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1 (yx4) OCR TeX 50 yx4

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5 i Jacobi Euler Fagnano Abel

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7 iii Jacobi sn cn,dn Landen Weierstrass ζ σ , ζ σ

8 iv σ 1, σ, σ ϑ ϑ σ sn J(τ) λ(τ) sn

9 1 1. { R z, } φ(z) dz (1) φ(z) z R z φ(z) φ(z) z (1) z φ(z) φ(z) (1) (1) φ(z), (1) () φ(z) = s, R(z, s) = f(z) (1) f(z) dz () φ(z) φ(z) = a 0 z 4 + a 1 z 3 + a z + a 3 z + a 4 (3) (i) α 0 = 0 α 1 = 0. α 0, α 1 0 φ(z) (ii) φ(z) = 0 z φ (z) = 0 (1) ()

10 φ(z) = φ (z) = 0 φ(z) = 0 z φ(z) () () f(z) = A 1 + A s B 1 + B s, A 1, A, B 1, B z. B 1 B s f(z) = C 1 + C s D C 1, C, D z f(z) = C 1 D + C s Ds = R 1 + R s R 1, R z (1) R 1 R s R 1 R s R ( ) R = h C h z h + k A k (z a) k + l B l (z b) l + C A B,, a, b, R s, z h I h = dz, h = 0, 1,,, s dz J k = (z a) l, k = 1,,. s I z h s z d dz (zh s) = hz h 1 s + 1 zh φ (z) s { } = zh 1 hφ(z) + zφ (z) s φ(z) (3)) d { dz (zh s) = zh 1 (h + 4)a 0 z 4 + (h + 3)a 1 z 3 + (h + )a z s } + (h + 1)a 3 z + ha 4 β, z h s = (h + 4)a 0 I h+3 + (h + 3)a 1 I h+ + (h + )a I h+1 + (h + 1)a 3 I h + ha 4 I h 1.. h = 0 s = 4a 0 I 3 + 3a 1 I + a I 1 + a 3 I 0

11 1. 3 I h a 0 =0 I, I 1, I 0, a 0 = 0, a 1 =0 I 1, I 0, J d dz s (z a) k = ks = (z a) k φ (z) (z a) k s 1 { s(z a) k+1 kφ(z) + (z a)φ (z) φ(z) (3) (3) (z α) d s dz (z a) k = 1 { s(z a) k+1 kφ(a) + (1 k)φ (a)(z a) + (1 k)φ (a)(z a) + 3 k φ (a)(z a) 3 + k 6 1 φ (a)(z a) 4}. z k = 1 s (z a) k = kφ(a)j k+1 + (1 k)φ (a)j k J k + (1 k)φ (a)j k k φ (a)j k + k 6 1 φ (a)j k 3. s z a = kφ(a)j φ (a)j φ (a)j φ (a)j φ(a) = 0 φ(a) = 0, φ (a) = 0 J 1, J 0, J 1, J J 0, J 1, J } J 0, J 1, J,

12 4 dz J 0 = s = I, z a J 1 = dz = I 1 ai 0, s (z a) J = dz = I ai 1 + a I 0 s I I k J k I 0, I 1, I, J 1 I 0 = I = dz, I 1 = φ(z) z dz, J 1 = φ(z) zdz φ(z), dz (z a) φ(z). α = 0 ( φ(z) ) I, φ(a) = 0 J 1 (4). φ(z) (i), (ii) φ(z) φ(z) { R z, } φ(z) dz (1) φ(z) φ(z) = a(z α 0 )(z α 1 )(z α )(z α 3 ) a α α (1) z = Aζ + B, AD BC = 0, Cζ + D A, B, C, D (1) R 1 {ζ, } φ 1 (ζ) dζ, φ 1 (ζ) = c(ζ β 0 )(ζ β 1 )(ζ β )(ζ β 3 ) ()

13 . 5 α i = Aβ i + B, i = 0, 1,, 3 Cβ i + D α 1 α 0 α 1 α α 3 α α 3 α 0 = β 1 β 0 β 1 β β 3 β β 3 β 0 (3) (3) α β β α 0, α 1, α β 0, β 1, β α 1 α 0 α 1 α z α z α 0 = β 1 β 0 β 1 β ζ β ζ β 0 α 1, α 0, α 3 0, 1, α 1 α 0 α 1 z α 3 z = 0 1 α 3 α 0 0 ζ ζ 1 = 1 ζ z = α 3(α 1 α 0 )ζ α 1 (α 3 α 0 ) (α 1 α 0 )ζ (α 3 α 0 ) { R z, } φ( ) dz = R 1 {ζ, } ζ(1 ζ)(1 λζ) dζ (4) λ = α 1 α 0 α 1 α α 3 α α 3 α 0 λ α 0, α 1, α, α 3 (α 0, α 1, α, α 3 ) 0, 1, α 1, α 0, α 3 4 P 3 = 4 λ λ λ λ, 1 λ, 1 λ, 1 1 λ, λ λ 1, λ 1 λ 0, 1, α 1, α 0, α 3 (4) λ (α 3, α 1, α, α 0 ) = α 1 α 3 α 1 α α 0 α α 0 α 3 = 1 α 1 α 0 α 1 α α 3 α α 3 α 0 = 1 λ α 1, α 0, α (α 0, α 1, α 3, α ) = λ λ 1 (5)

