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| − | A concept in value-distribution theory. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e0367901.png" /> be a meromorphic function in the whole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e0367902.png" />-plane and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e0367903.png" /> denote its number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e0367904.png" />-points (counting multiplicities) in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e0367905.png" />. According to R. Nevanlinna's first fundamental theorem (cf. [[#References|[1]]]), as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e0367906.png" />, | + | {{TEX|done}} |
| | + | 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. [[#References|[1]]]), as $r\to\infty$, |
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e0367907.png" /></td> </tr></table>
| + | $$N(r,a,f)+m(r,a,f)=T(r,f)+O(1),$$ |
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| − | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e0367908.png" /> is the characteristic function, which does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e0367909.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679010.png" /> is the counting function (the logarithmic average of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679011.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679012.png" /> is a function expressing the average proximity of the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679013.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679014.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679015.png" /> (cf. [[Value-distribution theory|Value-distribution theory]]). For the majority of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679016.png" /> the quantities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679018.png" /> are asymptotically equivalent, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679019.png" />. A (finite or infinite) number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679020.png" /> is called an exceptional value if this equivalence as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679021.png" /> is violated. One distinguishes several kinds of exceptional values. | + | 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|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. |
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| − | A number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679022.png" /> is called an exceptional value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679023.png" /> in the sense of Poincaré if the number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679024.png" />-points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679025.png" /> in the whole plane is finite (cf. [[#References|[1]]], [[#References|[2]]]), in particular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679026.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679027.png" />. | + | 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. [[#References|[1]]], [[#References|[2]]]), in particular if $f(z)\neq a$ for any $z$. |
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| − | A number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679028.png" /> is called an exceptional value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679029.png" /> in the sense of Borel if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679030.png" /> grows slower, in a certain sense, than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679031.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679032.png" /> (cf. [[#References|[1]]], [[#References|[2]]]). | + | 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. [[#References|[1]]], [[#References|[2]]]). |
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| − | A number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679033.png" /> is called an exceptional value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679034.png" /> in the sense of Nevanlinna (cf. [[#References|[1]]]) if its defect (cf. [[Defective value|Defective value]]) | + | A number $a$ is called an exceptional value of $f$ in the sense of Nevanlinna (cf. [[#References|[1]]]) if its defect (cf. [[Defective value|Defective value]]) |
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679035.png" /></td> </tr></table>
| + | $$\delta(a,f)=1-\lim_{r\to\infty}\sup\frac{N(r,a,f)}{T(r,f)}>0.$$ |
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| − | A number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679036.png" /> is called an exceptional value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679037.png" /> in the sense of Valiron if | + | A number $a$ is called an exceptional value of $f$ in the sense of Valiron if |
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679038.png" /></td> </tr></table>
| + | $$\Delta(a,f)=1-\lim_{r\to\infty}\inf\frac{N(r,a,f)}{(T(r,f)}>0.$$ |
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| − | A number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679039.png" /> for which | + | A number $a$ for which |
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679040.png" /></td> </tr></table>
| + | $$\beta(a,f)=\lim_{r\to\infty}\inf\frac{\max\limits_{|z|=r}\ln^+1/|f(z)-a|}{T(r,f)}>0$$ |
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| − | is also called an exceptional value for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679041.png" />. The quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679042.png" /> (the positive deviation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679043.png" />) characterizes the rate of the asymptotic approximation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679044.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679045.png" /> (cf. [[#References|[3]]]). | + | 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. [[#References|[3]]]). |
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| | ====References==== | | ====References==== |
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| | ====Comments==== | | ====Comments==== |
| − | An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679046.png" />-point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679047.png" /> is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679048.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036790/e03679049.png" />. | + | An $a$-point of $f$ is a point $z$ such that $f(z)=a$. |
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. [1]), as $r\to\infty$,
$$N(r,a,f)+m(r,a,f)=T(r,f)+O(1),$$
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. [1], [2]), 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. [1], [2]).
A number $a$ is called an exceptional value of $f$ in the sense of Nevanlinna (cf. [1]) if its defect (cf. Defective value)
$$\delta(a,f)=1-\lim_{r\to\infty}\sup\frac{N(r,a,f)}{T(r,f)}>0.$$
A number $a$ is called an exceptional value of $f$ in the sense of Valiron if
$$\Delta(a,f)=1-\lim_{r\to\infty}\inf\frac{N(r,a,f)}{(T(r,f)}>0.$$
A number $a$ for which
$$\beta(a,f)=\lim_{r\to\infty}\inf\frac{\max\limits_{|z|=r}\ln^+1/|f(z)-a|}{T(r,f)}>0$$
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. [3]).
References
| [1] | R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German) |
| [2] | A.A. Gol'dberg, I.V. Ostrovskii, "Value distribution of meromorphic functions" , Moscow (1970) (In Russian) |
| [3] | 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 |
An $a$-point of $f$ is a point $z$ such that $f(z)=a$.