Denjoy–Carleman–Ahlfors theorem explained

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

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

Examples

The function

\exp(z) is of order 1 and has only one asymptotic value, namely 0. The same is true of the function \sin(z)/z, but the asymptote is attained in two opposite directions.Si (z) = \int ^{z (\sin \zeta)/(\zeta)} _{0} 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

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,

f(z) = \int ^{z (\sin(\zeta ^{\rho)})/(\zeta ^{\rho})} _{0} 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.

Notes and References

1. Sur les fonctions entiéres de genre fini. Comptes Rendus de l'Académie des Sciences. July 8, 1907. 145. 106–8. Arnaud Denjoy. Arnaud Denjoy.
2. Sur les fonctions inverses des fonctions entières d'ordre fini. Arkiv för Matematik, Astronomi och Fysik. 1921. 15. 10. 7. T. Carleman. Torsten Carleman.
3. Über die asymptotischen Werte der ganzen Funktionen endlicher Ordnung. Annales Academiae Scientiarum Fennicae. 1929. 32. 6. 15. L. Ahlfors. Lars Ahlfors.
4. Sur une inégalité différentielle dans la théorie des fonctions analytiques. Comptes Rendus de l'Académie des Sciences. April 3, 1933. 196. 995–7. T. Carleman.