14 6 0, 1, α λ ( 0, 1, 1, 0, 3 α 1, α 0, α 3 0, 1,, (013) λ (α 0, α 1, α, α 3 ) ) (013) (103) 1 3 λ 3 0 (310) (301) (103) (130) (301) λ (031) (310) (130) λ 0 3 (013) (031) 0 1 (130) (103) (031) (301) (013) (103) (310) (301) (310) (130) (013) (031) 1 1 λ λ 1 λ λ 1 λ (5) λ f 1 (λ) = 1 λ, f (λ) = 1 1 λ f 1 f (λ) = λ = λ λ 1, f 1 f 1 (λ) = 1 (1 λ) = 1 λ, (5) λ λ 1 λ, 1 λ, x + y = 1, (x 1) + y = 1, x = 1. λ 1 λ, 1 λ, 1

15 3. 7 x + y 1 x 1 ( 1 ) (5) α 0 = 0, α 1 = 1, α =, α 3 = i (01i) = (10i) = (i10) = (i01) = 3 + i 4 (4) ζ = ξ R 1 {ξ, ξ } (1 ξ )(1 λξ ) ξ dξ = R {ξ, } (1 ξ )(1 λξ ) dξ R λ k (1 ξ )(1 k ζ ) 3. α β α β β k () α 0 α 1 α α k 1 1 k (3), φ(ζ) = c(1 ζ )(1 k ζ) α 1 α 0 α 1 α α 3 α α 3 α 0 = 1 k k + 1 k k 1 k λ λ = 1 k 1 1 k k k 1 = ( ) 1 k 1 + k

16 8 k = 1 λ 1 + λ 1 + λ 1 λ λ λ k α 1 z α 0 α 3 α = ζ 1 1 k + 1 z α α 3 α 0 ζ k 1 = ζ 1 1 k ζ k = ± ζ 1 λ ζ + 1 (1) R 1 {ζ, } c(1 ζ )(1 k ζ ) dζ. c c = 1 { R z, } (1 z )(1 k z ) dz (1) (1) dz I 0 = (1 z )(1 k z ), I 1 = I = z dz (1 z )(1 k z ), J 1 = zdz (1 z )(1 k z ), I 1 z = t I 1 = 1 dt (1 t)(1 k t) dz (z a) (1 z )(1 k z ). () I I = 1 k { } dz 1 (1 z )(1 k z ) k z 1 z dz. I 0 I 1 k z 1 z dz J 1 zdz J 1 = (z a ) (1 z )(1 k z ) dz + a (z a ) (1 z )(1 k z )

17 3. 9 z = t J 1 dz (z a ) (1 z )(1 k z ) () dz (1 z )(1 k z ), (1 k z ) (1 z dz, ) dz (z a ) (1 z )(1 k z ). φ(z) α φ(z) = 0 (3) ( α φ(z) φ(z) = 0 z 4 0 ) dz (1 z )(1 k z ), (1 k z ) (1 z dz, ) dz (z a ) (1 z )(1 k z ). k a β Legendre-Jacobi z(1 z)(1 λz) dz zdz,, z(1 z)(1 λz) z(1 z)(1 λz) Riemann dz (z a) z(1 z)(1 λz). Legendre-Jacobi 1. dz z4 + 1 φ(z) = z = 0 α 0 = 1 + i, α 1 = 1 + i, α = 1 i, α 3 = 1 i (3) λ = α 1 α 0 α 1 α α 3 α α 3 α 0 = 1, k = 1 λ λ =. + 1

18 10 z 1+i z 1 i dz z4 + 1 = ( )i = 1 + i ζ 1 ζ + 1. dζ 1 (1 ζ )(1 k ζ ), k = + 1 z ζ. dz 1 z 3 α 0 = 1, α 1 = 1 + 3i, α =, α 3 = 1 3i λ = α 1 α 0 = 1 3i, α 3 α 0 3 i λ =, k = 1 λ 1 + λ = ( 3)i. z α 0 = ζ 1 λ, α α 0 ζ + 1 z 1 = 3 ζ 1 λ ζ + 1 dz z + 1 = i 4 3(1 + 3) ζ dζ (1 ζ )(1 k ζ ), k = ( 3)i

19 { R z, } φ(z) dz φ(z) z z ζ φ(z) = 0 α 0, α 1, α, α 3 ζ k 0 < k < 1 φ(z) φ(z) = 0 φ(z) (I) α 0 < α 1 < α 3 < α. λ = α 1 α 0 α 1 α α 3 α α 3 α 0 > 0. ( 0 < k 1 ) λ = 1 + < 1 λ z ζ z α 0 α 3 α = λ ζ 1 z α α 3 α 0 ζ + 1 (1) z ζ

