2400:
3637:
in 1976. Another direction was concentrated on the study of various subclasses of the class of all meromorphic functions in the plane. The most important subclass consists of functions of finite order. It turns out that for this class, deficiencies are subject to several restrictions, in addition to
2702:
3368:
1027:
2054:
3628:
A substantial part of the research in functions of one complex variable in the 20th century was focused on
Nevanlinna theory. One direction of this research was to find out whether the main conclusions of Nevanlinna theory are best possible. For example, the
1791:
710:
350:
563:
1588:
142:
This article describes mainly the classical version for meromorphic functions of one variable, with emphasis on functions meromorphic in the complex plane. General references for this theory are
Goldberg & Ostrovskii, Hayman and
3010:
3188:
2867:
1407:
821:
2484:
2395:{\displaystyle {\begin{array}{lcl}T(r,fg)&\leq &T(r,f)+T(r,g)+O(1),\\T(r,f+g)&\leq &T(r,f)+T(r,g)+O(1),\\T(r,1/f)&=&T(r,f)+O(1),\\T(r,f^{m})&=&mT(r,f)+O(1),\,\end{array}}}
1195:
3579:
1972:
3477:
1299:
1638:
805:
1608:, in which the meromorphic function is defined, is finite, the Nevanlinna characteristic may be bounded. Functions in a disc with bounded characteristic, also known as functions of
586:
202:
432:
1492:
1071:
which is the area of the whole
Riemann sphere. The result can be interpreted as the average number of sheets in the covering of the Riemann sphere by the disc |
40:
called it "one of the few great mathematical events of (the twentieth) century." The theory describes the asymptotic distribution of solutions of the equation
3150:
of the
Riemann sphere. However, there is a very different explanations of this 2, based on a deep analogy with number theory discovered by Charles Osgood and
1612:, are exactly those functions that are ratios of bounded analytic functions. Functions of bounded type may also be so defined for another domain such as the
2878:
3363:{\displaystyle \delta (a,f)=\liminf _{r\rightarrow \infty }{\frac {m(r,a,f)}{T(r,f)}}=1-\limsup _{r\rightarrow \infty }{\dfrac {N(r,a,f)}{T(r,f)}}.\,}
3058:. Much better estimates of the error term are known, but Andre Bloch conjectured and Hayman proved that one cannot dispose of an exceptional set.
2726:
3131:). A similar proof also applies to many multi-dimensional generalizations. There are also differential-geometric proofs which relate it to the
1310:
1022:{\displaystyle \int _{0}^{r}{\frac {dt}{t}}\left({\frac {1}{\pi }}\int _{|z|\leq t}{\frac {|f'|^{2}}{(1+|f|^{2})^{2}}}dm\right)=T(r,f)+O(1),\,}
4247:
4213:
4179:
4145:
4112:
4031:
3997:
1036:
is the area element in the plane. The expression in the left hand side is called the
Ahlfors–Shimizu characteristic. The bounded term
2697:{\displaystyle N_{1}(r,f)=2N(r,f)-N(r,f')+N\left(r,{\dfrac {1}{f'}}\right)=N(r,f)+{\overline {N}}(r,f)+N\left(r,{\dfrac {1}{f'}}\right).\,}
3633:
of
Nevanlinna theory consists in constructing meromorphic functions with pre-assigned deficiencies at given points. This was solved by
3155:
3643:
1101:
4317:
3417:. The Second Fundamental Theorem implies that the set of deficient values of a function meromorphic in the plane is at most
76:
3054:)), outside a set of finite length i.e. the error term is small in comparison with the characteristic for "most" values of
4301:
4283:
3968:
3750:
3604:
Nevanlinna theory is useful in all questions where transcendental meromorphic functions arise, like analytic theory of
3140:
2425:) tends to infinity. These algebraic properties are easily obtained from Nevanlinna's definition and Jensen's formula.
4296:
4278:
3146:
The proofs of
Nevanlinna and Ahlfors indicate that the constant 2 in the Second Fundamental Theorem is related to the
1869:
3427:
1786:{\displaystyle \quad N(r,a,f)=N\left(r,{\dfrac {1}{f-a}}\right),\quad m(r,a,f)=m\left(r,{\dfrac {1}{f-a}}\right).\,}
4137:
3728:
3495:
1609:
1234:
1095:
The role of the characteristic function in the theory of meromorphic functions in the plane is similar to that of
3620:, and complex hyperbolic geometry, which deals with generalizations of Picard's theorem to higher dimensions.
3132:
4273:
3397:) tends to infinity (which is always the case for non-constant functions meromorphic in the plane). The points
729:
3772:
705:{\displaystyle m(r,f)={\frac {1}{2\pi }}\int _{0}^{2\pi }\log ^{+}\left|f(re^{i\theta })\right|d\theta .\,}
3843:
3100:
3061:
The Second
Fundamental Theorem allows to give an upper bound for the characteristic function in terms of
4291:
4236:
Arithmetic geometry. Lectures given at the C.I.M.E summer school, Cetraro, Italy, September 10--15, 2007
3605:
4234:(2011). "Diophantine approximation and Nevanlinna theory". In Corvaja, Pietro; Gasbarri, Carlo (eds.).
