Nevanlinna P. Boutroux 20 Nevanlinna Nevanlinna 1930 Malmquist 1950 H. Wittich Painlevé 2002 Nevanlinna Painlevé 1 Painlevé property Painlevé

Transcription

1 Nevanlinna i

2 Nevanlinna P. Boutroux 20 Nevanlinna Nevanlinna 1930 Malmquist 1950 H. Wittich Painlevé 2002 Nevanlinna Painlevé 1 Painlevé property Painlevé ii

3 Nevanlinna Clunie Riccati Riccati Painlevé (I), (II) Painlevé property Painlevé 47 Painlevé 15. Painlevé Painlevé 56 Painlevé 17. (I) (II) iii

4 ,, 1 log + x 1 m(r, f) 2 N(r, f), N(r, f), N 1 (r, f) 1 n(r, f), n(r, f), n 1 (r, f) 1 S(r, f) 19 T (r, f) 2 δ(, f), δ(α, f) 12 ϑ(, f), ϑ(α, f) 12 µ( ) 17 ϱ(f) 2 iv

5 Nevanlinna Nevanlinna Nevanlinna (1) r > r 0 χ(r), ψ(r) χ(r) = O(ψ(r)) (r ) χ(r) ψ(r) ψ(r) χ(r) χ(r) ψ(r) ψ(r) χ(r) χ(r) ψ(r) (2) x > 0 log + x := max{log x, 0}. 1. C f(z) f(z) a ι(a) n(r, f) := ι(a) a r f(a)= r f(z) (counting function) N(r, f) := r 0 ( n(ρ, f) n(0, f) ) dρ ρ + n(0, f) log r n(r, f) 1 2 n(r, f) := 1, n 1 (r, f) := n(r, f) n(r, f) = (ι(a) 1) a r f(a)= a r f(a)= r ( N(r, f) := ) dρ n(ρ, f) n(0, f) + n(0, f) log r, 0 ρ r ( N 1 (r, f) := n1 (ρ, f) n 1 (0, f) ) dρ 0 ρ + n 1(0, f) log r 1

6 α α- N(r, 1/(f α)) z r α- α- N(r, 1/(f α)), N 1 (r, 1/(f α)) f(z) (proximity function) m(r, f) := 1 2π log + f(re iφ ) dφ 2π 0 z = r f(z) z 0 = re iφ 0 log r(φ φ 0 ) r f(z) f(z) T (r, f) := m(r, f) + N(r, f) f(z) (characteristic function) r T (r, f) f(z) T (r, f) log T (r, f) ϱ(f) := lim sup r log r f(z) (growth order) Example 1.1. f 1 (z) = e z N(r, f 1 ) = 0, m(r, f 1 ) r T (r, f 1 ) r. f 2 (z) = e z N(r, f 2 ) = 0, m(r, f 2 ) r T (r, f 2 ) r. f 3 (z) = 1/(e z 1) N(r, f 3 ) r, m(r, f 3 ) = O(1) T (r, f 3 ) r f 1 (z), f 2 (z) f 3 (z) T (r, f j ) (j = 1, 2, 3) ϱ(f j ) 1 Example 1.2. T (r, e z ) r, T (r, exp(z 2 )) r 2, g(z) T (r, g) = O(log r) ( Proposition 3.1 ). 1, 2, 0 2

7 Example 1.3. f(z) = (exp(z 1/2 ) + exp( z 1/2 ))/2 = k=0 z k /(2k)! T (r, f) = m(r, f) r 1/2 ϱ(f) = 1/2 Example Example 1.1 T (r, e z ) T (r, 1/(e z 1)) Theorem 2.1. f(z) α T (r, f) = T (r, 1/(f α)) + O(1). Remark 2.1. O(1) R(r, f, α) R(r, f, α) C log + α C 0 f(z) Lemma 2.2. Jensen-Poisson R z R f(z) a j (j = 1,..., p), b k (k = 1,..., q) z < R log f(z) = 1 2π log f(re iφ ) R2 z 2 2π 0 Re iφ z dφ 2 p R(z a + log j ) q j=1 R 2 a j z R(z b log k ) k=1 R 2 b k z. Remark 2.2. f(z) z = R Proof. F (z) z < R + ε, ε > 0 F (z) 0, log F (z) log F (z) := log F (0) + 3 z 0 F (ζ) F (ζ) dζ

8 z < R + ε z < R log F (z) = 1 2πi ζ =R log F (ζ) ζ z dζ R 2 /z > R 0 = 1 log F (ζ) 2πi ζ =R ζ R 2 /z dζ ζζ = R 2, 1 ζ z 1 ζ R 2 /z = R 2 z 2 (R 2 zζ)(ζ z) = R2 z 2 ζ z 2 ζ log F (z) = 1 2πi = 1 2π 0 ζ =R 2π log F (ζ) R2 z 2 dζ (2.1) ζ z 2 ζ log F (Re iφ ) R2 z 2 Re iφ z 2 dφ, z < R log F (z) = 1 2π 2π 0 log F (Re iφ ) R2 z 2 dφ (2.2) Re iφ z 2 z = R f(z) 0,, F (z) = f(z) p j=1 R 2 a j z R(z a j ) q k=1 R(z b k ) R 2 b k z (2.3) ε z < R + ε F (z) 0, z = R F (z) = f(z) F (z) (2.2) z = R f(z) a a z a = ε, z < R z = R (2.1) z = R z = R (2.3) F (z) ε 0 4

9 z = R f(z) 0, f(z) = F (z) a log F (ζ) = O(log ε ) [] Proof of Theorem 2.1. f(z) z = 0 f(z) = z l g(z), g(0) 0,, l Z l = n(0, 1/f) n(0, f) Lemma 2.2 R, f(z) r, g(z) = z l f(z) z = 0 log g(0) = 1 2π (log f(re iφ ) l log r)dφ + 2π 0 p j=1 a j 0 log a j r q = 1 2π log f(re iφ ) dφ (n(0, 1/f) n(0, f)) log r + 2π 0 = 1 2π log + f(re iφ ) dφ 1 2π log + 1/f(re iφ ) dφ 2π 0 2π 0 ( p + log a ) ( j q n(0, 1/f) log r j=1 r a j 0 k=1 b k 0 log b k n(0, f) log r k=1 r b k 0 log b k r f(z) 0 g(z) p j=1 a j 0 log a j r = r 0 log ρ d(n(ρ, 1/f) n(0, 1/f)) r = [log ρ r (n(ρ, 1/f) n(0, 1/f) )] r = N(r, 1/f) + n(0, 1/f) log r 0 r 0 ( n(ρ, 1/f) n(0, 1/f) ) dρ ρ log g(0) = m(r, f) m(r, 1/f) + N(r, f) N(r, 1/f) T (r, f) = T (r, 1/f) + log g(0) (2.4) α C N(r, f α) = N(r, f) ) 5

10 2π m(r, f α) = 1 log + f(re iφ ) α dφ, 2π 0 log + f(re iφ ) α log + f(re iφ ) + log + α + log 2, log + f(re iφ ) α log + f(re iφ ) log + α log 2 Lemma 2.3 m(r, f α) = m(r, f) + O(1). T (r, f) = T (r, f α) + O(1). (2.4) [] Lemma 2.3. α j (j = 1,..., h) log + ( h j=1 log + ( h j=1 ) ( ) α j log + h max α j 1 j h ) α j h log + α j. j=1 h log + α j + log h, j=1 m(r, f), T (r, f) Proposition 2.4. f(z), g(z) α, β C h N m(r, αf + βg) m(r, f) + m(r, g) + O(1), m(r, fg) m(r, f) + m(r, g), m(r, f h ) = hm(r, f), T (r, αf + βg) T (r, f) + T (r, g) + O(1), T (r, fg) T (r, f) + T (r, g), T (r, f h ) = ht (r, f). 6

11 3. Proposition 3.1. f(z) T (r, f) = O(log r) (r ) Proof. f(z) f(z) N(r, f) = O(log r). γ r max z =r f(z) = O(r γ ) m(r, f) = O(log r). T (r, f) = O(log r) T (r, f) = O(log r) n(r, f) n(0, f) 1 log r r 2 r (n(ρ, f) n(0, f)) dρ ρ N(r2, f) log r + O(1) T (r2, f) log r + O(1) = O(1) f(z) b k (k = 1,..., q) Theorem 2.1 T (r, 1/f) = T (r, f) + O(1) = O(log r) a j (j = 1,..., p). g(z) = f(z) p j=1(z a j ) 1 q k=1 (z b k) g(z) 0 Lemma 2.2 r z r/2 log g(z) = 1 2π 2π 2π 0 log g(re iφ ) r 2 z 2 re iφ z 2 dφ 2 log + g(re iφ ) dφ m(r, g) = T (r, g) T (r, f)+log r log r π 0 g(z) f(z) [] Remark 3.1. f(z) A 0 T (r, f) = A 0 log r+o(1) f(z) C T (r, f) = log + C H. Cartan Theorem 3.2. H. Cartan f(z) z = 0 T (r, f) = 1 2π N(r, 1/(f e iθ ))dθ + log + f(0). 2π 0 7

12 Proof. θ [0, 2π] e iθ f(0) f(z) e iθ Lemma 2.2 z = 0 log f(0) e iθ (3.1) = 1 2π p log f(re iφ ) e iθ dφ + log aθ j q 2π 0 r log b k r = 1 2π 2π 0 j=1 k=1 log f(re iφ ) e iθ dφ N(r, 1/(f e iθ )) + N(r, f). a θ j, b k f(z) e iθ z r Theorem 2.1 f(z) = a z, a 0 Lemma 2.2 R = 1, z = 0 log a = 1 2π log a e iφ dφ 2π 0 log a = 1 2π log a e iφ dφ + log a 2π 0 ( a > 1), a = 0 a C 1 2π 2π 0 log a e iφ dφ = log + a (0 < a 1), (3.1) θ [0, 2π] ( f(0) = e iθ 0 θ = θ 0 ), log + f(0) = 1 ( 2π 1 ) 2π log f(re iφ ) e iθ dθ dφ 2π 0 2π 0 1 2π N(r, 1/(f e iθ ))dθ + N(r, f) 2π 0 = 1 2π log + f(re iφ ) dφ + N(r, f) 1 2π N(r, 1/(f e iθ ))dθ 2π 0 2π 0 = T (r, f) 1 2π N(r, 1/(f e iθ ))dθ 2π 0 [] f(z) z = 0 Theorem 3.2 dt (r, f) d log r = 1 2π n(r, 1/(f e iφ ))dφ 2π 0 8

13 z = 0 l f(z) = z l g(z), g(0) 0, (2.4) T (r, f) = T (r, 1/f) + log g(0). Theorem 3.3. T (r, f) log r Proposition 3.4. f(z) log r/t (r, f) = o(1) (r ) Proof. Proposition 3.1 f(z) T (r) = T (r, f) log r Theorem 3.3 (d/d log r)t (r) r T (r, f) = T (r) = O(log r) (r ) Proposition 3.1 f(z) (d/d log r)t (r) (r ) ε r ε r r ε (d/d log r)t (r) ε 1 r r ε T (r) ε 1 log r + O(1), log r/t (r, f) ε + O(1/ log r) ε log r/t (r, f) 0 (r ) [] Example f(z) = (1 e k z) k=1 k=1 e k z f(z) e k (k = 1, 2,... ) log + f(z) log + (1 + e k z ) k=1 k=1 n(t, 1/f) = n(t) n(t) log t r z = r log + (1 + e k z ) n(r) log(r + 1) (log r) 2, e k r 9

