2010 Mathematics Subject Classification: Primary: 30D35 [MSN][ZBL]
A concept in value-distribution theory. Let $f(z)$ be a meromorphic function in the whole $z$-plane and let $n(r,a,f)$ denote its number of $a$-points (counting multiplicities) in the disc $|z|\leq r$. According to R. Nevanlinna's first fundamental theorem (cf. ), as $r\to\infty$,
where $T(r,f)$ is the characteristic function, which does not depend on $a$, $N(r,a,f)$ is the counting function (the logarithmic average of $n(r,a,f)$) and $m(r,a,f)>0$ is a function expressing the average proximity of the values of $f$ to $a$ on $|z|=r$ (cf. Value-distribution theory). For the majority of values $a$ the quantities $N(r,a,f)$ and $T(r,f)$ are asymptotically equivalent, as $r\to\infty$. A (finite or infinite) number $a$ is called an exceptional value if this equivalence as $r\to\infty$ is violated. One distinguishes several kinds of exceptional values.
A number $a$ is called an exceptional value of $f$ in the sense of Poincaré if the number of $a$-points of $f$ in the whole plane is finite (cf. , ), in particular if $f(z)\neq a$ for any $z$.
A number $a$ is called an exceptional value of $f$ in the sense of Borel if $n(r,a,f)$ grows slower, in a certain sense, than $T(r,f)$, as $r\to\infty$ (cf. , ).
A number $a$ is called an exceptional value of $f$ in the sense of Nevanlinna (cf. ) if its defect (cf. Defective value)
A number $a$ is called an exceptional value of $f$ in the sense of Valiron if
A number $a$ for which
is also called an exceptional value for $f$. The quantity $\beta(a,f)$ (the positive deviation of $f$) characterizes the rate of the asymptotic approximation of $f(z)$ to $a$ (cf. ).
|||Rolf Nevanlinna, "Analytic functions" , Springer (1970) (Translated from German) Zbl 0199.12501|
|||A.A. Gol'dberg, I.V. Ostrovskii, "Value distribution of meromorphic functions" , Moscow (1970) (In Russian). English translation, Amer. Math. Soc. (2008) ISBN 978-0-8218-4265-2 Zbl 1152.30026|
|||V.P. Petrenko, "Growth of meromorphic functions of finite lower order" Math. USSR Izv. , 3 : 2 (1969) pp. 391–432 Izv. Akad. Nauk SSSR Ser. Mat. , 33 : 2 (1969) pp. 414–454 Zbl 0194.11001|
An $a$-point of $f$ is a point $z$ such that $f(z)=a$.
Exceptional value. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Exceptional_value&oldid=37040