3697:
3655:
2459:) but without taking multiplicity into account (i.e. we only count the number of distinct poles). Then
100:
3099:
Nevanlinna's original proof of the Second
Fundamental Theorem was based on the so-called Lemma on the
345:{\displaystyle N(r,f)=\int \limits _{0}^{r}\left(n(t,f)-n(0,f)\right){\dfrac {dt}{t}}+n(0,f)\log r.\,}
3835:
3613:
3147:
108:
29:
3848:
3609:
3483:
3093:
2042:
720:
92:
3951:
3823:
3685:
3639:
3482:
where the summation is over all deficient values. This can be considered as a generalization of
1043:
The geometric meaning of the
Ahlfors—Shimizu characteristic is the following. The inner integral
84:
123: = ∞). Subsequent generalizations extended Nevanlinna theory to algebroid functions,
4243:
4209:
4175:
4141:
4108:
4070:
4027:
3993:
3960:
3809:
3742:
3673:
3647:
1428:
124:
4253:
4219:
4185:
4163:
4151:
4060:
4051:
4037:
4003:
3897:
3853:
3174:
The defect relation is one of the main corollaries from the Second Fundamental Theorem. The
1613:
558:{\displaystyle N(r,f)=\sum _{k=1}^{n}\log \left({\frac {r}{|a_{k}|}}\right)+n(0,f)\log r.\,}
132:
128:
21:
4096:
3911:
1583:{\displaystyle \rho (f)=\limsup _{r\rightarrow \infty }{\dfrac {\log ^{+}T(r,f)}{\log r}}.}
4257:
4239:
4223:
4205:
4189:
4171:
4155:
4129:
4092:
4088:
4041:
4023:
4007:
3989:
3907:
3681:
3617:
1201:
136:
104:
33:
3839:
3677:
3486:. Many other Picard-type theorems can be derived from the Second Fundamental Theorem.
3136:
1860:
1056:
96:
4087:, Die Grundlehren der mathematischen Wissenschaften, vol. 162, Berlin, New York:
4311:
3764:
3651:
2059:
88:
3857:
3720:
3669:
3665:
3661:
3634:
3005:{\displaystyle (k-2)T(r,f)\leq \sum _{j=1}^{k}{\overline {N}}(r,a_{j},f)+S(r,f),\,}
1593:
Functions of finite order constitute an important subclass which was much studied.
80:
72:
37:
815:
A second method of defining the Nevanlinna characteristic is based on the formula
4082:
3135:. The Second Fundamental Theorem can also be derived from the metric-topological
17:
3077:
is a transcendental entire function, using the Second Fundamental theorem with
4231:
4197:
4015:
3981:
3889:
3688:. Intensive research in the classical one-dimensional theory still continues.
3489:
As another corollary from the Second Fundamental Theorem, one can obtain that
3151:
4074:
180:) be the number of poles, counting multiplicity, of the meromorphic function
3418:
2862:{\displaystyle \sum _{j=1}^{k}m(r,a_{j},f)\leq 2T(r,f)-N_{1}(r,f)+S(r,f).\,}
3092:
takes every value infinitely often, with at most two exceptions, proving
1402:{\displaystyle \log M(r,f)\leq \left({\dfrac {R+r}{R-r}}\right)T(R,f),\,}
2048:
The characteristic function has the following properties of the degree:
4065:
3902:
3884:
2474:) is defined as the Nevanlinna counting function of critical points of
355:
This quantity measures the growth of the number of poles in the discs |
2005:
tends to infinity, so the First Fundamental Theorem says that the sum
3826:(1982). "Meromorphic solutions of algebraic differential equations".
4049:
Nevanlinna, Rolf (1925), "Zur Theorie der Meromorphen Funktionen",
3638:
the defect relation (Norair Arakelyan, David Drasin, Albert Edrei,
71:
Other main contributors in the first half of the 20th century were
3955:
3676:. This extension is the main tool of Complex Hyperbolic Geometry.
1079:. Then this average covering number is integrated with respect to
3937:
Valiron, G. (1931). "Sur la dérivée des fonctions algébroïdes".
4105:
Nevanlinna Theory and Its Relation to Diophantine Approximation
3158:. On this analogy with number theory we refer to the survey of
68:) which measures the rate of growth of a meromorphic function.
3584:
which generalizes the fact that a rational function of degree
56:
varies. A fundamental tool is the Nevanlinna characteristic
2041:. The first Fundamental theorem is a simple consequence of
1190:{\displaystyle \log M(r,f)=\log \max _{|z|\leq r}|f(z)|\,}
2037:), tends to infinity at the rate which is independent of
4238:. Lecture Notes in Mathematics. Vol. 2009. Berlin:
4202:
Diophantine Approximations and Value Distribution Theory
1989:. For non-constant meromorphic functions in the plane,
3956:
Meromorphic functions of one complex variable. A survey
3373:
By the First Fundamental Theorem, 0 ≤
3154:. According to this analogy, 2 is the exponent in the
107:. In its original form, Nevanlinna theory deals with
3498:
3430:
3305:
3191:
2881:
2729:
2669:
2581:
2487:
2057:
1872:
1755:
1685:
1641:
1528:
1495:
1346:
1313:
1237:
1104:
824:
732:
589:
435:
292:
205:
3806:
Nevanlinna theory and complex differential equations
2707:The Second Fundamental theorem says that for every
1055:, counting multiplicity (that is, the parts of the
4132:; Gubler, Walter (2006). "13. Nevanlinna Theory".
3573:
3471:
3362:
3004:
2861:
2696:
2394:
1966:
1785:
1582:
1401:
1293:
1189:
1021:
799:
704:
557:
344:
3139:, which can be considered as an extension of the
4204:. Lecture Notes in Mathematics. Vol. 1239.
3289:
3214:
1967:{\displaystyle T(r,f)=N(r,a,f)+m(r,a,f)+O(1),\,}
1512:
1139:
1047:is the spherical area of the image of the disc |
3965:Distribution of values of meromorphic functions
3747:Distribution of values of meromorphic functions
3885:"The inverse problem of the Nevanlinna theory"
3472:{\displaystyle \sum _{a}\delta (a,f)\leq 2,\,}
1204:. In fact, it is possible to directly compare
3574:{\displaystyle T(r,f')\leq 2T(r,f)+S(r,f),\,}
8:
4136:. New Mathematical Monographs. Vol. 4.