14 log + (1 + e k z ) = log(1 + r/t)dn(t) e k r r >r [ ] n(t) n(t) r + r t r r t dt r 2 r r t dn(t) log t dt log r. t2 z = r log + f(z) (log r) 2 T (r, f) = m(r, f) (log r) 2 T (r, 1/f) N(r, 1/f) r 1 n(t) r dt t 1 log t dt (log r) 2 t T (r, f) (log r) 2 ϱ(f) = 0 Example 3.2. γ > 0 T (r, f) r γ, ϱ(f) = γ f(z) 0 < γ < 1 E k (z) := 1 z k 1/γ, γ 1 ( E k (z) := 1 z ) ( z exp k 1/γ k + 1( z ) 2 1 ( z ) ) [γ] + + 1/γ 2 k 1/γ [γ] k 1/γ f(z) = k=1 E k (z) k 1/γ (k = 1, 2,... ) γ 1 n(t, 1/f) = n(t) t γ z = r log + E k (z) ) (log(1 + rk 1/γ ) + [γ](2rk 1/γ ) [γ] k 1/γ 2r 2r 1 k 1/γ 2r log(1 + r/t)dn(t) + r [γ] 2r 1 dn(t) t [γ] 2r 1 n(t) dt + r γ r γ, t k 1/γ > 2r log E k (z) 2([γ]+1) 1 (rk 1/γ ) [γ]+1 k 1/γ >2r log + E k (z) k 1/γ >2r (rk 1/γ ) [γ]+1 r [γ]+1 t [γ] 1 dn(t) r γ. z = r log + f(z) r γ T (r, f) r γ. N(r, 1/f) r γ T (r, f) r γ 0 < γ < r

15 4. g(z) z = r Proposition 4.1. g(z) ϱ(g) = lim sup r log log M(r, g), M(r, g) := max log r g(z). z =r Lemma 4.2. r, R r < R g(z) M(r, g) 1 T (r, g) log M(r, g) R + r T (R, g). R r g(z) Liouville r M(r, g) 1. R = 2r T (r, g) log M(r, g) 3T (2r, g) Proposition 4.1 Proof of Lemma 4.2. T (r, g) = m(r, g) = 1 2π 2π 0 log + g(re iφ ) dφ 1 2π log M(r, g)dφ log M(r, g) 2π 0 M(r, g) = g(z r ) z r, z r = r g(z) a j r a j R(z r a j )/(R 2 a j z r ) 1 Lemma 2.2 2π log M(r, g) = log g(z r ) 1 log g(re iφ R 2 r 2 ) 2π 0 Re iφ z r dφ 2 1 2π R 2 r 2 2π 0 (R r) 2 log+ g(re iφ ) dφ = R + r R + r m(r, g) = T (R, g) R r R r 11

16 [] 5. f(z) δ(, f) := lim inf r m(r, f) T (r, f) (deficiency), α δ(α, f) := lim inf r m(r, 1/(f α)) T (r, f) α 0 δ(, f) 1, 0 δ(α, f) 1 Proposition 5.1. f(z) (1) δ(, f) < 1 (resp. δ(α, f) < 1) (resp. α- ) (2) (resp. α- ) δ(, f) = 1 (resp. δ(α, f) = 1) Proof. (2) N(r, f) log r f(z) Proposition 3.4 ( lim 1 r ) m(r, f) N(r, f) = lim T (r, f) r T (r, f) = 0. δ(, f) = 1. [] 2 ϑ(, f) = lim inf r N 1 (r, f) T (r, f), N 1 (r, 1/(f α)) ϑ(α, f) = lim inf r T (r, f), α Proposition 5.2. f(z) (1) ϑ(, f) > 0 (resp. ϑ(α, f) > 0) (resp. α- ) 2 12

17 (2) (resp. α- ) 2 ϑ(, f) = 0 (resp. ϑ(α, f) = 0) Example 5.1. f(z) simple ϑ(, f) = 0, double ϑ(, f) 1/2 ϕ(z) = 1/(e z 1) ϑ(, ϕ) = 0. Example 5.2. Weierstrass - (z) = z 2 + (j,k) (Z 2 ) ( ) (z Ωjk ) 2 Ω 2 jk, Ω jk = jω 1 + kω 2, Im (ω 2 /ω 1 ) > 0, (Z 2 ) = Z 2 \ {(0, 0)} ω 1, ω 2 (z) 2 = 4 (z) 3 g 2 (z) g 3 (5.1) g 2, g 3 ω 1, ω 2 g2 3 27g3 2 0 N(r, ) r 2 m(r, ) = O(log r) (Example 7.1 ) T (r, ) r 2. δ(, ) = 0. double ϑ(, ) = 1/2. 6. f (z)/f(z) (Theorem 6.5). Proposition 6.1. f(z) Lemma 2.2 z < R f (z) f(z) = 1 2π log f(re iφ Re iφ ) π 0 (Re iφ z) dφ 2 p ( ) 1 a j + + z a j R 2 a j z j=1 q ( 1 + k=1 z b k ) b k R 2 b k z 13

18 Proof. F (z), z = x + iy ( x i y ( x i y ) (Re F (z)) = F (z), ) F (z) = 0 ( x i ) F (z) = 2F (z), y Lemma 2.2 / x i / y R 2 z 2 Re iφ z 2 = Reiφ Re iφ z + ( x i ) R 2 z 2 y Re iφ z = 2 ( x i y ( x i ) log y z Re iφ z 2Re iφ (Re iφ z) 2 ) log f(z) = d dz log f(z) = f (z) f(z), R(z a j ) R 2 a j z = 1 + z a j a j R 2 a j z. [] Proposition 6.2. f(z) r 0 r 0 < r < R r, R m(r, f /f) 3 log + T (R, f) + 4 log + R R r + C 0 C 0 r, R. Lemma 6.3. z ρ f(z) c j (j = 1,..., ν) ν = ν(ρ, f) := n(ρ, f) + n(ρ, 1/f) D(z) := min 1 j ν { z c j }, r < ρ 1 2π 2π 0 log + r D(re iθ ) dθ 3 2 log ν(ρ, f)

19 Proof of Lemma 6.3. θ j = arg c j (0 θ j < 2π) ( c j = 0 θ j = 0 ). ν 0 θ 1 < θ 2 < < θ ν < 2π (ν ν) θ 0 = θ ν 2π, θ ν +1 = θ 1 + 2π k = 1,..., ν η k = min { π 2ν, 1 2 (θ k θ k 1 ) Θ k := [θ k ηk, θ k + η k + ], } 2π, η + k = min { π 2ν, 1 2 (θ k+1 θ k ) log + r ν dθ I(Θ 0 D(re iθ k ) + I(Θ ), (6.1) ) k=1 ( I(X) = log + r ν X D(re iθ ) dθ, ) Θ = [0, 2π] \ Θ k θ Θ k D(re iθ ) r sin θ θ k I(Θ k ) log + 1 Θ k sin θ θ k θ θ dθ k π/(2ν) 2 I(Θ ) π/(2ν) 0 2π 0 log log + k=1 log + 1 sin θ θ k dθ π 1/ν 2φ dφ = π log ψ dψ = π (log ν + 1). 0 ν 1 2π sin(π/(2ν)) dθ log νdθ = 2π log ν. 0 (6.1) [] Proof of Proposition 6.2. ρ = (r + R)/2 Proposition 6.1 R ρ z = re iθ f (z) f(z) 1 2π log f(ρe iφ ρ ) π 0 (ρ r) dφ + ( 1 2 c j D(z) + 1 ) ρ r ρ 2ρ ( ) ( 1 m(ρ, f) + m(ρ, 1/f) + ν(ρ, f) (ρ r) 2 D(z) + 1 ) (6.2) ρ r } 15

20 c j, D(z), ν(ρ, f) Lemma 6.3 N(R, f) R n(ρ, f) ρ ( ) dt n(t, f) n(0, f) t R ρ ( ) n(ρ, f) n(0, f) R R N(R, f) + n(0, f) = 2R R ρ N(R, f) + n(0, f). R r ν(ρ, f) 2R ( ) N(R, f) + N(R, 1/f) + O(1) R r 4R ( ) T (R, f) + O(1). (6.3) R r m(ρ, f) + m(ρ, 1/f) 2T (ρ, f) + O(1) 2T (R, f) + O(1). (6.4) (6.3), (6.4) (6.2) 2ρ(ρ r) 2 8R(R r) 2, (ρ r) 1 = 2(R r) 1, z = re iθ f (z) f(z) 8R ( ) 2T (R, f) + O(1) (R r) 2 + 4R R r ( T (R, f) + O(1) ) ( 1 D(z) + 2 log + log + f (re iθ ) f(re iθ ) 2 log + R r 4R ( 6 R r (T (R, f) + O(1)) R r + R R r + log+ T (R, f) + log + ) r ). D(z) r D(re iθ ) + O(1). θ Lemma 6.3 (6.3) [] 16

21 Lemma 6.4. Pólya T (r) [r 0, ) (r 0 > 0) T (r) 1 E [r 0, ), µ(e) 2 E, [r 0, ) \ E T ( r + 1/T (r) ) < 2T (r) µ( ) Lebesgue Proof. r 1 := min{r r 0 T (r + 1/T (r)) 2T (r)} E = r 1 := r 1 + 1/T (r 1 ) r 2 := min{r r 1 T (r + 1/T (r)) 2T (r)} r 2 := r 2 + 1/T (r 2 ) r j := min { r r j 1 T ( r + 1/T (r) ) 2T (r) }, r j := r j + 1/T (r j ) r j, r j r j r j 1, r j 1 r 1 < r 1 r 2 < r 2 r j < r j r j+1 < r j+1 r j r j r < r j r j = 1/T (r j ) 0 T (r ) = E := {r T (r + 1/T (r)) 2T (r)} E j 1[r j, r j] T (r j+1 ) T (r j) = T (r j + 1/T (r j )) 2T (r j ) T (r j ) 2 j 1 T (r 1 ) 2 j 1. µ(e) (r j r j ) = 1/T (r j ) j 1 j 1 2 (j 1) = 2. j=1 [] Theorem 6.5. f(z) µ(e) < E (0, ) r, r E m(r, f /f) log T (r, f) + log r. ϱ(f) < ( ) ( ) r m(r, f /f) log r. 17

22 Proof. ϱ(f) = γ < T (r, f) = O(r γ+1 ) (r ) Proposition 6.2 R = 2r r m(r, f /f) log + T (2r, f) + O(1) log r. Proposition 6.2 R = r + 1/T (r, f) Lemma 6.4 E, µ(e) < r E T (r + 1/T (r, f), f) < 2T (r, f) r, r E m(r, f /f) log + T (r + 1/T (r, f), f) + log T (r, f) + log r log(2t (r, f)) + log T (r, f) + log r log T (r, f) + log r. [] Corollary 6.6. f(z) k E k, µ(e k ) < r, r E k m(r, f (k) /f) log T (r, f) + log r. ϱ(f) < m(r, f (k) /f) log r Proof. k = 1 Theorem 6.5 k r, r E k, m(r, f (k) ) m(r, f (k) /f) + m(r, f) m(r, f) + O(log T (r, f) + log r). f(z) l f (k) (z) (l + k) r E k N(r, f (k) ) (k + 1)N(r, f). T (r, f (k) ) (k + 1)T (r, f) + O(log T (r, f) + log r) (6.5) r E k m(r, f (k+1) /f) m(r, f (k+1) /f (k) ) + m(r, f (k) /f) m(r, f (k+1) /f (k) ) + log T (r, f) + log r (6.6) 18

23 (6.5) f (k) (z) Theorem 6.5 E k+1 ( E k ), µ(e k+1 ) < r E k+1 m(r, f (k+1) /f (k) ) log T (r, f (k) ) + log r log T (r, f) + log r. (6.6) k + 1 k ϱ(f) < [] (6.5) Corollary 6.7. k E k, µ(e k ) < r, r E k T (r, f (k) ) (k + 1)T (r, f) + O(log T (r, f) + log r). ϱ(f) < T (r, f (k) ) (k + 1)T (r, f) + O(log r). [r 0, ) (r 0 > 0) χ(r) E, µ(e) < r, r E χ(r)/t (r, f) 0 χ(r) = S(r, f) Corollary 6.8. f(z) k m(r, f (k) /f) = S(r, f). Proof. f(z) Corollary 6.6, Proposition 3.4 z f (k) (z)/f(z) = O(z 1 ) Remark 3.1 [] Lemma 6.9. χ(r), ψ(r) [r 0, ) E, µ(e) < [r 0, ) \ E 19