20 1 (II) α 0 α α 0 α 3 } α 1 α 0 α 3 α 0 () α 1 α α 3 α λ = α 1 α 0 α 3 α α 1 α α 3 α 0, λ = 1. λ = e iθ (θ) k (3) k = 1 λ 1 + λ = 1 e iθ 1 + e iθ = eiθ e iθ e iθ = i tan θ. + e iθ z ζ (1) z α 0 α1 α α 3 α = ζ 1 z α α 1 α 0 α 3 α 0 ζ + 1 (3) () (3) z ζ 1 k ζ = t k < 0 t dζ (1 ζ )(1 k ζ ) = 1 dt k 1 (1 t )(1 h t ) h = 1 1 k 0 < h < 1 ( h (III) (III) α 0 α α 1 α 3 α 1 α 0 α 3 α 0 α 1 α α 3 α (3) k = 0 λ = 1 α k = 1 k 0 1

21 5. 13 λ = α 1 α 0 α 1 α α 3 α α 3 α 0 > 0, ( 0 < k 1 ) λ = 1 + < 1 λ z ζ (3) 1 z α 0 z α = ζ 1 ζ + 1 z 1 ζ 1 = ζ + 1 ζ 1 1 iζ 1 ζ = t t dζ (1 ζ )(1 k ζ ) = i dt (1 t )(1 h t ) h = 1 k 0 < h < 1 1 (III) (II) 0 < k < 1 k k a 5. φ(z) z 1 (i), (ii) R(z, s), s = φ(z)

22 14 φ(z) = 0 α 0, α 1, α, α 3 (φ(z) α ) R(z, s) α 0, α 1, α, α 3 Riemann (α 0, α 1 ) (α, α 3 ), α 0 α 1 α α 3 α 0 α 1 α 0 α 1 α 0 α 1 α α 3 R(z, s) Riemann R 4 5 R R(z, s) R(z, s) z (1) () t (1) R α α = z a = t α = 1 z = t () R α α = z a = t α = 1 z = t R(z, s) t t = 0 z = a z = α s = (z α 0 )(z α 1 )(z α )(z α 3 ) z = α 0 () z α 0 = t s = t (α 0 α 1 + t )(α 0 α + t )(α 0 α 3 + t )

23 s α 0 1 s α 0 R(z, s) R Cauchy R(z, s) R C R(z, s) dz = 0. (C) C R(z, s) a K K a R(z, s) 1 R(z, s) dz πi (K) a R(z, s) K ( a ) a a 8 t R(z, s) R(z, s) dz dt t t = 0 a R(z, s) = + = + c t c (z a) + c 1 z a + c 0 + c 1 (z a) + c (z a) + + c 1 t + c 0 + c 1 t + c t +, c R(z, s) z = a c 1 R(z, s) dz dt t = 0 a R(z, s) = + c z a + c 1 (z a) + c 1/ 0 + c 1 (z a) 1 + c (z a) + = + c t + c 1 t + c 0 + c 1 t + c t +, a R(z, s) c R(z, s) dz dt t = 0 R(z, s) R(z, s) 0 R R(z, s) 0 R(z, s) R 0 R(z, s) R

24 16 R R(z, s) R R R(z, s) R Liouville 6. R(z, s) R(z, s) R F (z) = z a R(z, s) dz (a ) R R R(z, s) 0 a z C, C R R(z, s) 0 A R(z, s) dz = R(z, s) dz ± πia (C) (C ) F (z) z F (z) F (z) R Cauchy R L R R ( R(z, s) ) R(z, s) dz = 0 (L). C 1, C L R(z, s) dz = R(z, s) dz (C 1) (C ) F (z) R

25 6. 17 R 6 ( ( α 0 α 1 α α α, α R R 9 R R B A,

26 18 10 R A, B R R A, B R R A, B A, B 1,,3,4 R R Cauchy F (z) R A B F (z) R(z, s) 0 R(z, s) 0 F (z) R A, B, F (z) R (R ) a z F (z) R R A B

27 A, B 1 B A q R(z, s) 0 c z R C 1 A p ) C F (z) F 1, F R F (z) c z R C 1, C 1 F (z) ( c z C 1 C 1 ) C 1 R A, B, p z F = = +, F 1 = (C ) (C 1 ) = c p p = (C 1 ) c + q p + + (B) q B q p p q A F 1 = p c p z + + = F +, (B) q (B) q + z p, F = F 1 (B) A A p (B) C 1 C A (B) A B F 3 F 3 = F 1 ± ( A) ( A) A B +

28 0 A A F (z) (Periodicity-modulus) (B) P (A) B F (z) ( A) P (B). P (A) = (B), P (B) = ( A) = (A) R A m 1 m B n 1 n F 1 + (m 1 m )P (A) + (n 1 n )P (B) R(z, s) 0 0 c 1 c, l 1, l, πi(l 1 c 1 + l c + ) z R(z, s) dz z R z c u z c R(z, s) dz = u + mp (A) + np (B) + πi(l 1 c 1 + l c + ), P (A), P (B) c 1, c, m, n, l 1, l, 7. R () 5 t R(z, s)dz ( R(z, s)dz = + c t + c ) 1 + c 0 + c 1 t + c t + dt, t c R c 1 = 0 R(z, s) 0 F (z) AB R c 1 R F (z) = R(z, s) dz = c t = 0 F (z) + c 1 log t + c 0 t + c 1 t + c 1 = 0 c 1 c, c 3, c 0 F (z) R