1294:{\displaystyle T(r,f)\leq \log ^{+}M(r,f)\,}
1855:of Nevanlinna theory states that for every
111:of one complex variable defined in a disc |
4064:
3986:Introduction to complex hyperbolic spaces
3901:
3847:
3725:Meromorphic functions and analytic curves
3570:
3497:
3468:
3435:
3429:
3359:
3304:
3292:
3229:
3217:
3190:
3001:
2962:
2939:
2933:
2922:
2880:
2858:
2813:
2767:
2745:
2734:
2728:
2693:
2668:
2626:
2580:
2492:
2486:
2387:
2332:
2258:
2058:
2056:
1963:
1871:
1782:
1754:
1684:
1640:
1535:
1527:
1515:
1494:
1398:
1345:
1312:
1290:
1263:
1236:
1186:
1181:
1164:
1151:
1143:
1142:
1103:
1018:
959:
949:
944:
935:
918:
913:
899:
896:
883:
875:
874:
860:
840:
834:
829:
823:
796:
731:
701:
675:
648:
635:
630:
611:
588:
554:
509:
503:
494:
488:
472:
461:
434:
402:repeated according to multiplicity. Then
341:
291:
236:
231:
204:
3030:For functions meromorphic in the plane,
1486:of a meromorphic function is defined by
1040:(1) is not important in most questions.
3709:
3178:of a meromorphic function at the point
800:{\displaystyle T(r,f)=m(r,f)+N(r,f).\,}
3715:
3713:
2409:is a natural number. The bounded term
3592: − 2 < 2
3143:to the coverings of infinite degree.
164:be a meromorphic function. For every
7:
3924:
3870:
3159:
144:
155:
3421:and the following relation holds:
3299:
3224:
3163:
1522:
14:
4170:. SpringerBriefs in Mathematics.
1067:times). This area is divided by
156:Nevanlinna's original definition
4134:Heights in Diophantine Geometry
4107:. World Scientific Publishing.
3941:. Vol. 59. pp. 17–39.
3858:10.1070/RM1982v037n04ABEH003967
3088: = ∞, we obtain that
2720:on the Riemann sphere, we have
1712:
1642:
119:or in the whole complex plane (
4020:Survey of Diophantine geometry
3684:extended Nevanlinna theory to
3672:extended Nevanlinna theory to
3564:
3552:
3543:
3531:
3519:
3502:
3456:
3444:
3413:) > 0 are called
3349:
3337:
3329:
3311:
3296:
3273:
3261:
3253:
3235:
3221:
3207:
3195:
2995:
2983:
2974:
2949:
2912:
2900:
2894:
2882:
2852:
2840:
2831:
2819:
2803:
2791:
2779:
2754:
2648:
2636:
2620:
2608:
2560:
2543:
2534:
2522:
2510:
2498:
2381:
2375:
2366:
2354:
2338:
2319:
2306:
2300:
2291:
2279:
2266:
2246:
2233:
2227:
2218:
2206:
2197:
2185:
2172:
2154:
2141:
2135:
2126:
2114:
2105:
2093:
2080:
2065:
1957:
1951:
1942:
1924:
1915:
1897:
1888:
1876:
1734:
1716:
1664:
1646:
1559:
1547:
1519:
1505:
1499:
1392:
1380:
1335:
1323:
1287:
1275:
1253:
1241:
1182:
1178:
1172:
1165:
1152:
1144:
1126:
1114:
1012:
1006:
997:
985:
956:
945:
936:
926:
914:
900:
884:
876:
790:
778:
769:
757:
748:
736:
684:
665:
605:
593:
539:
527:
510:
495:
451:
439:
394:in the punctured disc 0 < |
326:
314:
283:
271:
262:
250:
221:
209:
1:
3969:American Mathematical Society
3751:American Mathematical Society
3658:, Alan Weitsman and others).
3654:, Joseph Miles, Daniel Shea,
3141:Riemann–Hurwitz formula
811:Ahlfors–Shimizu version
32:. It was devised in 1925, by
3828:Russian Mathematical Surveys
2944:
2631:
194:Nevanlinna counting function
4297:Encyclopedia of Mathematics
4279:Encyclopedia of Mathematics
4274:"Value-distribution theory"
3027:) is a "small error term".
723:for meromorphic functions)
367:increases. Explicitly, let
127:, holomorphic maps between
4334:
4138:Cambridge University Press
4081:Nevanlinna, Rolf (1970) ,
3958:, appeared as appendix to
3729:Princeton University Press
3385:) ≤ 1, if
3182:is defined by the formula
3133:Gauss–Bonnet theorem
2429:Second fundamental theorem
1228:) for an entire function:
572: = max(log
3963:; Ostrovskii, I. (2008).
3745:; Ostrovskii, I. (2008).
1853:First Fundamental Theorem
1620:First fundamental theorem
717:Nevanlinna characteristic
151:Nevanlinna characteristic
28:is part of the theory of
4290:Petrenko, V.P. (2001) ,
4272:Petrenko, V.P. (2001) ,
3156:Thue–Siegel–Roth theorem
1479:is a rational function.
131:of arbitrary dimension,
3939:Bull. Soc. Math. France
3773:Oxford University Press
2413:(1) is negligible when
2001:) tends to infinity as
1977:where the bounded term
1800: = ∞, we set
3954:and J. Langley (2008).
3883:Drasin, David (1976).