24 χ(r) ψ(r) r [r 0, ) χ(r) ψ(r + r ) Proof. r µ(e) < r r [r 0, ) c r < r r + c r E χ(r) χ(r + c r ) ψ(r + c r ) ψ(r + r ). [] 7. Clunie Lemma 7.1. Clunie u Q(z, u) = q ι (z)u ι 0 (u ) ι1 (u (s) ) ιs, ι = (ι 0, ι 1,..., ι s ) ι I q ι (z) I ι h N ι I ι 0 + ι ι s h f(z) f h+1 = Q(z, f) m(r, f) = S(r, f) ϱ(f) < m(r, f) = O(log r). Proof. I := {φ [0, 2π] f(re iφ ) 1} m(r, f) = 1 2π log + f(re iφ ) dφ = 1 log + f(re iφ ) dφ 2π 0 2π I φ I log + f(re iφ ) = log + Q(z, f)f h ( log + f q ι ι 1 f (s) ι I f f ( log + f q ι ι 1 f (s) ι I f f ι s f ι 0 +ι 1 + +ι s h) ι s ) log r + log + f /f + + log + f (s) /f. 20 (7.1)

25 (7.1) Corollaries 6.6, 6.8 [] Lemma 7.2. A. Z. Mohon ko and V. D. Mohon ko F (z, u) z, u, u,..., u (s) f(z) F (z, f) = 0 c C F (z, c) 0 m(r, 1/(f c)) = S(r, f) ϱ(f) < m(r, 1/(f c)) = O(log r). Proof. g = f c F (z, c) = F (z, f) F (z, c) = F (z, g + c) F (z, c) = q ι (z)g ι0 (g ) ι1 (g (s) ) ιs, 1 ι 0 + +ι s d 0 ι = (ι 0,..., ι s ) q ι (z) d 0 φ I := {φ [0, 2π] g(re iφ ) 1} log + 1/g(re iφ ) log ( F + (z, c) 1 g q ι ι 1 g (s) ι s ) 1 ι 0 + +ι s d 0 g g log + F (z, c) 1 + log r + log + g /g + + log + g (s) /g φ I S(r, g) = S(r, f) m(r, F (z, c) 1 ) T (r, F (z, c) 1 ) = O(log r) [] Remark 7.1. Lemma 7.1 Lemma 7.2 S(r, f) log T (r, f) + log r Example 7.1. Weierstrass - (z) (cf. Example 5.2) ( ) 2 = 4 3 g 2 g 3 Lemma 7.1 m(r, ) = S(r, ). N(r, ) r 2 T (r, ) r 2 +S(r, ). µ(e) < E r E T (r, ) = O(r 2 )+o(t (r, )) K 0 T (r, ) K 0 r 2 r E Theorem 3.3 Lemma 6.9 r > 0 T (r, ) K 0 (r + r ) 2 = O(r 2 ) ϱ( ) 2 Lemma 7.1 m(r, ) = O(log r) 21

26 8. Theorem 8.1. f(z) q α 1,..., α q q m(r, f) + m(r, 1/(f α j )) + N 1 (r, f) + N(r, 1/f ) 2T (r, f) + S(r, f). j=1 Corollary 8.2. f(z), α 1,..., α q q ( δ(, f) + ϑ(, f) + δ(αj, f) + ϑ(α j, f) ) 2. j=1 Proof. q j=1 N 1 (r, 1/(f α j )) N(r, 1/f ) Theorem 8.1 q ( m(r, f) + N 1 (r, f) + m(r, 1/(f αj )) + N 1 (r, 1/(f α j )) ) j=1 2T (r, f) + S(r, f) T (r, f) r lim inf [] Corollary 8.2 n N δ(α, f) 1/n α C 2n ϑ(α, f) 1/n α δ(α, f) > 0 ϑ(α, f) > 0 α Corollary 8.2 Corollary 8.3. f(z) ( ) δ(α, f) + ϑ(α, f) 2, Ĉ = C { }. α Ĉ 22

27 α C { } f(z) = α z α Picard (exceptional value of Picard) δ(α, f) = 1 α Ĉ 2 Theorem 8.4. Picard Picard 2 Example 8.1. f(z) = e z Picard 0, f(z) = 1/ cos z Picard 0 Theorem 8.5. Valiron P (u), Q(u) C[u] R(u) = P (u)/q(u) P (u), Q(u) d P, d Q d R := max{d P, d Q } f(z) T (r, R(f)) = d R T (r, f) + O(1). (8.1) Remark 8.1. M z z u M z - P (z, u), Q(z, u) M z [u] M z - R(z, u) = P (z, u)/q(z, u) u d R = max{d P, d Q } {c λ (z) M z λ Λ} P (z, u), Q(z, u) ( ) T (r, R(z, f)) = d R T (r, f) + O T (r, c λ Λ λ) + O(1). Theorem 8.5 Proof of Theorem 8.5. R(u) d Q = 0, d R = d P = d 1 R(u) = P (u) = C 0 + C 1 u + + u d T (r, P (f)) T (r, C 0 ) + T (r, (C 1 + C 2 f + + f d 1 )f) + O(1) T (r, f) + T (r, C 1 + C 2 f + + f d 1 ) + O(1) dt (r, f) + O(1). 23

28 z = r log + P (f) d log + f + O(1) (8.2) z = r (8.2) P (f) f d /2 log + P (f) d log + f log 2 (8.2) P (f) < f d /2 z 0 f d + C d 1 f d C 1 f + C 0 = α(z 0 )f d, α(z 0 ) < 1/2 (8.3) ζ F (ζ) = (1 α(z 0 ))ζ d + C d 1 ζ d C 1 ζ + C 0 z 0 κ 0 = κ 0 (C 0,..., C d 1 ) ζ κ 0 F (ζ) 1 (8.3 f(z 0 ) κ 0 O(1) d log + κ 0 (8.2) (8.2) m(r, P (f)) dm(r, f) + O(1) N(r, P (f)) = dn(r, f) T (r, P (f)) dt (r, f) + O(1) d Q = 0 (8.1) d Q 1 T (r, R(f)) = T (r, 1/R(f)) + O(1) d R = d Q d P d Q = d R T (r, R(f)) d R T (r, f) + O(1) (8.4) d Q = 1 d P R(u) = B 0 + 1/(B 1 + B 2 u), B 2 0 (8.4) d d Q d P (8.4) d Q = d + 1 d P R(u) = P (u) Q(u) = B 0 + P 1(u) Q(u) = B U(u) + P 2 (u)/p 1 (u), d P1 d Q 1 = d, d U = d Q d P1, d P2 < d P1 24

29 T (r, R(f)) = T (r, P 1 (f)/q(f)) + O(1) = T (r, Q(f)/P 1 (f)) + O(1) T (r, U(f)) + T (r, P 2 (f)/p 1 (f)) + O(1). U(u) T (r, R(f)) d U T (r, f) + d P1 T (r, f) + O(1) = d Q T (r, f) + O(1). d Q = d+1 (8.4) (8.4) V (u), W (u) C[u] V (u)q(u) + W (u)p (u) = 1, d V + d Q = d W + d P d R = d Q d P d W d V (8.4) T ( r, 1/(Q(f)W (f)) ) = T ( r, P (f)/q(f) + V (f)/w (f) ) T (r, R(f))+T (r, V (f)/w (f))+o(1) T (r, R(f))+d W T (r, f)+o(1), T ( r, 1/(Q(f)W (f)) ) = T (r, Q(f)W (f))+o(1) = (d Q +d W )T (r, f)+o(1) T (r, R(f)) d Q T (r, f) + O(1). (8.4) (8.1) [] Proof of Theorem 8.1. q q ( m(r, 1/(f α j )) = T (r, 1/(f αj )) N(r, 1/(f α j )) ) j=1 j=1 q = qt (r, f) N(r, 1/(f α j )) + O(1) j=1 P (u) = q j=1(u α j ) Theorem 8.5 qt (r, f) = T (r, P (f)) + O(1) = T (r, 1/P (f)) + O(1) q = T (r, 1/P (f)) N(r, 1/(f α j )) + O(1) j=1 q = m(r, 1/P (f)) + N(r, 1/P (f)) N(r, 1/(f α j )) + O(1) j=1 m(r, 1/P (f)) + O(1). 25

30 α j (1 j q) 1/P (u) = q j=1 γ j (u α j ) 1, γ j 0 = m ( r, q j=1γ j /(f α j ) ) m ( r, q j=1γ j f /(f α j ) ) + m(r, 1/f ) + O(1) = m(r, 1/f ) + S(r, f) = T (r, 1/f ) N(r, 1/f ) + S(r, f) = T (r, f ) N(r, 1/f ) + S(r, f) = m(r, f ) + N(r, f ) N(r, 1/f ) + S(r, f) m(r, f /f) + m(r, f) + N(r, f ) N(r, 1/f ) + S(r, f). q m(r, f)+ m(r, 1/(f α j )) 2m(r, f)+n(r, f ) N(r, 1/f )+S(r, f) j=1 = 2T (r, f) ( 2N(r, f) N(r, f ) + N(r, 1/f ) ) + S(r, f) = 2T (r, f) ( N 1 (r, f) + N(r, 1/f ) ) + S(r, f) [] 26

31 Nevanlinna Airy w + zw = 0 Nevanlinna Mathieu w + (θ 0 + θ 1 (e iz + e iz ))w = 0 Nevanlinna 9. w = 1 + w 2, (9.1) w = z 1 w 2, (9.2) w = (1 + 2z)z 2 w w 2, (9.3) w = w 1 /2, (9.4) ( = d/dz) (9.1) w = tan(z C) z = C + π/2 + kπ (k Z) C (9.2) w = (C log z) 1 z = 0,, e C z = 0, e C (9.3) w = z 2 e 1/z (C e 1/z ) 1 z = 0 z = 1/ log C (9.4) w = (z C) 1/2 27

32 z =, C F (z, u 0, u 1,..., u n 1 ) z, u 0, u 1,..., u n 1 w (n) = F (z, w, w,..., w (n 1) ) (9.5) F (z, u 0, u 1,..., u n 1 ) (z 0, w0, 0 w1, 0..., wn 1) 0 w(z 0 ) = w 0 0, w (z 0 ) = w 0 1,..., w (n 1) (z 0 ) = w 0 n 1 (9.5) w(z) = w(w0, 0 w1, 0..., wn 1; 0 z) w0, 0 w1, 0..., wn 1 0 w0, 0 w1, 0..., wn 1 0 w(z) (movable singular point) (fixed singular point) (9.1), (9.2), (9.3) (9.4) ( (9.1), (9.2), (9.3)). Painlevé property Painlevé property universal covering Nevanlinna 28

33 10. Riccati w = P (z, w) Q(z, w) (10.1) P (z, w), Q(z, w) z, w P (z, w), Q(z, w) w p, q u = 1/w u = P (z, u) Q (z, u) (10.2) P (z, u), Q (z, u) P (z, u) := u d+2 P (z, 1/u), d := max{p 2, q} Q (z, u) := u d Q(z, 1/u), (10.3) z, u Z :=Z 0 Z 0 Z 1 Z 1, (10.4) Z 0 := { z 0 C Q(z 0, w) 0 }, Z 0 := { z 0 C Q (z 0, u) 0 }, Z 1 := { z 0 C P (z 0, w 0 ) = Q(z 0, w 0 ) = 0 for some w 0 C }, Z 1 := { z 0 C P (z 0, 0) = Q (z 0, 0) = 0 } Theorem w = φ(z) (10.1) φ(z) c Z Proof. c Z φ(z) c C : z = χ(t), t > 0 C \ {c} w = φ(z) 29

