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Cherry W., Ye Z. Nevanlinna's Theory of Value Distribution: The Second Main Theorem and Its Error Terms
Cherry W., Ye Z.  Nevanlinna's Theory of Value Distribution: The Second Main Theorem and Its Error Terms







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: Nevanlinna's Theory of Value Distribution: The Second Main Theorem and Its Error Terms

: Cherry W., Ye Z.

:

"On the one hand, this monograph serves as a self-contained introduction to Nevanlinna's theory of value distribution because the authors only assume the reader is familiar with the basics of complex analysis. On the other hand, it also serves as a valuable reference for the research specialist because the authors present, for the first time in book form, the most modern and refined versions of the Second Main Theorem with precise error terms, in both the geometric and logarithmic derivative based approaches. A unique feature of the monograph is its "number-theoretic digressions". These special sections assume no background in number theory and explore the exciting interconnections between Nevanlinna theory and the theory of Diophantine approximation."


: en

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ed2k: ed2k stats

: 2001

: 201

: 02.04.2005

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$dd^cl-4$      32
Absolute value      29
Absolute value at infinity      29
Absolute value for meromorphic functions      30
Absolute value, Archimedean      29
Absolute value, Archimedean on a number field      152
Absolute value, equivalence between      29
Absolute value, non-Archimedean      29
Absolute value, non-Archimedean on a number field      153
Absolute value, p-adic      29 153
Absolute value, trivial      29
Ahlfors Beurling Inequality      79
Ahlfors Shimizu characteristic (or height)      see Characteristic function
Ahlfors, L.      vi 15 39 47 49 61 62 74 79
Ahlforss theory of covering surfaces      74
Algebraic integer      153
Algebraic number      151
Archimedean absolute value      see Absolute value Archimedean
Artin Whaples Product Formula      30 152 154 156
AV 1-3      29
AV 3'      29
Baker, I.N.      146
Bergweiler, W.      74 150
Beurling, A.      79
Bohr, H.      146
Bonk, M.      139 183
Borel Nevanlinna growth lemma      98
Borel, E.      v 98
Boundary term (in Stokess Theorem)      33
Bryuno, A.      159
Calculus lemma      52 53
Caratheodory, C.      142
Carleman, T.      80
Carlson, J.      31
Cartan, H.      3 43 107
Cartans Theorem equating height or characteristic with the average of the counting function      43
Cauchy Riemann equations      31
Characteristic function      2
Characteristic function, expressed as average of counting function      43 45
Characteristic function, expressed as height transform      57
Characteristic function, geometric (Ahlfors Shimizu)      17 39
Characteristic function, invariance under linear fractional transformation      23
Characteristic function, Nevanlinna      17
Chem class      32
Chem operator      see First Chem operator
Chem, S.S.      vi 61 62
Chen, H.      117 172 173
Cherry, W.      139
Chordal distance      15
Compensation function      see Mean proximity function
Complex conjugate      ix
Concave      47
Concavity of the logarithm      48
Conjugate, complex      see Complex conjugate
Constants      x
Convergence class      27
Convex      41
Convexity of $\bar{T}(f, r)$ (in log r)      41
Convexity of T(f, r) (in log r)      44
Cosine, Nevanlinna functions for      27
Cosine, order of      28
Counting function      1 12
Counting function, associated to ramification divisor      22 125
Counting function, number theoretic analogue of      156
Counting function, truncated      13 125
Counting function, unintegrated      12
Counting function, unintegrated, compared to A      42
Counting function, upper bound on      20
Curvature      37 60
Curvature computation      37 61
Dcvaney, M.      159
Defect      124
Defect, ramification      125
Defect, truncated      125
Deficient      125
Density, upper      171
Differential equations      129
Dufresnoy, J.      76
Eremenko, A.      3
Error term      vi 3 60
Euclidean form      36
Eulers constant      180
Eulers gamma function      see Gamma function
Exceptional set      3 50
Exponential function      5 20
Exponential function, Lang type of      180
Exponential function, Nevanlinna functions for      25
Exponential function, order of      27
Exterior derivative      31
Finite order      see Order finite
Finite order, Second Main Theorem for functions of      117
First Chem operator      32
First Main Theorem      2 5 17
First Main Theorem constant      17
First Main Theorem, equivalent to Jensens Formula      19
First Main Theorem, unintegrated      42
Fixed point      146
Fixed point of order n      146
Frank, G.      136
Fubini Study form      35
Fubinis Theorem      33
Fundamental Theorem of Algebra, v, I      11 19 42 120
Gamnui function      180
Gamnui function, derivative of      180
Gamnui function, Lang type of      185
Gauss Bonnet formula      74
GJ 1-4      34
Goldberg Grinshtein inequality      3 92 96 98
Goldberg, A.A.      v 3 42 89 92 96 98 107
Green Jensen Formula      33
Greens function, relation to Poisson kernel      83
Greens function      83
Griffiths, P.      31
Grinshtein, A.      3 89 92 96 98 107
Growth lemma      52 53
Growth order      see Order growth
Growth type      see Type growth
Hadamard, J.      v
Harmonic      31
Harmonic on closed disc      6
Hayman Stewart Theorem      43
Hayman, W.      v 3 42 107 146 147
Height function      2
Height function, expressed as average of counting function      43 45
Height function, geometric (Ahlfors Shimizu)      17 39
Height function, invariance under linear fractional transformation      23
Height function, Nevanlinna      17
Height transform      54
Height transform, Ahlfors Shimizu characteristic expressed as      57
Height transform, fundamental estimate for      54