3575:
3473:
3364:
3101:logarithmic derivative
3006:
2938:
2863:
2750:
2698:
2396:
1968:
1787:
1584:
1403:
1295:
1191:
1023:
801:
706:
559:
477:
346:
241:
4318:Meromorphic functions
4292:"Nevanlinna theorems"
3769:Meromorphic functions
3576:
3474:
3365:
3007:
2918:
2864:
2730:
2699:
2447:) in the same way as
2397:
1969:
1788:
1585:
1404:
1296:
1192:
1024:
802:
707:
560:
457:
347:
227:
109:meromorphic functions
99:, Tatsujiro Shimizu,
30:meromorphic functions
4242:. pp. 111–224.
4140:. pp. 444–478.
4026:. pp. 192–204.
3614:holomorphic dynamics
3496:
3428:
3189:
3148:Euler characteristic
2879:
2727:
2485:
2055:
1870:
1639:
1493:
1311:
1235:
1102:
822:
730:
715:Finally, define the
587:
576:, 0). Then the
433:
203:
168: ≥ 0, let
3840:1982RuMaS..37...61E
3804:Ilpo Laine (1993).
3686:algebroid functions
3624:Further development
3081: = 3 and
3073:). For example, if
839:
643:
93:Frithiof Nevanlinna
4084:Analytic functions
4066:10.1007/BF02543858
3903:10.1007/BF02392314
3698:Vojta's conjecture
3674:holomorphic curves
3656:Oswald Teichmüller
3640:Alexandre Eremenko
3571:
3469:
3440:
3360:
3354:
3303:
3228:
3103:, which says that
3002:
2859:
2694:
2683:
2595:
2392:
2390:
1981:(1) may depend on
1964:
1783:
1772:
1702:
1580:
1575:
1526:
1399:
1371:
1291:
1187:
1163:
1063:times are counted
1019:
825:
797:
702:
626:
578:proximity function
555:
342:
306:
192:. Then define the
125:holomorphic curves
101:Oswald Teichmüller
85:Edward Collingwood
4249:978-3-642-15944-2
4215:978-3-540-17551-3
4181:978-981-10-6786-0
4168:Nevanlinna Theory
4164:Kodaira, Kunihiko
4147:978-0-521-71229-3
4114:978-981-02-4402-6
4033:978-3-540-61223-0
3999:978-0-387-96447-8
3810:Walter de Gruyter
3648:Anatolii Goldberg
3596:critical points.
3431:
3353:
3288:
3277:
3213:
3137:theory of Ahlfors
2947:
2682:
2634:
2594:
1771:
1701:
1574:
1511:
1475:) if and only if
1429:rational function
1370:
1200:in the theory of
1138:
966:
868:
853:
624:
515:
381:, ...,
305:
133:quasiregular maps
129:complex manifolds
26:Nevanlinna theory
4325:
4304:
4286:
4261:
4227:
4193:
4159:
4130:Bombieri, Enrico
4118:
4103:Ru, Min (2001).
4099:
4077:
4068:
4052:Acta Mathematica
4045:
4011:
3973:
3972:
3949:
3943:
3942:
3934:
3928:
3922:
3916:
3915:
3905:
3880:
3874:
3868:
3862:
3861:
3851:
3820:
3814:
3813:
3801:
3795:
3792:
3786:
3783:
3777:
3776:
3761:
3755:
3754:
3739:
3733:
3732:
3717:
3618:minimal surfaces
3580:
3578:
3577:
3572:
3518:
3484:Picard's theorem
3478:
3476:
3475:
3470:
3439:
3415:deficient values
3369:
3367:
3366:
3361:
3355:
3352:
3332:
3306:
3302:
3278:
3276:
3256:
3230:
3227:
3162:and the book by
3094:Picard's Theorem
3042:) = o(
3011:
3009:
3008:
3003:
2967:
2966:
2948:
2940:
2937:
2932:
2868:
2866:
2865:
2860:
2818:
2817:
2772:
2771:
2749:
2744:
2711:distinct values
2703:
2701:
2700:
2695:
2689:
2685:
2684:
2681:
2670:
2635:
2627:
2601:
2597:
2596:
2593:
2582:
2559:
2497:
2496:
2438:
2401:
2399:
2398:
2393:
2391:
2337:
2336:
2262:
2043:Jensen's formula
1973:
1971:
1970:
1965:
1792:
1790:
1789:
1784:
1778:
1774:
1773:
1770:
1756:
1708:
1704:
1703:
1700:
1686:
1614:upper half-plane
1596:When the radius
1589:
1587:
1586:
1581:
1576:
1573:
1562:
1540:
1539:
1529:
1525:
1416: >
1408:
1406:
1405:
1400:
1376:
1372:
1369:
1358:
1347:
1300:
1298:
1297:
1292:
1268:
1267:
1202:entire functions
1196:
1194:
1193:
1188:
1185:
1168:
1162:
1155:
1147:
1070:
1028:
1026:
1025:
1020:
978:
974:
967:
965:
964:
963:
954:
953:
948:
939:
924:
923:
922:
917:
911:
903:
897:
895:
894:
887:
879:
869:
861:
854:
849:
841:
838:
833:
806:
804:
803:
798:
721:Jensen's formula
711:
709:
708:
703:
691:
687:
683:
682:
653:
652:
642:
634:
625:
623:
612:
564:
562:
561:
556:
520:
516:
514:
513:
508:
507:
498:
489:
476:
471:
390:be the poles of
351:
349:
348:
343:
307:
301:
293:
290:
286:
240:
235:
137:minimal surfaces
22:complex analysis
4333:
4332:
4328:
4327:
4326:
4324:
4323:
4322:
4308:
4307:
4289:
4271:
4268:
4250:
4240:Springer-Verlag
4230:
4216:
4206:Springer-Verlag
4196:
4182:
4172:Springer-Verlag
4162:
4148:
4128:
4125:
4123:Further reading
4115:
4102:
4089:Springer-Verlag
4080:
4048:
4034:
4024:Springer-Verlag
4014:
4000:
3990:Springer-Verlag
3980:
3977:
3976:
3959:
3950:
3946:
3936:
3935:
3931:
3923:
3919:
3882:
3881:
3877:
3869:
3865:
3849:10.1.1.139.8499
3822:
3821:
3817:
3803:
3802:
3798:
3793:
3789:
3784:
3780:
3763:
3762:
3758:
3741:
3740:
3736:
3719:
3718:
3711:
3706:
3694:
3682:Georges Valiron
3631:Inverse Problem
3626:
3602:
3511:
3494:
3493:
3426:
3425:
3333:
3307:
3257:
3231:
3187:
3186:
3172:
3170:Defect relation
3087:
2958:
2877:
2876:
2809:
2763:
2725:
2724:
2719:
2674:
2661:
2657:
2586:
2573:
2569:
2552:
2488:
2483:
2482:
2465:
2434:
2431:
2389:
2388:
2346:
2341:
2328:
2313:
2312:
2274:
2269:
2240:
2239:
2180:
2175:
2148:
2147:
2088:
2083:
2053:
2052:
1868:
1867:
1760:
1747:
1743:
1690:
1677:
1673:
1637:
1636:
1622:
1563:
1531:
1530:
1491:
1490:
1451: log
1359:
1348:
1341:
1309:
1308:
1259:
1233:
1232:
1100:
1099:
1093:
1068:
955:
943:
925:
912:
904:
898:
870:
859:
855:
842:
820:
819:
813:
728:
727:
671:
661:
657:
644:
616:
585:
584:
499:
493:
484:
431:
430:
389:
380:
373:
294:
246:
242:
201:
200:
158:
153:
105:Georges Valiron
34:Rolf Nevanlinna
12:
11:
5:
4331:
4329:
4321:
4320:
4310:
4309:
4306:
4305:
4287:
4267:
4266:External links
4264:
4263:
4262:
4248:
4228:
4214:
4194:
4180:
4160:
4146:
4124:
4121:
4120:
4119:
4113:
4100:
4078:
4046:
4032:
4012:
3998:
3975:
3974:
3944:
3929:
3917:
3875:
3863:
3815:
3796:
3794:Ru (2001) p.61
3787:
3778:
3756:
3734:
3708:
3707:
3705:
3702:
3701:
3700:
3693:
3690:
3678:Henrik Selberg
3664:, Joachim and
3644:Wolfgang Fuchs
3625:
3622:
3601:
3598:
3582:
3581:
3569:
3566:
3563:
3560:
3557:
3554:
3551:
3548:
3545:
3542:
3539:
3536:
3533:
3530:
3527:
3524:
3521:
3517:
3514:
3510:
3507:
3504:
3501:
3480:
3479:
3467:
3464:
3461:
3458:
3455:
3452:
3449:
3446:
3443:
3438:
3434:
3371:
3370:
3358:
3351:
3348:
3345:
3342:
3339:
3336:
3331:
3328:
3325:
3322:
3319:
3316:
3313:
3310:
3301:
3298:
3295:
3291:
3290:lim sup
3287:
3284:
3281:
3275:
3272:
3269:
3266:
3263:
3260:
3255:
3252:
3249:
3246:
3243:
3240:
3237:
3234:
3226:
3223:
3220:
3216:
3215:lim inf
3212:
3209:
3206:
3203:
3200:
3197:
3194:
3171:
3168:
3119:) =
3085:
3013:
3012:
3000:
2997:
2994:
2991:
2988:
2985:
2982:
2979:
2976:
2973:
2970:
2965:
2961:
2957:
2954:
2951:
2946:
2943:
2936:
2931:
2928:
2925:
2921:
2917:
2914:
2911:
2908:
2905:
2902:
2899:
2896:
2893:
2890:
2887:
2884:
2870:
2869:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2833:
2830:
2827:
2824:
2821:
2816:
2812:
2808:
2805:
2802:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2770:
2766:
2762:
2759:
2756:
2753:
2748:
2743:
2740:
2737:
2733:
2715:
2705:
2704:
2692:
2688:
2680:
2677:
2673:
2667:
2664:
2660:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2633:
2630:
2625:
2622:
2619:
2616:
2613:
2610:
2607:
2604:
2600:
2592:
2589:
2585:
2579:
2576:
2572:
2568:
2565:
2562:
2558:
2555:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2503:
2500:
2495:
2491:
2463:
2430:
2427:
2403:
2402:
2386:
2383:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2345:
2342:
2340:
2335:
2331:
2327:
2324:
2321:
2318:
2315:
2314:
2311:
2308:
2305:
2302:
2299:
2296:
2293:
2290:
2287:
2284:
2281:
2278:
2275:
2273:
2270:
2268:
2265:
2261:
2257:
2254:
2251:
2248:
2245:
2242:
2241:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2179:
2176:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2149:
2146:
2143:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2119:
2116:
2113:
2110:
2107:
2104:
2101:
2098:
2095:
2092:
2089:
2087:
2084:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2061:
2060:
2021:) +
1975:
1974:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1861:Riemann sphere
1836:) =
1812:) =
1794:
1793:
1781:
1777:
1769:
1766:
1763:
1759:
1753:
1750:
1746:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1711:
1707:
1699:
1696:
1693:
1689:
1683:
1680:
1676:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1648:
1645:
1621:
1618:
1591:
1590:
1579:
1572:
1569:
1566:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1538:
1534:
1524:
1521:
1518:
1514:
1513:lim sup
1510:
1507:
1504:
1501:
1498:
1467:) =
1447:) ~
1410:
1409:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1375:
1368:
1365:
1362:
1357:
1354:
1351:
1344:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1302:
1301:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1266:
1262:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1198:
1197:
1184:
1180:
1177:
1174:
1171:
1167:
1161:
1158:
1154:
1150:
1146:
1141:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1092:
1089:
1083:with weight 1/
1057:Riemann sphere
1030:
1029:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
977:
973:
970:
962:
958:
952:
947:
942:
938:
934:
931:
928:
921:
916:
910:
907:
902:
893:
890:
886:
882:
878:
873:
867:
864:
858:
852:
848:
845:
837:
832:
828:
812:
809:
808:
807:
795:
792:
789:
786:
783:
780:
777:
774:
771:
768:
765:
762:
759:
756:
753:
750:
747:
744:
741:
738:
735:
713:
712:
700:
697:
694:
690:
686:
681:
678:
674:
670:
667:
664:
660:
656:
651:
647:
641:
638:
633:
629:
622:
619:
615:
610:
607:
604:
601:
598:
595:
592:
580:is defined by
566:
565:
553:
550:
547:
544:
541:
538:
535:
532:
529:
526:
523:
519:
512:
506:
502:
497:
492:
487:
483:
480:
475:
470:
467:
464:
460:
456:
453:
450:
447:
444:
441:
438:
385:
378:
371:
353:
352:
340:
337:
334:
331:
328:
325:
322:
319:
316:
313:
310:
304:
300:
297:
289:
285:
282:
279:
276:
273:
270:
267:
264:
261:
258:
255:
252:
249:
245:
239:
234:
230:
226:
223:
220:
217:
214:
211:
208:
157:
154:
152:
149:
97:Henrik Selberg
13:
10:
9:
6:
4:
3:
2:
4330:
4319:
4316:
4315:
4313:
4303:
4299:
4298:
4293:
4288:
4285:
4281:
4280:
4275:
4270:
4269:
4265:
4259:
4255:
4251:
4245:
4241:
4237:
4233:
4229:
4225:
4221:
4217:
4211:
4207:
4203:
4199:
4195:
4191:
4187:
4183:
4177:
4173:
4169:
4165:
4161:
4157:
4153:
4149:
4143:
4139:
4135:
4131:
4127:
4126:
4122:
4116:
4110:
4106:
4101:
4098:
4094:
4090:
4086:
4085:
4079:
4076:
4072:
4067:
4062:
4059:(1–2): 1–99,
4058:
4054:
4053:
4047:
4043:
4039:
4035:
4029:
4025:
4021:
4017:
4013:
4009:
4005:
4001:
3995:
3991:
3987:
3983:
3979:
3978:
3970:
3966:
3962:
3957:
3953:
3948:
3945:
3940:
3933:
3930:
3926:
3921:
3918:
3913:
3909:
3904:
3899:
3896:(1): 83–151.
3895:
3892:
3891:
3886:
3879:
3876:
3872:
3867:
3864:
3859:
3855:
3850:
3845:
3841:
3837:
3833:
3829:
3825:
3819:
3816:
3811:
3807:
3800:
3797:
3791:
3788:
3785:Ru (2001) p.5
3782:
3779:
3774:
3770:
3766:
3760:
3757:
3752:
3748:
3744:
3738:
3735:
3730:
3726:
3722:
3716:
3714:
3710:
3703:
3699:
3696:
3695:
3691:
3689:
3687:
3683:
3679:
3675:
3671:
3667:
3663:
3659:
3657:
3653:
3652:Walter Hayman
3649:
3645:
3641:
3636:
3632:
3623:
3621:
3619:
3615:
3611:
3607:
3599:
3597:
3595:
3591:
3587:
3567:
3561:
3558:
3555:
3549:
3546:
3540:
3537:
3534:
3528:
3525:
3522:
3515:
3512:
3508:
3505:
3499:
3492:
3491:
3490:
3487:
3485:
3465:
3462:
3459:
3453:
3450:
3447:
3441:
3436:
3432:
3424:
3423:
3422:
3420:
3416:
3412:
3408:
3404:
3400:
3396:
3392:
3388:
3384:
3380:
3376:
3356:
3346:
3343:
3340:
3334:
3326:
3323:
3320:
3317:
3314:
3308:
3293:
3285:
3282:
3279:
3270:
3267:
3264:
3258:
3250:
3247:
3244:
3241:
3238:
3232:
3218:
3210:
3204:
3201:
3198:
3192:
3185:
3184:
3183:
3181:
3177:
3169:
3167:
3165:
3161:
3157:
3153:
3149:
3144:
3142:
3138:
3134:
3130:
3126:
3122:
3118:
3114:
3110:
3106:
3102:
3097:
3095:
3091:
3084:
3080:
3076:
3072:
3068:
3064:
3059:
3057:
3053:
3049:
3045:
3041:
3037:
3033:
3028:
3026:
3022:
3018:
2998:
2992:
2989:
2986:
2980:
2977:
2971:
2968:
2963:
2959:
2955:
2952:
2941:
2934:
2929:
2926:
2923:
2919:
2915:
2909:
2906:
2903:
2897:
2891:
2888:
2885:
2875:
2874:
2873:
2872:This implies
2855:
2849:
2846:
2843:
