(z^{2})/z^{2}}$ 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, ${\displaystyle f(z)=\int _{0}^{z}{\frac {\sin(\zeta ^{\rho })}{\zeta ^{\rho }}}d\zeta ,}$ 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

1. Arnaud Denjoy (July 8, 1907). "Sur les fonctions entiéres de genre fini". Comptes Rendus de l'Académie des Sciences. 145: 106–8.
2. T. Carleman (1921). "Sur les fonctions inverses des fonctions entières d'ordre fini". Arkiv för Matematik, Astronomi och Fysik. 15 (10): 7.
3. L. Ahlfors (1929). "Über die asymptotischen Werte der ganzen Funktionen endlicher Ordnung". Annales Academiae Scientiarum Fennicae. 32 (6): 15.
4. 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.