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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.<ref>{{cite journal|title=Sur les fonctions entiéres de genre fini|journal=[[Comptes Rendus de l'Académie des Sciences]]|date=July 8, 1907|volume=145|pages=106–8|url=http://gallica.bnf.fr/ark:/12148/bpt6k3099v/f106.image.langFR|author=Arnaud Denjoy|authorlink=Arnaud Denjoy}}
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[[Torsten Carleman]] showed that the number of asymptotic values was less than or equal to (5/2)ρ in 1921.<ref>
{{cite journal|title=Sur les fonctions inverses des fonctions entières d'ordre fini|journal=[[Arkiv för Matematik, Astronomi och Fysik]]|year=1921|volume=15|issue=10|page=7|author=T. Carleman|authorlink=Torsten Carleman}}
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In 1929 [[Lars Ahlfors]] confirmed Denjoy's conjecture of 2ρ.<ref>
{{cite journal|title=Über die asymptotischen Werte der ganzen Funktionen endlicher Ordnung|journal=[[Annales Academiae Scientiarum Fennicae]]|year=1929|volume=32|issue=6|page=15|author=L. Ahlfors|authorlink=Lars Ahlfors}}
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Finally, in 1933, Carleman published a very short proof.<ref>
{{cite journal|title=Sur une inégalité différentielle dans la théorie des fonctions analytiques|journal=Comptes Rendus de l'Académie des Sciences|date=April 3, 1933|volume=196|pages=995–7|url=http://gallica.bnf.fr/ark:/12148/bpt6k3148d/f995.image.langFR|author=T. Carleman}}
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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>\exp(z)</math> approaches zero, but the quotient <math>0/\exp(z)</math> does not go to 1.

==Examples==
The function <math>\exp(z)</math> is of order 1 and has only one asymptotic value, namely 0. The same is true of the function <math>\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>\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>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>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==
{{reflist}}

[[Category:Analytic geometry]]