34 Γ := Γ(τ) C { }, Γ(τ) := { φ(χ(t)) t τ }. τ>0 Claim 1: γ Γ (a) γ P (c, γ) 0 Q(c, γ) = 0, (b) γ = Q (c, 0) = 0 P (c, 0) 0. γ Γ z ν c, φ(z ν ) γ {z ν } C Q(c, γ) 0 Theorem A.3 φ(z) c c Q(c, γ) = 0 c Z 1 P (c, γ) 0 γ = c Z 1 (b) Claim 2: Γ γ 0 C { } C z c φ(z) γ 0 c Z 0 Q(c, w) 0 Q(c, w) γ 1,..., γ q (q q) Claim 1 Γ {γ 1,..., γ q, }, j := {w w γ j < ρ j } (j = 1,..., q ), := {w w > ρ } k l = k l k, l {1,..., q, } Γ γ, γ Γ, γ γ k l γ k, γ l {zν} C z ν c, φ(z ν) λ C \ ( 1 q ) C \ {γ 1,..., γ q, } Q(c, λ) 0 Theorem A.3 φ(z) c Γ γ 0 Case 1: γ 0 Claim 1 (a), C c, C \ {c} φ (z) 0 C \ {c} φ (z) w = φ(z) z = φ (w) w- C := φ(c \ {c}) z = φ (w) dz dw = Q(z, w) P (z, w) 30 (10.5)

35 w γ 0, w C φ (w) c Theorem A.3 φ (w) w = γ 0 (dφ /dw)(γ 0 ) = 0 w = γ 0 φ (w) = c + a m (w γ 0 ) m + O((w γ 0 ) m+1 ), a m 0, m N, m 2 φ(z) = γ 0 + a 1/m m (z c) 1/m +. c z = c φ(z) Case 2: γ 0 = u = ψ(z) = 1/φ(z) (10.2) z c, z C ψ(z) 0 Q (c, 0) = 0 c Z 0 Claim 1, (b) u = ψ(z) z = ψ (u) C := ψ(c \ {c}) dz du = Q (z, u) P (z, u) u 0, u C ψ (u) c u = 0 ψ (u) (dψ /du)(0) = 0 ψ (u) = c + a m u m + O(u m+1 ), a m 0, m N, m 2 φ(z) = 1/ψ(z) = a 1/m m (z c) 1/m + z = c Q (c, 0) 0 u = ψ(z) z = c ψ(c) = 0 1/φ(z) = ψ(z) = b n (z c) n + O((z c) n+1 ), b n 0, n N z = c φ(z) [] Remark u = z + u u = (c + 1)e z c z 1 z = c 31

36 u = c(z c) + (c + 1)(z c) 2 /2 + w = 1/u w = zw 2 w w = φ(c, z) = ( (c + 1)e z c z 1 ) 1 φ(c, z) = (z c) 1( c + (c + 1)(z c)/2 + ) 1 z = c c 0 simple c = 0 double Painlevé property Theorem R C \ Z universal covering (10.1) R (10.1) Riccati w = p 0 (z) + p 1 (z)w + p 2 (z)w 2 (10.6) p j (z) (j = 0, 1, 2) Proof. (10.1) p 2, q = 0 q 1 c C \ Z Q(c, w) 0 Q(c, λ) = 0 P (c, λ) 0 dz Q(z, w) = dw P (z, w) z = Φ(w) Φ(λ) = c (dφ/dw)(λ) = 0 z = Φ(w) = c + a m (w λ) m + O((w λ) m+1 ), a m 0, m N, m 2 Φ(w) w = φ(z) z = c (10.1) R q = 0 Q(z, w) = Q(z) p 3 d = p 2 1 u = 1/w (10.2) Q (z, u) = Q(z)u d 32

37 c C \ Z P (c, 0) 0 dz du = Q(z)ud P (z, u) z = Φ (u) Φ (0) = c p 2 [] Riccati (10.6) (10.4) Z Z = { z 0 1/pj (z 0 ) = 0 for some j } Theorem 10.2 Theorem R 0 C \ Z universal covering Riccati (10.6) R 0 p j (z) (j = 0, 1, 2) (10.6) C Proof. p j (z) c c C φ(z) C c c A := lim inf z c, z C φ(z) 0 A < + {z ν } C z ν c, φ(z ν ) γ C Theorem A.3 φ(z) z = c A = + z c, z C φ(z) v = p 2 (z) p 1 (z)v p 0 (z)v 2 v = ψ(z) = 1/φ(z) z c, z C ψ(z) 0 Theorem A.3 z = c ψ(z) φ(z) φ(z) R 0 [] Riccati Malmquist Nevanlinna Yosida 1930 Theorem Malmquist-Yosida n R(z, w) z, w (w ) n = R(z, w) (10.7) 33

38 w = φ(z) R(z, w) w 2n R(z, w) = R 2n (z)w 2n + + R 1 (z)w + R 0 (z), R j (z) C(z) φ(z) R(z, w) w n Remark n = 1 (10.7) Riccati Theorem 10.2 Example w = (z) (w ) 2 = 4w 3 g 2 w g 3 (Example 5.2). w = 1/ (z) (w ) 2 = g 3 w 4 g 2 w 3 + 4w Example n = 1 φ(z) (10.7) n = 2 w = cos z = (e iz + e iz )/2 (w ) 2 = 1 w 2 φ(z) R(z, w) w n w = tan z = i(e iz e iz )/(e iz + e iz ) (w ) 2 = 1 + w 2 Proof. φ(z) R(z, w) = P (z, w)/q(z, w) P (z, w), Q(z, w) z, w w p, q P (z, w), Q(z, w) w j z φ(z) z 0 φ(z) = c l (z z 0 ) l +, l N p q w = φ(z) (10.7) z = z 0 R(z, φ(z)) = O((z z 0 ) (p q)l ) = O(1) p > q φ(z) (10.7) z z 0 n(l + 1) = (p q)l l = n/(p q n) N n + 1 p q 2n Q(z, φ)(φ ) n = P (z, φ) φ p = Q(z, φ, φ ) ( Q(z, u, v) u, v (p 1) ) Lemma 7.1 m(r, φ) = S(r, φ) (10.8) 34

39 q 1 P (z, w) Q(z, w) R(z, w) = U(z, w) + P 1 (z, w)/q(z, w) U(z, w) P 1 (z, W ) z w p q, q 1 φ(z) (10.7) Φ(z) := φ (z) n U(z, φ(z)) = P 1(z, φ(z)) Q(z, φ(z)) (10.8) Theorem 6.5 m(r, Φ) = m ( r, (φ ) n U(z, φ) ) (10.9) m(r, φ /φ) + m(r, φ) + O(log r) = S(r, φ) (10.10) (10.9) Φ(z) φ(z) φ(z) Φ(z) Φ(z) N(r, Φ) log r N(r, 1/Φ) N(r, φ)+o(log r) (10.10) T (r, Φ) = S(r, φ) T (r, Φ) = T (r, 1/Φ) + O(1) N(r, 1/Φ) + O(1) N(r, φ) + O(log r) N(r, φ) = S(r, φ) (10.8) T (r, φ) = S(r, φ) q = 0 R(z, w) w 2n φ(z) Corollary 8.2 δ(γ, φ) = 0 γ φ(z) φ(z) γ- v = ψ(z) = 1/(φ(z) γ) (v ) n = R(z, v) := ( v) 2n R(z, γ + 1/v), R(z, v) = R 2n (z)v 2n + + R 1 (z)v + R 0 (z) R(z, w) = (γ w) 2n R(z, 1/(w γ)) w 2n R(z, w) w n+1 Lemma 7.1 m(r, φ) = S(r, φ) φ(z) N(r, φ) = O(log r) = S(r, φ). T (r, φ) = S(r, φ) R(z, w) w n [] 11. Riccati 35

40 Riccati w = p 0 (z) + p 1 (z)w + p 2 (z)w 2, p j (z) C(z) (11.1) Riccati (11.1) w = v/p 2, v = u /u u (p 2(z)/p 2 (z) + p 1 (z))u + p 0 (z)p 2 (z)u = 0 Painlevé (11.1) Theorem Riccati (11.1) φ(z) T (r, φ) = O(r θ ) θ p j (z) C(z) (j = 0, 1, 2) Proof. p 2 (z) 0 p j (z) (j = 0, 1, 2) κ Z, c 0 0, µ 1 z p 2 (z) = z κ (c 0 + O(z 1 )), p 0 (z) + p 1 (z) = O( z µ ) φ(z) z 0 φ(z) ψ(z) = 1/φ(z) z 0 ψ(z) u = ψ(z) u = p 2 (z) p 1 (z)u p 0 (z)u 2 z 0 z z 0 < 1 ψ(z) z 0 2µ κ ψ (z) = z0 κ ( c 0 + O( z 0 1 )) z 0 z = z 0 + z0 κ t Ψ(t) = ψ(z 0 + z0 κ t) Ψ(0) = 0 t < z 0 κ Ψ(t) z 0 2µ κ (d/dt)ψ(t) = c 0 + O( z 0 1 ) (11.2) t τ 0 := ( c 0 1 /3) z 0 2µ κ c 0 t /2 Ψ(t) 2 c 0 t (11.3) 36

41 τ = sup { } τ t < τ Ψ(t) z 0 2µ κ, τ τ 0 (11.2) t τ Φ(t) c 0 + O( z 0 1 ) τ 0 (1/2) z 0 2µ κ τ τ > τ 0 t τ 0 (11.2) t τ 0 Ψ(t) = ( c 0 + O( z 0 1 ) ) t (11.3) z, φ(z) (11.3) φ(z) z z 0 < z 0 κ τ 0 = C 0 z 0 λ, C 0 := c 0 1 /3, λ := κ + κ + 2µ 2 z 0 z 0 r z r φ(z) n(r, φ) πr 2 (πc 2 0r 2λ ) 1 r 2+2λ N(r, φ) = O(r 2+2λ ) (11.1) w = φ(z) Lemma 7.1 m(r, φ) = S(r, φ) T (r, φ) = O(r 2+2λ ) + o(t (r, φ)) r E, µ(e) < Lemma 6.9 T (r, φ) = O(r 2+2λ ) (Example 7.1 ). p 2 (z) 0 φ (z) = p 0 (z) + p 1 (z)φ(z) φ(z) z φ(z) = O(exp(C 1 z µ+1 )) (C 1 > 0) T (r, φ) = m(r, φ)+ O(log r) r µ+1 [] Theorem φ(z) (11.1) (1) p 2 (z) 0 m(r, φ) = O(log r), δ(, φ) = 0 p 2 (z) 0 δ(, φ) = 1. (2) α C p 0 (z)+αp 1 (z)+α 2 p 2 (z) 0 m(r, 1/(φ α)) = O(log r), δ(α, φ) = 0 p 0 (z) + αp 1 (z) + α 2 p 2 (z) 0 φ(z) α- δ(α, φ) = 1. 37

42 (3) 2 p j (z) (j = 0, 1, 2) p 2 (z) (4) 2 α- p j (z) p 0 (z) + αp 1 (z) + α 2 p 2 (z) = 0 Remark δ(, φ) = 0, δ(α, φ) = 0 α- (Proposition 5.1). Proof. p 2 (z) 0 Lemma 7.1 Theorem 11.1 T (r, φ) = O(r θ ) m(r, φ) = O(log r) φ(z) Proposition 3.4 δ(, φ) = 0 p 2 (z) 0 φ(z) w = p 0 (z)+p 1 (z)w φ(z) p 0 (z), p 1 (z) N(r, φ) = O(log r) δ(, φ) = 1 lim r N(r, φ)/t (r, φ) = 1 (1) v = 1/(φ(z) α) v = p 2 (z) (p 1 (z) + 2αp 2 (z))v (p 0 (z) + αp 1 (z) + α 2 p 2 (z))v 2 (1) (2) (3) φ(z) (11.1) v (3) (4) [] 12. Painlevé (10.1) Painlevé property Riccati (10.1) F (z, w, w ) = 0 F (z, w 0, w 1 ) z, w 0, w 1 Painlevé property - (Example 5.2) (w ) 2 = 4w 3 g 2 w g 3 Painlevé property w = R(z, w, w ) (12.1) 38