29 7. 1 c, c 3, 0 F (z) R R R R ) R F (z) 3 F 1 (z) = dz (1 z )(1 k z ) R dz (1 z )(1 k z ) = (c 0 + c 1 t + c t + )dt R(z) F (z) = 1 k z 1 z dz R F (z) z = 1 t 1 k z 1 z dz = 1 t (c 0 + c t + )dt F (z) F 3 (z) = dz (z a ) (1 z )(1 k z ) z = ±a F 1 (z) z = ±a dz (z a ) (1 z )(1 k z ) = 1 t (c 0 + c 1 t + )dt F 3 (z)

30 8. φ(z) z dz s (s = φ(z)) F 1 (z) dz s R(z, s) dz s = φ(z) t R a z a = t, dz = dt, α z α = t, dz = tdt, z α = 1, dz = dt t t, R(z, s)dz t R(z, s) dz R(z, s) z = a z = α z = s z = a z = α z = s R(z, s) R 5 C R(z, s) = C s (1) φ(z) z = 1, dz = dt t t 3

31 9. 3 (1) dz C s C s dz s z s φ(z) z z φ(z) φ(z) dz s R(z, s) dz z z s 9. dz (1 z )(1 k z ) 4 0 < k < 1 ( 1 ) k a, b, b a = 1 a 1 b 1, z u = 1 z dz (1 z )(1 k z ) (1) k < 1 z 1 k ()

32 4. () k z < k z = k z k4 z (n 1) k n z n +. 4 n z 1 k < 1 k (1) u = 1 z dz z k z dz z 4 k4 z (4) 1 z 1 z 1 z + dz 1 z (n 1) 1 k n dz 4 n +. z 1 z dz 1 z = 1 i log (z + i 1 z ), z dz 1 z = 1 z 4 dz = z 4 1 z 1 z dz + z 1 z, 1 z dz + z 1 z 1 z 4 ( z + 3 ), 1 z z n dz 1 3 (n 1) = 1 z 4 n + z 1 z n 1 z dz 1 z { z n + n 1 n (n 1)(n 3) + (n )(n 4) zn 6 + } (n 1) n 4 u = 1 i L 0 log (z + i 1 z ) (3) + ( 1 z L 1 z + 3 L 3z ) 3 5 L 5z 5 + (4) (4) z = sin θ

33 9. 5 ( ) ( ) ( ) L 0 = 1+ k + k 4 + k 6 +, ( ) ( ) ( ) L 1 = k + k 4 + k 6 +, ( ) ( ) L 3 = k 4 + k 6 +, ( ) L 5 = k 6 +, 4 6 k < 1 L u (3) 1 z u (3) log ( )u πl 0 (3) u 0 nπl 0 ± u 0 (n ) (5) (1) P (A), P (B) u 0 u 0 + mp (A) + np (B) (m, n ) (6) (5) (6) (5) u 0 (6) 6 z R z R (1) z R z z z (1) (5) 13 (3) u 1 k () 1 R ( 1, 1 k ) ( 1, 1 k ) (1 z )(1 k z ) u () z 1 C 1, C, C 3 C 1 u 0 C C 1 C 3 C 1

34 6 1, 1 (C ) = (C 1) dz dz 1 0 dz (1 z )(1 k z ) = K 1 1 dz 1 dz = = K. 1 (C 3 ) = u 0 + 4K. 1 4K u 0 + 4nK (n ) z Riemann (3) u 4πK±u 0 (n ) (5) K π L 0 (5) 6 P (B) = ( A) = 1 1 dz 1 dz = 1 = 4K. (5) u (6) P (A) P (A) P (B) A ( 1, 1 k ), ( 1, 1 k ) ± 1 k B 1 1 u A B (3) 1 k 1 k P (A) ±P (A)

35 () dζ c 1 (c 1 ) (1) ζ(1 ζ)(1 λζ) (1) λ x + y 1, x 1 ζ = ξ (1) dξ c ξ(1 ξ )(1 λξ ) (c ) () ζ ξ R(ξ 4 λ) 0 (3) 4 λ () 1 4 λ λ = h hη = 1 4 λξ λξ. (4) dη c 3 (c 3 ) (5) (1 η )(1 h η ) (5) () (3) 4 λξ 4 λξ = X + Y i X 0. (4) hη = 1 X Y i 1 + X + Y i = (1 X) + Y (1 + X) + Y 1.