Height transform, second derivative expressed as integral on circle      54
Height, absolute logarithmic      155
Height, number theoretic      154 155
Hempel, J.      142
Hinkkanen, A.      107 110 162 163
Holomorphic on closed disc      6
HT 1-3      53
Huber, A.      78
Imaginary part      ix
Infinite order      see Order infinite
Infinite order, example      28
Initial Laurent coefficient      9
Integers, in a number field      153
Interior term (in Stokess Theorem)      33
Isoperimctric inequality, planar      79
Isoperimctric inequality, spherical      81
Jank, G.      vi 3
Jankowski, M.      171
Jenkins, J.      142
Jensen formula      11
Jensen, J.      9
Jensens Inequality      48
Khinchin convergence condition      49 159
Khinchin function      49 159
Khinchin, A.      159
Khinchins Theorem      159 177
Kolokolnikov, A.      90
Kolokolnikovs inequality      90
Laguerres Theorem      186
Laine, I.      vi
Landau type theorem      see Theorem of Landau type
Landau, E.      141
Lang type      177
Lang type of exponential function      180
Lang type of gamnui function      185
Lang type of rational function      178
Lang type of trigonometric functions      180
Lang type, greater than or equal      178
Lang type, Lang type 1      178
Lang type, less than or equal      177
Lang type, precise      178
Lang, S.      vi 3 50 53 60 61 151 159 177
Lemma on the Logarithmic Derivative      89 104
Lemma on the Logarithmic Derivative, sharpness of      161
Li, P.      136
Logarithmic derivative      89
Logarithmic measure      50
Lower order      see Order growth lower
LT $\geq$      178
LT $\leq$      177
Malmquist, J.      130
Malmquists theorem      129
Maximal type      27
Maximum modulus function      28
Mean proximity function, difference between Nevanlinna and geometric      15
Mean proximity function, geometric      15
Mean proximity function, Nevanlinna      14
Mean proximity function, number theoretic analog of      155
Mean proximity function, sometimes called compensation function      20
Mean type      27
Mean-proximity function      2
Meromorphic on closed disc      6
Metropolis, N.      159
Miles, J.      42 43 89 162
Minimal type      27
Monic polynomial      153
Moving targets      144
Mues, E.      136
Nevanlinna characteristic function      see Characteristic function
Nevanlinna defect      see Defect
Nevanlinna functions      17
Nevanlinna functions for exponential function      25
Nevanlinna functions for rational functions      24 121
Nevanlinna functions for sine and cosine      27
Nevanlinna functions for sums and products      23
Nevanlinna functions, how they are related on average      43
Nevanlinna growth lemma      52 53
Nevanlinna Inverse Problem      127
Nevanlinna theory      1 17
Nevanlinna, F.      vi 2 3 49 61
Nevanlinna, R.      v vi 24 14 49 53 61 89 98 107 127 129 135 144
Ngoan, V.      106
Non-archimedean      see Absolute value non-Archimedean
Number field      30
Number theoretic digression      4 29
Numbering conventions      x
o notation      ix
Order of a zero or pole      9
Order, finite      27
Order, growth      27
Order, growth, lower      27
Order, infinite      27
Osgood, C.F.      vi 30 144 151
Ostrovskii, I, V.      v 3 106
Ostrowski, A.      29
p-adic absolute value      see Absolute value p-adic
Picard, E.      v 128
Picards Theorem      20 128
Picards Theorem, number theory analogue of      158
Poincane, H.      31
Poisson formula      7
Poisson kernel      6
Poisson kernel, harmonic      6
Poisson kernel, relation to Greens function      83
Poisson kernel, transofmation under Mobius automorphism      6
Poisson Jensen formula      9
Poisson, S.      7
Polar coordinate form of $d^c$      31
Polya Laguerre class      186
Product formula      29
Product into sum      16 67
Projective line      6
Proximity function      see Mean proximity function
Pseudo area form      37
Pseudo volume (area) form      59
Pull back      32
Radius bound      138
Ramification      20
Ramification for polynomials and rational functions      21
Ramification Theorem      58 71 111
Ramification Theorem, coefficient 1/2 best possible      60
Ramification Theorem, open question about improving error term      72 112
Ramification, alternative to Wronskian      22
Ramification, counting function      see Counting function associated to ramification divisor
Ramification, defect      125
Ramification, divisor      21
Ramification, index      21
Ramification, point      20
Ramification, term      see Counting function asssociated to ramification divisor
Rational function, Lang type of      178
Rational functions, Nevanlinna functions for      24 121
Real part      ix
Reinders, M.      136
Riccati equation      130
Richtmyer, R.      159
Riemann sphere      6
Riemann zeta function      186
Riemann Hurwitz Theorem      74
Ring of integers      153
Roth, K.      151
Roths Theorem      151 154 156 158
Roths Theorem, conjectural improvement of      159
Ru, M.      144
Schottky type theorem      see Theorem of Schottky type
Schottky, F.      142 143
Second Main Theorem      vi 2 60 72 77 108
Second Main Theorem for functions of finite order      117
Second Main Theorem for moving targets      144
Second Main Theorem, sharpness of      161
Second Main Theorem, uniform      77
Second Main Theorem, via curvature      60 72
Second Main Theorem, via logarithmic derivatives      108
Sharpness of the Lemma on the Logarithmic Derivative      161
Sharpness of the Second Main Theorem      161
Shimizu, T.      vi 15 39
Sine, Nevanlinna functions for      27
Sine, order of      28
Singular term (in Stokess Theorem)      33
Singular term (in Stokess Theorem)contribution from $\log|g|^2$      37
Singular volume (area) form      62
Slowly moving      144
Small function      144
Smirnov, V.      89 90
Smirnovs inequality      90
Soas, L.      177 183 186
Sodin, M.      3
Solution to the Dirichlet Problem      8
Spherical area form      35
Spherical derivative      37
Steinmetz, N.      144
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