43 R(z, w 0, w 1 ) z, w 0, w 1 Painlevé property w = 6w 2 g 2 /2 - Painlevé property Painlevé property (12.1) (12.1) 1900 Painlevé Gambier w = 6w 2 + z, w = 2w 3 + zw + α, (I) (II) w = (w ) 2 w w z + 1 z (αw2 + β) + γw 3 + δ w, (III) w = (w ) 2 2w w3 + 4zw 2 + 2(z 2 α)w + β w, (IV) ( 1 w = 2w + 1 )(w ) 2 w (V) w 1 z ( (w 1)2 + αw + β ) + γw δw(w + 1) +, z 2 w z w 1 w = 1 ( 1 2 w + 1 w ) (w ) 2 (VI) w z ( 1 z + 1 z ) w w z ( w(w 1)(w z) + α + βz ) γ(z 1) δz(z 1) + +. z 2 (z 1) 2 w2 (w 1) 2 (w z) 2 α, β, γ, δ C Painlevé Painlevé (α-method) (12.1) α (II),..., (VI) α 0 < α < ε α = 0 39

44 Painlevé Painlevé property Painlevé property Painlevé (I) 1960 Hukuhara (VI) Theorem (I), (II), (IV) C C \ {0}, C \ {0, 1} universal covering R 0, R 0,1, (III), (V) R 0 (VI) R 0,1 (I), (II), (IV) z = (III), (V) z = 0, (VI) z = 0, 1, (I), (II) Theorem 12.1 Painlevé property w 13. (I), (II) Painlevé property Painlevé w = 6w 2 + z, w = 2w 3 + zw + α (I) (II) Painlevé property Riccati Painlevé property (Theorem 10.3) Ljapunov Theorem (I) Theorem 12.1 Step 1: w(z) (I) 40

45 Proposition w(z) z = σ w(z) = (z σ) 2 σ(z σ) 2 /10 (z σ) 3 /6 + b(z σ) 4 + b Proof. w(z) = c l (z σ) l +, l N, c l 0 (I) l + 2 = 2l, (l + 1)lc l = 6c 2 l l = 2, c 2 = 1 j= 2 c j (z σ) j (z σ) j 2 (j 4)(j + 3)c j = G j (σ, c k ; k j 1), j 1 G j (σ, c k ;... ) σ, c k c j G 4 (σ, c k ; k 3) = 0 c 4 = b [] Remark σ, b Remark G 4 (σ, c k ; k 3) = 0 c 4 G 4 (I) Painlevé test Painlevé test Painlevé property σ w(z) = u(z) 2 u(z) z = σ u(z) ξ = z σ σ = z ξ w(z) = ξ 2 zξ 2 /10 ξ 3 /15 + bξ 4 + (13.1) w (z) = 2ξ 3 zξ/5 3ξ 2 /10 + 4bξ 3 + (13.2) (13.1) ξ = ±u(z)(1 zu(z) 4 /20 u(z) 5 /30+bu(z) 6 /2+ ) (13.2) w (z) = 2u(z) 3 zu(z)/2 u(z) 2 /2 ± 7bu(z)

46 b (u, v) w = u 2, w = u 2 /2 (2u 3 + zu/2 u 3 v) (13.3) w (I) w (u, v) ± u = 1 + zu 4 /4 ± u 5 /4 u 6 v/2, v = ±z 2 u/8 + 3zu 2 /8 ± (1/4 zv)u 3 5u 4 v/4 ± 3u 5 v 2 /2 (13.4) (13.3) (u, v) (I) Step 2: (13.3) w + w 1 /2 = u(2w 2 + z/2 w 1 v) (w ) 2 + w w 4w3 2zw = 1 4w 2 + z2 4w + v2 w 3 zv w 2 4v. (13.5) w(z) Φ(z) := w (z) 2 + w (z) w(z) 4w(z)3 2zw(z) (13.6) (13.5) Φ(z) w(z) w(z) (13.3) u(z), v(z) (13.5) Φ(z) (I) w(z) 2w (z) (d/dz)(φ(z) w (z)/w(z)) = 2w(z) (I) Φ (z) + 2w(z) = w (z) w(z) w (z) 2 w(z) 2 Φ(z) = 6w(z) + z w(z) Φ(z) w(z) + w (z) 2 w(z) 4w(z) 2z 3 w(z). Φ (z) + Φ(z) w(z) = z 2 w(z) + w (z) (13.7) w(z) 3 42

47 z γ(z 0, z) w(t) 0 [ Φ(z) = E(z 0, z) 1 Φ(z 0 ) E(z 0, z) 2w(z) w(z 0 ) 2 ] E(z 0, t) γ(z 0,z) 2w(t) 4 (2tw(t)3 1)dt (13.8) ( ) E(z 0, t) = exp w(τ) 2 dτ, t γ(z 0, z), γ(z 0, t) γ(z 0, z). γ(z 0,t) Φ(z) Ljapunov Step 3: w(z) z = a 0 a 0 Γ w(z) A = lim inf w(z) z a 0, z Γ 0 < A < + Γ w(z) A/2 (13.8) Φ(z) Γ {a n } Γ a n a 0, w(a n ) w 0 C \ {0} (13.6) {w (a n )} {a n } w (a n(k) ) (k ) Theorem A.3 w(z) z = a 0 A = + w(z) (z a 0, z Γ) (13.8) Φ(z) Γ (13.5), (13.6) v(z) v(z) Γ w(z) 1/2 ±u(z) u(z), v(z) {a n} Γ a n a 0, u(a n) 0, v(a n) v 0 C Theorem A.3 a 0 u(z) w(z) Γ Step 4: A = 0 Painlevé Hukuhara (II) w = f 0 (z, w) (13.9) 43

48 f 0 (z, w) z, w z a 0 1, w 1 f 0 (z, w) K Lemma θ 0 := min{1/2, K 1 /2} c c a 0 < 1/2 (13.9) w(z) z = c w(c) θ 0 /6, w (c) 2 w(z) w (c) z c < θ 0 w (c) z c = θ 0 /2 w(z) θ 0 /4 Proof. ξ = ρ(z c), ρ = w (c), w(z) = η(ξ) (13.9) η = ρ 2 g 0 (ξ, η), g 0 (ξ, η) := f 0 (z, w) = f 0 (c + ρ 1 ξ, η) ( = d/dξ) η(ξ) η(0) = w(c), η(0) = 1 η(ξ) η(ξ) = η(0) + ξ + ξ 2 h(ξ), (13.10) h(ξ) := ρ 2 ξ 2 ξ r 0 := sup { r M(r) < 1 }, 0 τ 0 g 0 (t, η(t))dtdτ M(r) := max ξ r η(ξ) η(0) = w(c) θ 0 /6 1/12, r 0 > 0 M( ξ ) < 1, ξ < 1 z a 0 z c + c a 0 < 1/ w (c) + 1/2 1 (13.10) f 0 (z, w) h(ξ) ρ 2 ξ 2 ( ξ 2 /2)K K/8, (13.11) η(ξ) θ 0 /6 + ξ + K ξ 2 /8 (13.12) r 0 < θ 0 (13.11) θ 0 ξ < r 0 < 1 η(ξ) θ 0 /6 + θ 0 + (Kθ 0 /8)θ 0 1/12 + 1/2 + 1/32 < 2/3 r 0 r 0 θ 0 ξ < θ 0 (13.11), (13.12) η(ξ) ξ = θ 0 /2 η(ξ) ξ ξ 2 h(ξ) η(0) ( 1 (θ 0 /2)K/8 ) (θ 0 /2) θ 0 /6 (1 1/32)θ 0 /2 θ 0 /6 > θ 0 /4. 44

49 z, w(z) [] Step 5: A = 0 Γ = [c 0, a 0 ] z a 0 < 1/2 (13.9) (I) z a 0 1, w 1 f 0 (z, w) K = 7 + a 0 Lemma 13.2 θ 0 K 1 /2 θ 0 Γ 0 := {z Γ w(z) θ 0 /6} Γ A = 0 ε > 0 Γ 0 {z z a 0 < ε} = Γ 0 w (z) 2 {a n } Γ 0, a n a 0 {w(a n )}, {w (a n )} Theorem A.3 w(z) z = a 0 Γ = [c 0, a 0 ] c 0 a 0 Γ 0 c 1 Lemma 13.2 D 1 : z c 1 w (c 1 ) 1 θ 0 /2 D 1 w(z) θ 0 /4 z = a 0 D 1 D 1 [c 1, a 0 ] a 0 Γ Γ 0 c 2 Γ 0 D 2 : z c 2 w (c 2 ) 1 θ 0 /2 D 2 w(z) θ 0 /4 D 2 [c 2, a 0 ] a 0 c 3 Γ 0 {D n } n=1 {c n } n=1 Γ 0 (1) D n : z c n r n =: w (c n ) 1 θ 0 /2. (2) D n w(z) θ 0 /4 (3) n a 0 D n (4) c 1 a 0 > c 2 a 0 > > c n a 0 >. (5) {z z c n > r n } Γ 0 D n [c n, a 0 ] c n+1 (6) n c n+1 c n > r n n 1 r n a 0 c 0 c n c Γ \ {a 0 } (6) (1) r n = w (c n ) 1 θ 0 /2 0 (n ) w(c ) θ 0 /6, w (c ) = w(z) Γ \ {a 0 } c n a 0 n D n D j D := Γ ( ) n=1 D n Γ := D { z Im((z c 0 )/(a 0 c 0 )) 0 } 45

50 Γ a 0 Γ 0 n=1 D n (2) Γ w(z) θ 0 /6 lim inf z a0,z Γ w(z) θ 0 /6 > 0 (6) Γ (1 + π) a 0 c 0 A = 0 0 < A + (Step 3 ). a 0 (I) Theorem (II) Theorem 12.1 (II) Painlevé property (I) w(z) (II) Proposition σ w(z) w(z) = ±(z σ) 1 σ(z σ)/6 (α ± 1)(z σ) 2 /4 + b(z σ) 3 +. b (I) w = u 1, w = u 2 (1 + zu 2 /2 + (α ± 1/2)u 3 u 4 v) (13.13) (II) ± u = 1 + zu 2 /2 + (α ± 1/2)u 3 u 4 v, ± v = (z/2 + (α ± 1/2)u u 2 v)(α ± 1/2 2uv) (13.14) (13.13) u ± Ljapunov Φ(z) := w (z) 2 + w (z) w(z) w(z)4 zw(z) 2 2αw(z) v 2 v 2 ( w z w 2 2α w 3 ) v Φ(z) + α2 1/4 + αz w 2 w + z2 4 = 0 46

51 Φ(z) Φ (z) + Φ(z) w(z) = α 2 w(z) + w (z) w(z) 3 w(z) 1 Φ(z) w(z) Γ a 0 A = lim inf z a0, z Γ w(z) (I) a Painlevé Painlevé (I), (II) (I) z = w(z) = c p z p + c p 1 z p 1 +, p Z, c p 0 (I) z p 2 = 2p 1, p 2 = 1 2p, p 2 2p = 1 Theorem 12.1 Theorem (I) (II) α α = 0 w 0 Theorem (II) α Z Proof. (II) w 0 (z) Proposition 13.3 simple ±1 p w 0 (z) = ε j (z σ j ) 1 + P 0 (z), ε j = ±1, P 0 (z) C[z] j=1 z = ( p w 0 (z) = P 0 (z) + ε j )z 1 + O(z 2 ) (14.1) j=1 P 0 (z) d 1 (14.1) (II) z 3d = d+1 P 0 (z) c 0 47