36 8 () h < 1 h = 1 4 λ λ < 1. λ h λ h h < 1 λ λ = cos θ + i sin θ, 4 λ = cos θ 4 + i sin θ 4 π 3 θ π 3 14 ( ) cos 1 θ h = 4 + sin θ 4 ( ) cos 1 + θ 4 + sin θ 4 = tan θ 8 tan π 4 < 1. λ λ = 1 + i tan θ, π 3 θ π 3 θ λ h 0 θ π 3 h θ d dθ h > 0 h = e R(log h) d dθ R(log ( d h) = R dθ log ) h > 0 log h = log(1 λ 1 4 ) log(1 + λ 1 4 )

37 10. 9 d dθ log h = 1 ( λ s 4 + λ s λ λ 1 4 λ s 4 = 1 1 λ 1 dλ dθ = 1 λ λ s 4 dλ 1 λ dθ. ) dλ dθ { dλ dθ = 1 sec θ = i (1 + tan θ) = i 1 + = iλ(1 λ). ( ) } λ 1 i d dθ log h = 1 λ λ s 4 iλ(1 λ) 1 λ = i(λ s λ 4 ). λ = 1 + i tan θ = cos θ + i sin θ cos θ cos θ λ 1 4 = 4 + i sin θ 4 cos 3θ 4, λ s 4 = 4 + i sin 3θ 4 cos θ 4 (cos θ) 3 ( d R dθ log ) h = sin θ cos θ 3θ sin 4 4 (cos θ) 3 0 θ = 0 h 0 θ π 3 θ θ = π 3 tan π 4 1 (5)

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39 31 Jacobi 11. φ(z) u = z dz φ(z) (1) z u 0 u 0 + mp (A) + np (B) Q u 0 + mp (A) + np (B) (m, n ) P (A), P (B) Q z u z = f(u) f(u) u f (u) f (u) = dz du = φ(z) () z f(u) z f(u) = f(u) () u u = f(u) f(u) u = f(u) F (u) ω F (u + ω) = F (u) u ω f(u) mp (A) + np (B) (m, n ) u u z f{u + mp (A) + np (B)} = f(u)

40 3 Jacobi f(u) 1. f(u) 1. f(u) ω ω f(u) f(u + ω) = f(u), f(u + ω) = f(u + ω + ω) = f(u + ω) = f(u). ) f(u) ω 1 ω, m 1 ω 1 + m ω (m 1, m ) f(u) m 1 ω 1 m ω 1 f(u + m ω ) = f(u) u u + m 1 ω 1 f(u + m 1 ω 1 + m ω ) = f(u + m 1 ω 1 ) = f(u). f(u) ω 1, ω, ω 3, m 1 ω 1 + m ω + m 3 ω 3 + f(u) sin u u = nπ(n ) 3. f(u)

41 1. 33 a ϵ ω 1 a < ϵ, ω a < ϵ ω 1, ω ω 1 ω < ϵ ω 1 ω ϵ f(u) G u 0 ω, ω, f(u 0 ) = f(u 0 + ω ) = f(u 0 + ω ) = u 0 + ω, u 0 + ω, G c f(u) = c f(u) 3 f(u) f(u 0 ), f(u 0 + ω ), f(u) f(u) = c f(u) ω ω 0 ω ω, 3ω, 3 ω 0 ω 0 ω 0 Ω Ω = nω 0 + ω, 0 < ω < ω 0 n ω ω Ω nω 0 ω ω 0 ω f(u) ±ω 0

42 34 Jacobi 4. ω 0 ω 0 (.) 0ω 0 ω 1 0ω 0, 0ω ω 0 0, ω 0 ω 0ω 0 0ω 0ω 0 ω 0 0 ω 0ω 0 mω 0 + nω (m, n ) ω 0 ω 0ω 0, 0ω f(u) ω 0, ω 5. f(u) (ω, ω ) (Ω Ω ) f(u) m, n, µ, ν mω + nω µω + νω (Ω Ω ) (ω, ω ) (ω, ω ) (Ω Ω ) } } Ω = aω + bω ω = αω + βω Ω = cω + dω (1) ω = γω + δω () a, b, c, d α, β, γ, δ () (1) } Ω = (aα + bγ)ω + (aβ + bδ)ω Ω = (cα + dγ)ω + (cβ + dδ)ω (3)

43 1. 35 Ω Ω µω + νω f(u) (3) (aα + bδ) aβ + bδ 0 Ω Ω aβ + bδ aα + bν = 1 ( ) aα + bγ aβ + bδ = cα + dγ cβ + dδ ( ) aα + bγ cα + dγ aβ + bδ cβ + dδ = a c b d α γ β δ = 1,, ad bc = ±1 αδ βγ = ±1 f(u) (ω, ω ) (aω + bω, cω + dω ) (a, b, c, d ad bc = ±1) Ω = aω + bω, Ω = cω + dω ±ω = dω bω, ±ω = cω + aω mω + nω Ω, Ω (Ω Ω ) ad bc = 1 a, b, c, d 6. (ω, ω ) (aω + bω, cω + dω ) a, b, c, d ad bc = ±1

44 36 Jacobi mω+nω (ω, ω ) 0ω, 0ω z dz u = z = f(u) φ(z) f{u + mp (A) + np (B)} = f(u) (m, n ) f(u) P (A) P (B) f(u) f(u) ω R z u z c dz φ(z) = u + uω (n ) Φ(z) = e πi u ω z R z R u u + nω πi u ω nπi Φ(z) u

45 14. sn 37 z Φ(z) R 5, u P (A) P (B) 0 f(u) u R f(u) P (A), P (B) 0 P (A) : P (B). u u 0 + P (A) u 0 + P (B) f(u) u 0 = 0 0, P (A), P (B), P (A) + P (B) z = f(u) 14. sn u = z 0 dz (1 z )(1 k z ) (1) z = sn u z = 0 +1 sin sine sn k k sn (Modulus) k z = sn(u, k) 1 k = k k (complementary modulus) k = 0 (1) u = z 0 dz 1 z = sin 1 z sn(u, 0) = sin u.

