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.Torsten Carleman showed that the number of asymptotic values was less than or equal to (5/2)ρ in 1921. In 1929 Lars Ahlfors confirmed Denjoy's conjecture of 2ρ. Finally, in 1933, Carleman published a very short proof.

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.

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.

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