2837:
2834:
2828:
2825:
2822:
2814:
2810:
2806:
2800:
2797:
2794:
2788:
2785:
2782:
2776:
2773:
2768:
2764:
2760:
2757:
2751:
2746:
2741:
2738:
2735:
2731:
2723:
2722:
2721:
2718:
2714:
2710:
2690:
2686:
2678:
2675:
2671:
2665:
2662:
2658:
2654:
2651:
2645:
2642:
2639:
2628:
2623:
2617:
2614:
2611:
2605:
2602:
2598:
2590:
2587:
2583:
2577:
2574:
2570:
2566:
2563:
2556:
2553:
2549:
2546:
2540:
2537:
2531:
2528:
2525:
2519:
2516:
2513:
2507:
2504:
2501:
2493:
2489:
2481:
2480:
2479:
2477:
2473:
2469:
2462:
2458:
2454:
2450:
2446:
2442:
2437:
2428:
2426:
2424:
2420:
2416:
2412:
2408:
2384:
2378:
2372:
2369:
2363:
2360:
2357:
2351:
2348:
2343:
2333:
2329:
2325:
2322:
2316:
2309:
2303:
2297:
2294:
2288:
2285:
2282:
2276:
2271:
2263:
2259:
2255:
2252:
2249:
2243:
2236:
2230:
2224:
2221:
2215:
2212:
2209:
2203:
2200:
2194:
2191:
2188:
2182:
2177:
2169:
2166:
2163:
2160:
2157:
2151:
2144:
2138:
2132:
2129:
2123:
2120:
2117:
2111:
2108:
2102:
2099:
2096:
2090:
2085:
2077:
2074:
2071:
2068:
2062:
2051:
2050:
2049:
2046:
2044:
2040:
2036:
2032:
2028:
2024:
2020:
2016:
2012:
2008:
2004:
2000:
1996:
1992:
1988:
1984:
1980:
1960:
1954:
1948:
1945:
1939:
1936:
1933:
1930:
1927:
1921:
1918:
1912:
1909:
1906:
1903:
1900:
1894:
1891:
1885:
1882:
1879:
1873:
1866:
1865:
1864:
1862:
1858:
1854:
1849:
1847:
1843:
1839:
1835:
1831:
1827:
1823:
1819:
1815:
1811:
1807:
1803:
1799:
1779:
1775:
1767:
1764:
1761:
1757:
1751:
1748:
1744:
1740:
1737:
1731:
1728:
1725:
1722:
1719:
1713:
1709:
1705:
1697:
1694:
1691:
1687:
1681:
1678:
1674:
1670:
1667:
1661:
1658:
1655:
1652:
1649:
1643:
1635:
1634:
1633:
1632:, and define
1631:
1628: ∈
1627:
1619:
1617:
1615:
1611:
1607:
1603:
1600:of the disc |
1599:
1594:
1577:
1570:
1567:
1564:
1556:
1553:
1550:
1544:
1541:
1536:
1532:
1516:
1508:
1502:
1496:
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1485:
1480:
1478:
1474:
1470:
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1462:
1458:
1454:
1450:
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1426:
1421:
1419:
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1395:
1389:
1386:
1383:
1377:
1373:
1366:
1363:
1360:
1355:
1352:
1349:
1342:
1338:
1332:
1329:
1326:
1320:
1317:
1314:
1307:
1306:
1305:
1284:
1281:
1278:
1272:
1269:
1264:
1260:
1256:
1250:
1247:
1244:
1238:
1231:
1230:
1229:
1227:
1223:
1219:
1215:
1211:
1207:
1203:
1175:
1169:
1159:
1156:
1148:
1135:
1132:
1129:
1123:
1120:
1117:
1111:
1108:
1105:
1098:
1097:
1096:
1090:
1088:
1086:
1082:
1078:
1074:
1066:
1062:
1058:
1054:
1050:
1046:
1041:
1039:
1035:
1015:
1009:
1003:
1000:
994:
991:
988:
982:
979:
975:
971:
968:
960:
950:
940:
932:
929:
919:
908:
905:
891:
888:
880:
871:
865:
862:
856:
850:
846:
843:
835:
830:
826:
818:
817:
816:
810:
793:
787:
784:
781:
775:
772:
766:
763:
760:
754:
751:
745:
742:
739:
733:
726:
725:
724:
722:
718:
698:
695:
692:
688:
679:
676:
672:
668:
662:
658:
654:
649:
645:
639:
636:
631:
627:
620:
617:
613:
608:
602:
599:
596:
590:
583:
582:
581:
579:
575:
571:
551:
548:
545:
542:
536:
533:
530:
524:
521:
517:
504:
500:
490:
485:
481:
478:
473:
468:
465:
462:
458:
454:
448:
445:
442:
436:
429:
428:
427:
425:
421:
417:
413:
409:
405:
401:
397:
393:
388:
384:
377:
370:
366:
362:
358:
338:
335:
332:
329:
323:
320:
317:
311:
308:
302:
298:
295:
287:
280:
277:
274:
268:
265:
259:
256:
253:
247:
243:
237:
232:
228:
224:
218:
215:
212:
206:
199:
198:
197:
195:
191:
187:
184:in the disc |
183:
179:
175:
171:
167:
163:
150:
148:
146:
140:
138:
134:
130:
126:
122:
118:
114:
110:
106:
102:
98:
94:
90:
89:Otto Frostman
86:
82:
78:
74:
69:
67:
63:
59:
55:
51:
47:
43:
39:
35:
31:
27:
23:
19:
4295:
4277:
4235:
4201:
4167:
4133:
4104:
4083:
4056:
4050:
4019:
3988:. New York:
3985:
3964:
3961:Goldberg, A.
3947:
3938:
3932:
3920:
3893:
3888:
3878:
3866:
3834:(4): 61–95.
3831:
3827:
3824:Eremenko, A.
3818:
3805:
3799:
3790:
3781:
3768:
3759:
3746:
3743:Goldberg, A.
3737:
3731:. p. 8.