46 38 Jacobi k = 1 (1) u = z 0 dz = 1 1 z log 1 + z 1 z sn(u, 1) = eu 1 e u = tanh u. + 1 sn u sn u P (A) P (B) P (A), P (B) 6 P (A) = (B), P (B) = ( A) P (B) P (B) (B) A B (A) A 1, B 1, 1 k R R = (A) dz 1 (1 z )(1 k z ) = 4 dz (1 z )(1 k z ). 0 (B) 1 = 1 k 1 dz (1 z )(1 k z ) = k 1 dz (1 z )(1 k z ), z = 1 1 k t, k = 1 k (B) 1 dt = i 0 (1 t )(1 k t ), 1 0 dz 1 (1 z )(1 k z ), dz (1 z )(1 k z ) 0 K, K 4K, ik sn u

47 14. sn 39 4K ik sn u (1) z z u u sn( u) = z = sn u. () sn u u u = 4K + ik u u u K + ik sn u = sn u ( K 19 ) sn u (1) z u z ±1, ± 1 k R R z ( ±1, ± 1 k (1) 0 z u + 4mK + nik (m, n ) 0 z K u + 4mK + nik 9 0 z R z 1 0 dz dz z + 0 dz 1 = 0 = K dz z 0 z dz 0 dz. z u u u = K + ik u K ik (0, K) (0, ik )

48 40 Jacobi sn u = sn u = sn(k + ik u) = sn(k u) sn(k u) = sn u, (3) u u () sn(k + u) = sn u. (4) K sn K = 1, sn 0 = 0 (4) sn K = 0, sn 3K = 1, sn 4K = 0. sn u sn u 0, K 4K 0 sn u U = 0 dz (1 z )(1 k z ) z = it 1 t 1 dt U = i 0 (1 t )(1 k t ) = ik (4) sn ik = sn(k + ik ) =. ik K + ik (B) 1 k 1 dz (1 z )(1 k z ) = ik 1 k 0 1 = 0 1 k 0 K 1 k 0 dz (1 z )(1 k z ) = K + ik,

49 15. cn,dn 41 sn(k + ik ) = 1 k. (4) sn(3k + ik ) = 1 k. sn u 0. 0 sn u ±1, ± 1 k cn,dn (1) u = z 0 dz (1 z )(1 k z ) (1) du dz = 1 (1 z )(1 k z ). d dz sn u = du du = (1 z )(1 k z ) = (1 sn u)(1 k sn u) () 1 sn u (3) 1 sn u = 0 u = K, 3K u = K (3) Taylor φ(u) = 1 sn u () φ (u) = sn u (1 sn u)(1 k sn u), φ (u) = sn{1 (1 + k ) sn u + 3k sn 4 u} φ(k) = 0, φ (K) = 0, φ (K) = k

50 4 Jacobi φ(u) = k (u K) + c 4 (u K) 4 + (c 4, ) 1 sn u = k (u K) 1 + c 4 k (u K) + { = ±k (u K) 1 + c } 4 k (u K) + u = K (3) 0 u = 3K (3) 1 sin u ± cos u 1 sn u = cn u u = k sn u = dn u ( dn 0 = 1 cn, dn,sn () d sn u = cn u dn u. (4) du cn u, dn u d du cn u = d sn u cn u dn u 1 sn u = du 1 sn = sn u dn u, u d du dn u = d 1 k sn du u = k sn u cn u dn u 1 k sn u = k sn u dn u. (5) (4) (5) sn u = u (1 + k ) u3 3! + (1 + 14k + k 4 ) u5 5!, cn u = 1 u! + (1 + 4k ) u4 4!, dn u = 1 k u! + (4k + k 4 ) u4 4!. 16. f(u) f(u) f(v) f(u + v) () u, v

51 e u e u+v = e u +e v sin u sin(u+v) = sin u cos v + cos u sin v sin cos u = ± 1 sin u sn u dx (1 x )(1 k x ) + dy = 0. (1) (1 y )(1 k y ) x dx u = (1 x )(1 k x ), v = 0 y 0 dy (1 y )(1 k y ) () u + v = c (c ) (3) (1) () dx du = (1 x )(1 k x ), dy dv = (1 y )(1 k y ) (3) dy du = dy dv = (1 y )(1 k y ). d x du = (1 + k )x + k x 3, d y du = (1 + k )x + k y 3, y d x du y ( d x du ) x x d y du k xy ( ) = d y 1 k x y du y d x du x d y du y dx du x dy du ( ) k xy y dx du + x dy du = 1 k x y, u ( log y dx du x dy ) = log(1 k x y ) + c (c ) du y dx du x dy du 1 k x y = C (C ) (4)