52 c 0 0 (14.1) (II) c 0 z 0 P 0 (z) 0 w 0 (z) w 0 (z) 0 α = 0 (14.1) (II) p α + ε j = 0 (14.2) j=1 α Z [] Theorem α Z (II) Bäcklund (II) α (II) α Lemma (1) α 1/2 w α (II) α w α+1 = T + α (w α ) := w α α + 1/2 w α + w 2 α + z/2 (14.3) (II) α+1 (2) α 1/2 w α (II) α w α 1 = T α (w α ) := w α + α 1/2 w α w 2 α z/2 (14.4) (II) α 1 (3) α ±1/2 (II) α w α (Tα+1 T α + )(w α ) = (T α 1 + Tα )(w α ) = w α Remark (II) α w = εw 2 + εz/2 + v, v = 2εwv ε/2 + α, (14.5) ε = ±1 α 1/2 (resp. α 1/2) (14.3) (resp. (14.4)) 0 w α (14.5) 48

53 H(z, w, v) := εw 2 v + v 2 /2 (α ε/2)w + εzv/2 Hamilton Hamilton dw dz = H v, dv dz = H w Proof. (14.5) U ε,α : ŵ = w + ε(α ε/2)/v, v = v ŵ = εŵ 2 εz/2 + v, v = 2εŵ v + ε/2 + (α ε) (w, v) (II) α (14.5) (ŵ, v) (II) α ε v w (1), (2) (II) α (w α, v α ) U 1,α ( w α + (α 1/2)/v α, v α ) (II) α 1 U 1,α 1 (w α, v α ) (3) [] Proof of Theorem α = 0 W 0 (z) 0 (II) 0 w(z) 0 z = w(z) = c p z p + c p 1 z p 1 +, p Z, c p 0 (II) 0 z 3p = p + 1 (II) 0 W 0 (z) 0 α = m N (II) m W m (z) Lemma 14.4 w(z) = (T 1 T m 1 T m)(w m (z)) (II) 0 w(z) 0 W m (z) = (T + m 1 T + 1 T + 0 )(0) (II) m m N [] Theorem m Z \ {0} (II) m m 2 W m (z) = ε j (z σ j ) 1, ε j = ±1 j=1 49

54 ε j = 1 l + = (m 2 m)/2 ε j = 1 l = (m 2 + m)/2 Proof. Theorem 14.2 p W m (z) = ε j (z σ j ) 1 = mz 1 + O(z 2 ) j=1 (z ), (14.6) p m = ε j j=1 (14.7) Proposition 13.3 W m (z) 2 σ j (z σ j ) 2 p W m (z) 2 = (z σ j ) 2 + P (z), P (z) C[z] j=1 (14.6) P (z) 0 p W m (z) 2 = (z σ j ) 2 = pz 2 + O(z 3 ) (z ). j=1 (14.8) (14.6), (14.8) p = m 2 (14.7) l + + l = m 2, l + l = m l + = (m 2 m)/2, l = (m 2 + m)/2 [] 50

55 Painlevé Painlevé (I) (Theorems 12.1, 14.1). (I) Painlevé T (r, w) Painlevé ϱ(w) (I) (II), (IV) (III), (V) (I) (V) 15. Painlevé Theorem (I) w(z) T (r, w) = O(r C ) C > 0 w(z) Remark (I) w(z) w(z) (I) Theorem 15.1 Proposition N(r, w) = O(r C ). Lemma 7.1 m(r, w) = S(r, w) µ(e) < E r, r E T (r, w) = N(r, w) + m(r, w) = O(r C ) + o(t (r, w)) Lemma 6.9 Theorem 15.1 Proposition 15.2 r w(z) 13 Ljapunov Lemma Lemma 13.2 w (c) Lemma θ := 2 4 c 5 c w(c) θ 2 c 1/2 /6 51

56 (1) w(z) z c < δ c (2) (5/6)δ c z c δ c w(z) θ 2 c 1/2 /5. δ c θ c 1/4 min{1, θ c 3/4 / w (c) } < δ c 3θ c 1/4 (15.1) Proof. z c = ρt, ρ = c 1/4, w(z) = w(c + ρt) = θc 1/2 W (t) (I) Ẅ (t) = 6θW (t) 2 + θ 1 (1 + ρ 5 t) ( = d/dt) W (t) = W (0) + Ẇ (0)t + θ 1 t 2 /2 + g(t), (15.2) W (0) = θ 1 c 1/2 w(c), g(t) := θ 1 ρ 5 t 3 /6 + 6θ Ẇ (0) = θ 1 c 3/4 w (c), t τ 0 0 W (s) 2 dsdτ. Case 1: Ẇ (0) 1. η 0 := sup{η M(η) 8θ}, M(η) := max t η W (t) W (0) = θ 1 c 1/2 w(c) θ/6 η 0 > 0 η 0 < 3θ t η 0, c 5 g(t) θ 1 ρ 5 t θ t τ 0 0 W (s) 2 ds dτ θ 1 ρ 5 (3θ) 3 /6 + 6θ(8θ) 2 (3θ) 2 /2 < θ/4 (15.3) (15.2) W (t) W (0) + t + θ 1 t 2 /2 + θ/4 (1/ /2 + 1/4)θ < 7.92θ (15.4) t η 0 η 0 η 0 3θ (15.3) t 3θ 2.5θ t 3θ W (t) θ 1 t 2 /2 t W (0) g(t) ( / /6 1/4 ) θ > θ/5. 52

57 2 θ 1 x 2 /2 x x θ (1), (2) δ c = 3θ c 1/4 Case 2: Ẇ (0) = κ > 1. η 1 := sup{η M(η) 5θ} η 1 < (2/κ)θ (15.3), (15.4) t η 1 g(t) θ 1 ρ 5 (2θ) 3 /6 + 6θ(5θ) 2 (2θ) 2 /2 < θ/24, W (t) W (0) + κ t + θ 1 t 2 /2 + θ/24 (1/ /24)θ < 4.3θ η 1 η 1 (2/κ)θ t (2/κ)θ g(t) < θ/24 (0.8/κ)θ t (1.2/κ)θ W (t) κ t θ 1 t 2 /2 W (0) g(t) ( /2 1/6 1/24 ) θ > θ/5. δ c = 1.2θ c 1/4 /κ = 1.2θ c 1/4 θ c 3/4 / w (c) (1), (2) [] Remark Lemma 15.3 c 5 (2) (2 ) (5/6)δ c z c δ c w(z) θ 2 z 1/2 /5.5. Lemma z = R 0 (5 < R 0 < 10) w(z) σ σ > 10 Γ σ : z = ϕ(x) x Γ σ z (1) Γ σ z = R 0 σ (2) ϕ(x) x (3) Γ σ dz (6/ 11)d z. (4) Γ σ w(z) 2 11 z 1/2. Proof. R 0 e i arg σ σ Γ 0 Γ 0 Γ := {z w(z) θ 2 z 1/2 /6} Γ = θ 2 /6 > 2 11 R 0 e i arg σ σ Γ 0 Γ 53

58 c 1 Γ 1 : z c 1 (5/6)δ c1 γ 1 := {z = ξe i arg σ ξ > 0, z c 1 δ c1 } δ c1 Lemma γ 1 γ A(c 1 ) γ 1 σ b 1 A(c 1 ) Lemma 15.3 Remark 15.2 σ A(c 1 ) (Γ 0 A(c 1 )) {z z b 1 } ( b 1 ) w(z) θ 2 z 1/2 /6 w(b 1 ) θ 2 b 1 1/2 /5.5 b 1 Γ 0 \ Γ Γ 0 σ Γ c 2 Γ Lemma 15.3 δ c2 c 1 A(c 2 ) A(c 2 ) Γ 0 σ b 2 A(c 2 ) b 2 Γ 0 {c n } Γ, {δ cn } {A(c n )}. c n c Γ 0 \ {σ}, n δ cn 0 w (c ) = + w(c n ) c Γ 0 \ {σ} c n σ Γ σ ( ( Γ σ := Γ 0 A(c n )) ) { z z R 0, Im(z/σ) 0 } n=1 (1), (2) Γ σ w(z) θ 2 z 1/2 /6 (4) A(c n ) (3) [] Proposition 15.2 σ, σ > 10 w(z) 13 (13.6) Ljapunov Φ(z) = w (z) 2 + w (z) w(z) 4w(z)3 2zw(z) Lemma 15.4 σ Φ(z) w(z) Riccati 54

59 Step 1: R 0, 5 < R 0 < 10 z = R 0 w(z) 0, max Φ(z) < +, z =R 0 max 1/w(z) < + (15.5) z =R 0 Lemma 15.4 Γ σ z z Γ σ (z) := {t Γ σ t z } Γ σ z 0 (σ) ( z 0 (σ) = R 0 ) Φ(z) (13.8) Φ(z) Lemma 15.4 (3), (4) ( ) E(z 0 (σ), z) ±1 dt exp Γ σ(z) w(t) 2 ( z exp 11 z 0 (σ) ) d t = O( z C 0 ), t C 0 = / 11 (15.5) (13.8) ( ) Φ(σ) σ C 0 σ C0 1 + t C0+1/2 d t = O( σ 2C0+3/2 ) Γ σ (15.6) σ U(σ) := { z z σ < η(σ) }, η(σ) := sup { η 2 z 1/2 < w(z) < + in 0 < z σ < η ( 1) } [σ, z] U(σ) (13.8) E(σ, z) ±1 = O(1) (15.6) Φ(z) Φ(σ) t 1/2 d t = O( z 2C0+3/2 ), U(σ) [σ,z] Φ(z) K 0 z Λ (15.7) Λ = > 2C 0 + 3/2 K 0 σ Step 2: Φ(z) (13.6) w(z) = u(z) 2, z = σ + σ Λ/6 s v(s) := u(σ + σ Λ/6 s) (σ + σ Λ/6 s) Λ/6 v(s) < ε0 (15.8) 55

60 dv ds (s) = σ Λ/6( 1 + h(s, v(s)) ), (15.9) h(s, v(s)) < 1/2, v(0) = 0 u(z) u (σ) = σ Λ/6 (dv/ds)(0) = 1 ε 0 = ε 0 (K 0 ) σ (15.8) z = σ + σ Λ/6 s U(σ) η := sup{η s < η (15.8) } < ε 0 /4 (15.9) s η (< ε 0 /4) s /2 σ Λ/6 v(s) 3 s /2 3ε 0 /8 (15.10) M 0 σ M 0 s η (σ + σ Λ/6 s) Λ/6 v(s) σ Λ/6 v(s) (1 + M0 1 ) Λ/6 ε 0 /2 σ M 0 η σ M 0 η ε 0 /4 (15.10) s < ε 0 /4 w(z) 0 < z σ < (ε 0 /4) σ Λ/6 Theorem 11.1 n(r, w) = O(r C ), C = 2 + Λ/3 Proposition Painlevé (I) T (r, w) = O(r C ) C C C = ( Theorem 6.5), Theorem Painlevé (I) w(z) r 5/2 log r T (r, w) r5/2 56

61 Corollary (I) w(z) ϱ(w) 5/ Theorem 16.1 Theorem 15.1 w(z) (I) Ljapunov Φ(z) = w (z) 2 + w (z) w(z) 4w(z)3 2zw(z) Ljapunov Ψ(z) := w (z) 2 4w(z) 3 2zw(z) (I) 2w (z) Ψ (z) = 2w(z) z 0 z γ(z 0, z) w(z) Ψ(z) = Ψ(z 0 ) 2 w(t)dt (16.1) γ(z 0,z) z = r 0, r 0 > 10 w(z) K K > 2r 1/2 0 M(w; r 0 ) + M(Ψ; r 0 ) , (16.2) M(f; r) := max z =r f(z), D K := {z w(z) > K z 1/2 } Proposition a R(θ) := {z = xe iθ x r 0 } (a) w(a) = K a 1/2, a D K. (b) δ a R(θ) {z a δ a < z a } C \ D K, R(θ) {z a < z < a + δ a } D K. 57