3724:
3670:Lars Ahlfors
3666:Hermann Weyl
3662:Henri Cartan
3660:
3635:David Drasin
3630:
3627:
3606:differential
3603:
3600:Applications
3593:
3589:
3585:
3583:
3488:
3481:
3414:
3410:
3406:
3402:
3398:
3394:
3390:
3386:
3382:
3378:
3374:
3372:
3179:
3175:
3173:
3145:
3128:
3124:
3120:
3116:
3112:
3108:
3104:
3098:
3089:
3082:
3078:
3074:
3070:
3066:
3062:
3060:
3055:
3051:
3047:
3043:
3039:
3035:
3031:
3029:
3024:
3020:
3016:
3014:
2871:
2716:
2712:
2708:
2706:
2475:
2471:
2467:
2460:
2456:
2452:
2448:
2444:
2440:
2435:
2432:
2422:
2418:
2414:
2410:
2406:
2404:
2047:
2038:
2034:
2030:
2026:
2022:
2018:
2014:
2010:
2006:
2002:
1998:
1994:
1990:
1986:
1982:
1978:
1976:
1856:
1852:
1850:
1845:
1841:
1837:
1833:
1829:
1825:
1821:
1817:
1813:
1809:
1805:
1801:
1797:
1795:
1629:
1625:
1623:
1610:bounded type
1605:
1601:
1597:
1595:
1592:
1483:
1481:
1476:
1472:
1468:
1464:
1460:
1456:
1452:
1448:
1444:
1440:
1436:
1432:
1424:
1422:
1417:
1413:
1411:
1303:
1225:
1221:
1217:
1213:
1209:
1205:
1199:
1094:
1084:
1080:
1076:
1072:
1064:
1060:
1052:
1048:
1044:
1042:
1037:
1033:
1031:
814:
716:
714:
577:
573:
569:
567:
423:
419:
415:
411:
407:
403:
399:
395:
391:
386:
382:
375:
368:
364:
360:
356:
354:
193:
189:
185:
181:
177:
173:
169:
165:
161:
159:
141:
120:
116:
112:
81:Henri Cartan
73:Lars Ahlfors
70:
65:
61:
57:
53:
49:
45:
41:
38:Hermann Weyl
25:
18:mathematical
15:
4232:Vojta, Paul
4198:Vojta, Paul
4016:Lang, Serge
3982:Lang, Serge
3952:A. Eremenko
3925:Lang (1987)
3871:Lang (1987)
3160:Lang (1987)
1455:; in fact,
145:Lang (1987)
77:André Bloch
4258:1258.11076
4224:0609.14011
4190:1386.30002
4156:1115.11034
4042:0869.11051
4008:0628.32001
3890:Acta Math.
3808:. Berlin:
3765:Hayman, W.
3704:References
3612:equations
3610:functional
3401:for which
3152:Paul Vojta
2478:, that is
2433:We define
1471:(log
1431:of degree
1091:Properties
4302:EMS Press
4284:EMS Press
4075:0001-5962
3844:CiteSeerX
3523:≤
3460:≤
3442:δ
3433:∑
3419:countable
3300:∞
3297:→
3286:−
3225:∞
3222:→
3193:δ
3164:Ru (2001)
2945:¯
2920:∑
2916:≤
2889:−
2807:−
2783:≤
2732:∑
2632:¯
2538:−
2178:≤
2086:≤
1765:−
1695:−
1568:
1542:
1523:∞
1520:→
1497:ρ
1364:−
1339:≤
1318:
1270:
1257:≤
1157:≤
1136:
1109:
889:≤
872:∫
866:π
827:∫
696:θ
680:θ
655:
640:π
628:∫
621:π
546:
482:
459:∑
333:
266:−
229:∫
20:field of
4312:Category
4200:(1987).
4166:(2017).
4018:(1997).
3984:(1987).
3767:(1964).
3723:(1943).
3692:See also
3516:′
2679:′
2591:′
2557:′
1412:for any
1059:covered
909:′
719:by (cf.
4097:0279280
3912:0585644
3836:Bibcode
3721:H. Weyl
2443:,
1997:,
1859:in the
1435:, then
568:Let log
426:), and
374:,
16:In the
4256:
4246:
4222:
4212:
4188:
4178:
4154:
4144:
4111:
4095:
4073:
4040:
4030:
4006:
3996:
3927:ch.VII
3910:
3846:
3176:defect
3015:where
2405:where
1216:) and
1032:where
103:, and
3588:has 2
1484:order
1427:is a
363:, as
52:, as
4244:ISBN
4210:ISBN
4176:ISBN
4142:ISBN
4109:ISBN
4071:ISSN
4028:ISBN
3994:ISBN
3873:p.39
3680:and
3668:and
3608:and
1985:and
1851:The
1796:For
1624:Let
1604:| ≤
1482:The
1304:and
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1051:| ≤
418:) -
398:| ≤
359:| ≤
188:| ≤
160:Let
135:and
115:| ≤
48:) =
4254:Zbl
4220:Zbl
4186:Zbl
4152:Zbl
4061:doi
4038:Zbl
4004:Zbl
3898:doi
3894:138
3854:doi
1848:).
1832:,∞,
1824:),
1808:,∞,
1565:log
1533:log
1423:If
1315:log
1261:log
1140:max
1133:log
1106:log
646:log
543:log
479:log
422:(0,
330:log
196:by
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4314::
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4294:,
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4252:.
4218:.
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4036:.
4022:.
4002:.
3992:.
3967:.
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3906:.
3887:.
3852:.
3842:.
3832:37
3830:.
3771:.
3749:.
3727:.
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3646:,
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2017:,
2015:a
2013:,
2011:r
2009:(
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1987:a
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