52 44 Jacobi (4) u, v x = sn u, y = sn v, dx dy = cn u dn u, du (4) = cn v dn v du sn u cn v dn v + sn v cn u dn u 1 k sn u sn v = C (5). (5) (1) (1) (3) (5) (5) f(u + v) v = 0 sn u = f(u) f sn sn(u + v) = sn (I) D sn u cn v dn v + sn v cn u dn u 1 k sn u sn. (I) v D = cn u + sn u dn v = cn v + sn v dn u cn (u + v) = 1 sn (u + v) = D (sn u cn v dn v + sn v cn u dn u) D, D = (cn u + sn u dn v)(cn v + sn v dn u) cn(u + v) = cn cn u cn v sn u sn v dn u dn v 1 k sn u sn. (II) v D = dn u + k sn u cn v = dn v + k sn v cn u dn(u + v) = dn u dn v k sn u sn v cn u cn v 1 k sn sn v (III) dn sn K = 1, cn K = 0, dn K = k

53 sn(u + K) = cn u dn u sn(u + K) = sn u cn(u + K) = k sn u dn u cn(u + K) = cn u dn(u + K) = k dn(u + K) = dn u dn u sn(u + 3K) cn(u + 3K) = cn u dn u = k sn u dn u dn(u + 3K) = k dn u sn(u + 4K) cn(u + 4K) dn(u + 4K) = sn u = cn u = dn u (I) (II) (III) sn u sn v sn(u + v) = sn u cn v dn v sn v cn u dn u sn u cn u dn v sn v cn v dn u cn(u + v) = sn u cn v dn v sn v cn u dn u sn u cn v dn u sn v cn u dn v dn(u + v) = sn u cn v dn v sn v cn u dn u sn u = sn u cn u dn u 1 k sn 4 u cn u = 1 sn u + k sn 4 u 1 k sn 4 u dn u = 1 k sn u + k sn 4 u 1 k sn 4 u sn u = 1 cn u 1 + dn u cn u = dn u + cn u 1 + dn u dn u = k + dn u + k cn u 1 + dn u sn(u + v) sn(u v) = sn u sn v 1 k sn u sn v cn(u + v) cn(u v) = cn u sn v dn u 1 k sn u sn v dn(u + v) dn(u v) = dn u k sn v cn u 1 k sn u sn v

54 46 Jacobi 17. u = v sn u, cn u, dn u sn u, cn u, dn u v = u sn 3u sn u n n sn nu = xa n(x ) D n (x ) cn nu = yb n(x ) D n (x ) dn nu = zc n(x ) D n (x ) (1) n sn nu = xyza n(x ) D n (x ) cn nu = B n(x ) D n (x ) dn nu = C n(x ) D n (x ) () x = sn u, y = cn u, z = dn u, A n, B n, C n, D n x n = 1, A 1 = 1, B 1 = 1, C 1 = 1, D n = 1, A = B = 1 x + k x 4 C = 1 k x + k x 4 D = 1 k x 4 u, v nu, nu nu, (n + 1)u (1) () sn u, v nu n sn nu = sn nu cn nu dn nu 1 k sn 4, nu xyza n D n = xan D n yb n D n zc n D n 1 k x 4 A n 4 D n 4 = xyza nb n C n D n D n 4 k x 4 A n 4,

55 A n = A n B n C n D n, D n = D 4 n k x 4 A 4 n. n B n D n C n D n xyza n = xyzan D n D n 1 k x 4 y 4 z 4 A 4 n D n 4 = xyza nb n C n D n D 4 n k x 4 y 4 z 4 A n 4, A n = A n B n C n D n, D n = D 4 n k x 4 y 4 z 4 A 4 n.. n A n = A n B n C n D n B n = y B n D n x z A n C n C n = z C n D n k x y A n B n D n = D 4 n k x 4 4 A n A n+1 = A n B n C n D n + y z A n+1 B n C n D n+1 B n+1 = B n B n+1 D n D n+1 x z A n A n+1 C n C n+1 C n+1 = C n C n+1 D n D n+1 k x y A n A n+1 B n B n+1 D n+1 = D n D n+1 k x 4 y z A n A n+1 n A n = A n B n C n D n B n = B n D n x y z A n C n C n = C n D n k x y z A n B n D n = D 4 n k x 4 y 4 z 4 4 A n A n+1 = y z A n B n+1 C n+1 D n + A n+1 B n C n D n+1 B n+1 = B n B n+1 D n D n+1 x z A n A n+1 C n C n+1 C n+1 = C n C n+1 D n D n+1 k x y A n A n+1 B n B n+1 D n+1 = D n D n+1 k x 4 y z A n A n+1