62 (c) Ψ(a) 4K a 3/2. w(z) σ a (σ a ) : z σ a < (11/10)K 1/2 σ a 1/4 (1) a 1/10 < σ a < a + 1/10. (2) a (σ a ). (3) w(z) (σ a ) \ {σ a } (4) (σ a ) w(z) < K z 1/2, (σ a ) C \ D K. Proposition a, a R(θ) (a), (b), (c) (σ a, (σ a )), (σ a, (σ a )) Proposition 16.3 (σ a ) (σ a ) = (σ a ) = (σ a ) Propositions 16.3, 16.4 σ σ > 2r 0 w(z) σ 0 < z σ < (11/10)K 1/2 σ 1/4 w(z) n(r, w) r 5/2 (16.3) N(r, w) = O(r 5/2 ) σ R σ := R(arg σ) = {z = xe i arg σ x r 0 } (16.2) z 0 := r 0 e i arg σ R σ w(z 0 ) < Kr 1/2 0 /2, Ψ(z 0 ) < K z 0 D K z 0 R σ σ D K Proposition 16.3 (a), (b) a = a 1 R σ [z 0, a 1 ] C \ D K (16.1) Ψ(a 1 ) Ψ(z 0 ) +2 w(t) dt K +2K a 1 1/2 a 1 z 0 3K a 1 3/2 [z 0,a 1 ] 58

63 Proposition 16.3 (c) Proposition 16.3 σ 1 := σ a1, (σ 1 ) (1),..., (4) σ 1 = σ (σ)\{σ} (σ 1 ) [0, σ] Γ 1, b 1, b + 1, b 1 < b + 1 [b 1, b + 1 ] R σ Γ 1 R σ (1) := (R σ \ [b 1, b + 1 ]) Γ 1 z 0 (σ 1 ) [z 0, b + 1 ] Γ 1 z r 0 b + 1 R σ (1) (a), (b) a = a 2 R σ (1) a 2 R σ (1) [a 2 ] R σ (1) [a 2 ] π a 2 /2 R σ (1) [a 2 ] w(z) K z 1/2 K a 2 1/2 Ψ(a 2 ) Ψ(z 0 ) + 2 w(t) dt K + K a 2 1/2 π a 2 4K a 2 3/2 R (1) σ [a 2 ] (c) Proposition 16.3 σ 2 := σ a2, (a 2 ) Proposition 16.4 (σ 1 ) (σ 2 ) = σ 2 σ σ 1 R σ (2) σ j σ {σ j }, { (σ j )} (σ j ) πk 1 σ j 1/2 ( πk 1 ( σ + 1) 1/2 ) (a), (b), (c) a R σ, (1),..., (4) w(z) σ = σ a, (σ ) [a, σ] \ {a } σ σ w(z) (σ) \ {σ} Propositions 16.3, Proposition 16.3 a Proposition 16.3 a σ a Ljapunov Φ(z) a r 0 > 10 (16.4) (16.2), (a), (c) Ψ(z) w (a) 2 4 w(a) a w(a) + Ψ(a) (4K 3 + 2K + 4K) a 3/2 < 10K 3 a 3/2 59

64 w (a)/w(a) 10K 1/2 a 1/4 (16.4) K > 10 2 Φ(a) Ψ(a) + w (a)/w(a) < 4K a 3/2 (1 + K 1/2 a 5/4 ) < 5K a 3/2 (16.5) U(a) := { z z a < 2K 1/2 a 1/4} Lemma a z γ(z) U(a) (1) γ(z) := dt 1/10. γ(z) (2) t γ(z) a z w(t) Φ(z) < 6K z 3/2 Proof. Φ(z) (13.8) z 0 = a (2) (16.2), (16.4) t γ(z), w(t) w(a) = K a 1/2 300 E(a, t) ±1 exp( γ(z) /300 2 ) < (1) 0.99 a t 1.01 a (16.6) (16.5) [ Φ(z) E(a, z) 1 E(a, z) + 1 Φ(a) + 2 w(a) 2 E(a, t) + γ(z) w(t) ( t + 1 w(t) 3 ) ] dt [ 5K a 3/ ( (1.01 a ) )] < 6K z 3/2 [] steepest descent path Lemma f(z) D c 0 D f(c 0 ) 0 D D γ 0 60

65 (1) γ 0 c 0 c (2) f(c ) = 0 c D (3) γ 0 d f(z) /ds = df(z)/ds s γ 0 c 0 z Proof. X := Ref(z), Y := Imf(z) ( d f(z) ds ) 2 df(z) ds 2 ( d = ds X2 + Y 2 ) 2 (X 2 s +Y 2 s ) = (X sy XY s ) 2 f(z) 2 (X s = dx/ds). f(z) = C (d/ds ) f(z) = 0 XX s + Y Y s = 0 f(z)f (z) 0 X s Y XY s = 0 f(z) = C, C > 0 c 0 c 1 (3) f(z) c 1 (i) c 1 D, (ii) f(c 1 ) = 0, (iii) f (c 1 ) = 0. c 1 D f (c 1 ) = 0 f(c 1 ) 0 c 1 f(z) ( ), c 1 (3) γ 0 [] Lemma w(z) σ a σ a a 1.01K 1/2 a 1/4 < 1/10, (16.7) Φ(σ a ) < 6K σ a 3/2 (16.8) Proof. w(z) = u(z) 2 u(z) f(z) = u(z), D = U(a), c 0 = a Lemma 16.6 a a γ 0 u(a ) = 0 or a U(a) (16.9) d u(z) /ds = du(z)/ds on γ 0 (16.10) γ 0 u(z) u(z) u(a) = K 1/2 a 1/4 (16.11) 61

66 Φ(z) w(z) = u(z) 2 4u (z) 2 2u(z) 5 u (z) 4 2zu(z) 4 u(z) 6 Φ(z) = 0 u (z) F (z), F (z) := zu(z) 4 /2 + u(z) 6 Φ(z)/4 + u(z) 10 /16 (16.12) γ 0 z γ 0 (z) γ 0 (z) 1/10 Lemma 16.5, (16.6), (16.11), (16.2) F (z) (1/2)K 2 z/a + (3/2)K 2 z/a 3/2 + (1/16)K 5 a 5/2 2K /2 + (1/16)K 5 < 10 3 /3 (16.13) u(z) u (z) = u(z) 5 /4 + (1 + F (z)) 1/2 (16.14) γ 0 (z) 1/10 (1+F (z)) 1/2 1 < 10 3 /3 u (z), u(z) γ 0 (z) 1/10 u (z) = 1 + g(z), g(z) < 10 3 (16.15) (16.15) γ 0 (z) ds (16.10) ds/dt = 1 γ 0 (z) 1/10 u (t)ds γ 0 (z) γ du(t) 0 (z) ds ds = γ 0 (z) (1 + g(t))ds γ 0 (z) γ 0 (z) ds γ 0 (z) d u(t) ds = u(a) u(z) ds g(t)ds ( ) γ 0 (z) (16.2), (16.4) γ 0 (z) 1.01(K 1/2 a 1/4 u(z) ) < 0.06 γ 0 (z 1 ) = 1/10 z 1 γ 0 γ 0 (z 1 ) < 0.06 γ 0 < 1/10, (16.9) a γ (K 1/2 a 1/4 u(a ) ) < 1/10 (16.16) 62

67 a U(a) γ 0 a a = 2K 1/2 a 1/4 (16.16) u(a ) = 0 σ a = a w(z) (16.16) γ 0 a σ a (16.7) Lemma 16.5 (16.8) [] σ a w(z) Lemma z σ a < (6/5)K 1/2 σ a 1/4 Φ(z) < 7K z 3/2, (16.17) u (z) 1 < 10 2, (16.18) 0.99 z σ a u(z) 1.01 z σ a (16.19) u(z) (16.18) w(z) 1/2 Proof. µ 0 := sup { µ } z σa < µ (16.17), (16.18),(16.19) Lemma 16.7 u(z) γ 0 (16.18) (16.15) u (σ a ) 1 < 10 3 σ a (16.18) σ a ( γ 0 ) u(z) (16.19) (16.8) µ 0 > 0 µ 0 < (6/5)K 1/2 σ a 1/4 < 1/14 U 0 := {z z σ a < µ 0 } U σ a < z < 1.01 σ a (16.19) u(z) 1.01µ 0 < 1.3K 1/2 σ a 1/4, w(z) 1 < 10 2 (16.20) (13.8) z 0 = σ a, σ a z Φ(σ a ) < 6K σ a 3/2 (16.20) Lemma 16.5 U 0 Φ(z) < 6.5K σ a 3/2 6.7K z 3/2 (16.21) (16.12), (16.20), (16.21) U 0 F (z) (1/2)1.3 4 K 2 z/σ a K 2 z/σ a 3/2 + (1/16) K 5 < K /2 + (1/16) K 5 < 10 2 /6 63

68 (16.13) U 0 u (z) 1 < 10 2 /3, (16.22) ( /3) z σ a u(z) ( /3) z σ a (16.23) (16.21), (16.22), (16.23) µ 0 µ 0 (6/5)K 1/2 σ a 1/4 [] Remark w(z) 1/2 u (z) z σ a < (6/5)K 1/2 σ a 1/4 u (z) + 1 < 10 2 (16.7) Proposition 16.3 (1) σ a /a ( a + 1/10)/ a < 1.01 σ a a 1.01K 1/2 a 1/ /4 K 1/2 σ a 1/4 < (11/10)K 1/2 σ a 1/4, a (σ a ) (16.19) (σ a ) z/σ a 1 < 10 2 u(z) K 1/2 σ a 1/4 1.08K 1/2 (0.99 z 1 ) 1/4 > 1.07K 1/2 z 1/4 (σ a ) w(z) < K z 1/2 Proposition Proposition 16.4 (σ a ) (σ a ) u(z) (σ a ) (σ a ) (16.18) z (σ a ) (σ a ) [σ a, σ a ] u(z ) (z σ a ) 10 2 z σ a, u(z ) (z σ a ) 10 2 z σ a σ a σ a 10 2 σ a σ a σ a = σ a [] Theorem

69 (16.3) w(z) {σ j } j=1 double 2 σ 1 σ 2 σ j w(z) Lemma 7.1 κ κ > 1 S({σ j }, κr, z) := σ j <κr z σ j 2 Lemma r 0 (κ) r > r 0 (κ) r r/ 2 < z r < r z r S({σ j }, κr, z r ) 8n(κr)r 2 log r. (16.24) n(r) := n(r, w) = n(r, w)/2 Proof. 0 < δ < 1, ω(t) := min{1, t 1/4 } (t > 0) δ (r) := {z z < r} \ D δ, D δ := j=1 {z z σ j < δω( σ j )} (16.3) n(r) = n(r, w)/2 r 5/2 Ω(r) := 1 σ j r+1 ω( σ j ) 2 r+1 1 t 1/2 dn(t) r 1/2 n(r + 1) + r+1 1 t 3/2 n(t)dt r 2. δ Ar(D δ {z z < r}) < πδ 2 (Ω(r) + O(1)) < πr 2 /16 Ar(. ) z σ j = ρ, 65

