The Denjoy–Carleman–Ahlfors theorem is a mathematical theorem that 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.
For example, as one moves along the real axis toward negative infinity, the function
is of order 1 and has only one asymptotic value, namely 0.
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
, 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.
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.
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.