70 z = x + yi z σ j 2 dxdy = δω( σ j ) z σ j (κ+1)r δω( σ j ) ρ (κ+1)r 0 θ 2π 2π ( log(r/ω( σ j )) + B κ ), ρ 1 dρdθ Bκ := log((κ + 1)/δ). σ j κr r/ω( σ j ) κr 5/4 δ (r) S({σ j }, κr, z)dxdy (5/2)πn(κr)( log r + B κ + log κ) (16.25) F r := {z δ (r) S({σ j }, κr, z) 8n(κr)r 2 log r} (16.25) 8n(κr)r 2 log r Ar(F r ) (5/2)πn(κr) ( log r + B κ + log κ ) r 0 (κ) r > r 0 (κ) Ar(F r ) 3πr 2 /8 H r := F r (D δ {z z < r}) Ar(H r ) (3/8 + 1/16)πr 2 < πr 2 /2 z r r/ 2 < z r < r, z r δ (r) \ F r [] Lemma κ 8 κ K 0 r > 1, z < r r, z χ(κr, z) := σ j κr σ j κr z σ j 4 K 0, (z σ j ) 2 σ 2 j K 0 κ 1/2 r 1/2, Proof. σ j κr, z < r z/σ j 1/8 (z σ j ) 2 σj 2 = 2 z σ j 3 1 (z/σ j )/2 1 z/σ j 2 3r σ j 3 (16.3) r > 1 n(r) K 1 r 5/2 K 1 χ(κr, z) 3r σ j κr σ j 3 = 3r t 3 dn(t) κr 9r 66 κr t 4 n(t)dt 18K 1 κ 1/2 r 1/2

71 r > 1, z < r z < r σ j κr z σ j 4 (8/7) 4 σ j κr σ j 4 [] Lemma µ(e) < E (0, ) z (0, ) \ E z 0< σ j < (z σ j ) 2 σ 2 j z 9. Proof. E := (0, σ 1 + 1) ( j=2 ( σ j σ j 3, σ j + σ j 3 ) ) (16.3) µ(e) σ j 3 t 3 dn(t) t 4 n(t)dt < j Lemma z E ( + ) (z σ j ) 2 σj 2 ( z 6 +1)n(8 z, w)+ z 1/2 z 9 0< σ j <8 z σ j 8 z [] Lemma κ 8, 0 < η < 1 κ η L 0 1 r > 1 r γ(κr) = 0< σ j κr Proof. r > 1 1 σ j κr σ j 2 = κr 1 σ j 2 η 4 n(κr)r 2 + L 0 (ηr 1/2 + 1) ( η 2 t 2 dn(t) (κr) 2 r n(κr) κr ( κr ) η 2 (κr) t 3 r dt n(κr) + 2 t 3 n(t)dt η 2 r 1 = η 4 n(κr)r η 2 r η 2 r 1 ) t 3 n(t)dt t 3 n(t)dt

72 L < σ j <1 σ j 2 L 0 t 1 n(t) L 0 t 5/2 /4 [] Proposition 13.1 w(z) σ j (z σ j ) 2 Mittag-Leffler Lemma w(z) ( ) w(z) = h(z) + G(z), G(z) := (z σj ) 2 σj 2 j=1 h(z) σ 1 = 0 G(z) z 2 r, κ r > 1, κ 8 G(z) S({σ j }, κr, z) + γ(κr) + χ(κr, z) Lemma χ(κr, z) K 0 κ 1/2 r 1/2 z < r Lemma 16.9 r > r 0 (κ) r/ 2 < z r < r z r S({σ j }, κr, z r ) 8n(κr)r 2 log r Lemma η = η 0 (κ) := κ 1/2 L 1 0 r > 1 γ(κr) η 0 (κ) 4 n(κr)r 2 + κ 1/2 r 1/2 + L 0 r (κ) := r 0 (κ) + exp(η 0 (κ) 4 ) r > r (κ) r/ 2 < z r < r z r G(z r ) 9n(κr)r 2 log r + (K 0 + 1)κ 1/2 r 1/2 + L 0 r 1/2 log r (16.26) σ j <κr(z r σ j ) 4 S({σj }, κr, z r ) 2 Lemma r > r (κ) G (z r ) 64n(κr) 2 r 4 (log r) 2 + 6K 0 r(log r) 2 (16.27) (16.3), Lemma 7.1 Lemma r, r E T (r, h) = m(r, h) m(r, w) + m(r, G) = O(log r) Lemma 6.9 T (r, h) = O(log r) Proposition 3.1 h(z) w(z r ) = (1/ 6)(w (z r ) z r ) 1/2 (16.26), (16.27) h(z r ) G(z r ) + ( h (z r ) + G (z r ) + z r ) 1/2 r 1/2 log r + h (z r ) 1/2 68

73 z r h(z) C 0 C z r = w (z r ) 6w(z r ) 2 (16.26), (16.27) 2 1/4 r 1/2 z r 1/2 ( G (z r ) + 6( G(z r ) + C 0 ) 2) 1/2 G (z r ) 1/2 + 3( G(z r ) + C 0 ) 35n(κr)r 2 log r + 3(K 0 + 1)κ 1/2 r 1/2 + O(1) r r (κ) κ = κ 0 > 8 2 1/4 3(K 0 + 1)κ 1/2 0 > 0 n(κ 0 r) r 5/2 / log r, n(r, w) r 5/2 / log r, N(r, w) r 5/2 / log r Painlevé (II), (IV) Painlevé (II), (IV) (I) (II), (IV) simple ±1 2 Airy Weber (I) Theorem (II) T (r, w) = O(r 3 ) (IV) T (r, w) = O(r 4 ) (II) w(z) (II) Painlevé property Ljapunov Φ(z) = w (z) 2 + w (z) w(z) w(z)4 zw(z) 2 2αw(z) Ψ(z) := w (z) 2 w(z) 4 zw(z) 2 2αw(z) Φ (z) + Φ(z) w(z) = α 2 w(z) + w (z) w(z), 3 Ψ (z) = w(z) 2 69

74 z 0 z γ(z 0, z) Ψ(z) = Ψ(z 0 ) w(t) 2 dt, γ(z 0,z) [ Φ(z) = E(z 0, z) 1 Φ(z 0 ) 1 ( E(z0, z) 1 ) 2 w(z) 2 w(z 0 ) 2 E(z 0, t) ( αw(t) 3 1/2 ) ] dt, γ(z 0,z) w(t) 4 ( E(z 0, t) := exp w(τ) 2 dτ γ(z 0,t) ), γ(z 0, t) γ(z 0, z) z = r 0 > 10 w(z) K K > 2r 1/2 0 M(w; r 0 ) + M(Ψ; r 0 ) α D K := {z w(z) > K z 1/2 } n(r, w) = O(r 3 ), T (r, w) = O(r 3 ) (I) Proposition a R(θ) = {z = xe iθ x r 0 } Proposition 16.3 (a), (b) (c ) Ψ(a) 2K 2 a 2 w(z) σ a (σ a ) : z σ a < (11/10)K 1 σ a 1/2 (1) a 1/10 < σ a < a + 1/10. (2) a (σ a ). (3) w(z) (σ a ) \ {σ a } (4) (σ a ) w(z) < K z 1/2, (σ a ) C \ D K. Proposition a, a R(θ) (a), (b), (c ) (σ a, (σ a )), (σ a, (σ a )) (σ a ) (σ a ) = (σ a ) = (σ a ) 70

75 Proposition Proposition 16.3 (a), (c ) K Φ(a) 3K 2 a 2 {z z a < 2K 1 a 1/2 } a z γ(z) γ(z) 1/10 w(t) γ(z) u(z) = 1/w(z) Φ(z) < 4K 2 z 2 u (z) 2 u(z) 3 u (z) 1 zu(z) 2 2αu(z) 3 u(z) 4 Φ(z) = 0 steepest descent path w(z) σ a σ a a 1.01K 1 a 1/2, Φ(σ a ) < 4K 2 σ a 2 σ a Lemma z σ a < (6/5)K 1 σ a 1/2 Φ(z) < 5K 2 z 2, u (z) 1 < 10 2 or u (z) + 1 < 10 2, 0.99 z σ a u(z) 1.01 z σ a u (z) (I) (16.18) Proposition (I) Proposition Proposition

76 Painlevé Nevanlinna Painlevé (I), (II), (IV) Nevanlinna (I), (II) (III), (V) C \ {0} universal covering z = e t 17. (I) w(z) w = 6w 2 + z (I) (I) 6w 2 = w z w(z) (Theorem 15.1 Theorem 16.1) Lemma 7.1 Theorem m(r, w) = O(log r). w(z) Corollary δ(, w) = 0. a C w a (I) Lemma 7.2 Theorem a C m(r, 1/(w a)) = O(log r). Corollary a C δ(a, w) = 0. (I) C { } (Proposition 5.1). Picard Corollary (I) Nevanlinna 2 72

77 Lemma m(r, w ) = O(log r). Proof. Theorem 6.5 Theorem 17.1 m(r, w ) m(r, w /w) + m(r, w) = O(log r) [] w (z) 3 N(r, w ) = (3/2)N(r, w) = O(r 5/2 ) Corollary T (r, w ) = O(r 5/2 ). (I) 2 (Theorem 8.1) Theorem N(r, 1/w ) + N 1 (r, w) = 2T (r, w) + O(log r). Proof. (I) w (3) = 12ww + 1 1/w = w (3) /w 12w. Corollary 17.7 w (z) Corollary 6.6 m(r, 1/w ) m(r, w (3) /w ) + m(r, w) = O(log r). Lemma 17.6 N(r, 1/w ) = T (r, 1/w ) m(r, 1/w ) = T (r, w ) + O(log r) = N(r, w ) + m(r, w ) + O(log r) = (3/2)N(r, w) + O(log r) = (3/2)T (r, w) + O(log r) N 1 (r, w) = (1/2)N(r, w) = (1/2)T (r, w) + O(log r) [] Corollary N(r, 1/w ) = (3/2)T (r, w) + O(log r). a- a a a- ϑ(a, w) = 0 73

78 Theorem a C N 1 (r, 1/(w a)) (1/6)T (r, w) + O(log r), (17.1) ϑ(a, w) 1/6 N 1 (r, w) = (1/2)T (r, w) + O(log r), (17.2) ϑ(, w) = 1/2 Proof. Theorem 17.1 N 1 (r, w) = (1/2)N(r, w) (17.2) w(z) ϑ(, w) = 1/2 (17.1) a- a- X := {z w(z) = a, w (z) = 0} X N 1 (r, 1/(w a)) = O(log r) (17.1) X z X 3 a- w (z ) = 0 z = w (z ) 6w(z ) 2 = 6a 2 X 1 2 a- 16 Ljapunov Ψ(z) = w (z) 2 4w(z) 3 2zw(z) (16.1) Ψ(z) Ψ(z 0 ) = 2 z w(t)dt z 0 (17.3) z 0 X w(z 0 ) = a, w (z 0 ) = 0 (17.3) z G(z) = 2a(z z 0 ) 2 w(t)dt, (17.4) z 0 G(z) := w (z) 2 4(w(z) 3 a 3 ) 2z(w(z) a) (17.5) (17.4) G (z) = 2(a w(z)), G (z) = 2w (z) τ X \ { 6a 2 } w(τ) = a, w (τ) = 0, w (τ) 0 G(τ) = G (τ) = G (τ) = 0 74

79 τ G(z) 3 N 1 (r, 1/(w a)) (1/3)N(r, 1/G) + O(log r) (1/3)T (r, G) + O(log r) (17.6) w(z) z = σ w(z) = (z σ) 2 + o(1) (17.4) 2(z σ) 1 + O(1) σ G(z) simple (17.4) G(z) w(z) N(r, G) = (1/2)N(r, w) = (1/2)T (r, w) + O(log r) (17.5) Lemma 17.6 m(r, G) m(r, w ) + m(r, w) + O(log r) = O(log r). T (r, G) = (1/2)T (r, w) + O(log r) (17.6) [] 18. (II) w = 2w 3 + zw + α (II) (I) Theorems 14.2, 14.3 α Z (II) α Z w(z) (II) Theorem w(z) Lemma 7.1 w(z) (I) Theorem m(r, w) = O(log r) δ(, w) = 0. Theorem 14.5 Corollary (II) α = 0 w 0 a C w a (II) 2a 3 + az + α 0 a = α = 0 Lemma