56 48 Jacobi A 3,, A 4, A 3 = 3 4(1 + k )x + 6k x 4 k 4 x 8 B 3 = 1 4x + 6k x 4 4k 4 x 6 + k 4 x 8 C 3 = 1 4k x + 6k x 4 4k x 6 + k 4 x 8 D 3 = 1 6k x 4 + 4k (1 + k )x 6 3k 4 x 8 A 4 = 4 8(1 + k )x + 0k x 4 0k 4 x 8 + 8k 4 (1 + k )x 10 4k 6 x 1 B 4 = 1 8x + 4( + 5k )x 4 8k (3 + 4k )x 6 + k 4 (7 + 8k )x 8 8k 4 (3 + 4k )x k 4 ( + 5k )x 1 8k 6 x 14 + k 8 x 16 C 4 = 1 8k x + 4k (5 + k )x 4 8k (4 + 3k )x 6 + k (8 + 7k )x 5 8k 4 (4 + 3k )x k 6 (5 + k )x 1 8k 8 x 14 + k 8 x 16 D 4 = 1 0k x 4 + 3k (1 + k )x 6 k (8 + 9k + 8k 4 )x 8 + 3k 4 (1 + k )x 10 0k 6 x 1 + k 8 x 16 A n, A n+1, A n+1 D n+1, 18. sn 15 A(x, y, z) B(x, y, z) (1) x = sn u, y = cn u, z = dn u, A, B y = 1 x, z = 1 k x A B y, z (1) B(x, y, z)b(x, y, z)b(x, y, z) x y, z (1) R 1 (x) + yr (x) + zr 3 (x) + yzr 4 (x) () R () yzr 4 (x) du = R 4 (x) dx ( dx ) du = yz.

57 R (x) yr (x) du = 1 k x dx, R 3 (x) zr 3 (x) du = dx 1 x R 1 (x) x m, (x a) m, m 0 a d du (xm 3 yz) = (m 3)x m 4 y z x m z k x m y = (m 1)k x m (m )(1 + k )x m + (m 3)x m 4, I m = x m du x m 3 yz = (m 1)k I m (m )(1 + k )I m + (m 3)I m 4, I m I = sn u du, I 1 = sn u du (3) x a = t d du (t m+1 yz) = ( m + 1)t m y z t m+1 x(z + k y ) = (m 1)(1 a )(1 k a )t m + (m 3)(1 + k k a )at m+1 + (m )(1 + k 6k a )t m+ (m 5)k at m+3 (m 3)k t m+4 (4) J m = du t m J m J 1, J 0, J 1, J J 0 = du, J 1 = (x a) du, J = (x a) du

58 50 Jacobi (3) du J 1 = sn u a (5) a (1 a )(1 k a ) = 0 (4) J 1 J 0, J 1, J (5) (3) (5) I 1 u = v I 1 = sn du = sn v cn v dn v sn v dv = 1 k sn 4 v dv. sn v = t I 1 = dt 1 k t = kt log k 1 kt. I 1 = 1 k log 1 + k sn u 1 k sn u. sn u = x I 1 = sn u du = xdx (1 x )(1 k x ). x = t I 1 = 1 dt (1 t)(1 k t) = 1 k log(k 1 t + 1 k t) = 1 log(k cn u + dn u). k I J 1 sn u = x I = x dx (1 x )(1 k x ), J xdx 1 = (x a)(1 x )(1 k x ) ( 3 ) < k < 1 4 sn, cn, dn u = x 0 xdx (1 x )(1 k x ), 0 < k < 1

59 u = φ 0 x = sin φ (1) dφ 1 k sin φ φ x 0 x 1 u φ du dφ φ u (Amplitude) φ = am u () K am K = π am(nk) = nπ (1), () (n ) x = sin φ = sin am u, sn u = sin am u cn u = cos am u, dn u = am u 1 (t) 1 k sin t u am u u sn u, cn u, dn u sin u, cos u, u. u sn u x = it 1 t (3)

60 5 Jacobi x 0 dx t (1 x )(1 k x ) = i dt (1 t )(1 k t ) 0 iu (3) x = sn(iu, k), t = sn(u, k ) sn(iu, k) = i sn(u, k ) cn(u, k ), cn(iu, k) = 1 sn (iu, k) = 1 cn(u, k ), dn(iu, k) = 1 k sn (iu, k) = dn(u, k ) cn(u, k ) dn (u, k ) = 1 k sn (u, k ) sn(u, k) sn u sn(u, k ) snu sn(iu) = i snu cnu, cn(iu) = 1 dnu, dn(iu) = cnu cnu. sn(u + iv) sn u dnv + i cn u dn u snv cnv sn(u + iv) = cn v + k sn u sn, v cn u cnv i sn u dn u snv dnv cn(u + iv) = cn v + k sn u sn, v dn(u + iv) = dn u cnv dnv ik sn u cn u snv cn v + k sn u sn. v v = K, K sn(u + ik ) = 1 1 k sn u sn(u + ik ) = sn u cn(u + ik ) = i dn u cn(u + ik ) = cn u k sn u dn(u + ik ) = i cn u dn(u + ik ) = dn u sn u sn 4K, ik cn, dn 4K, K + ik 4K, ik

61 0. Landen 53 sn, cn, dn K ik. sn K = 1, sn ik =, sn(k+ik ) = 1 k, 3 u 0 ik R(sn u) = 0, I(sn u) > Landen 4 C c C c AB R, r δ δ < R r T F F ACF = φ, ACF = φ F T = F c r = (R + δ + Rδ cos φ) r = (R + δ) r 4Rδ sin φ. 4Rδ (R + δ) r = k (1) F T = (R + δ) r 1 k sin φ F T = (R + δ) r 1 k sin φ, F T : F T = 1 k sin φ : 1 k sin φ () F F GG ( 5 φ φ dφ, dφ F F GG L ( dφ : dφ = F G : ( F G.

62 54 Jacobi 4 5 dφ, dφ ( F G : ( F G = F G : F G = F L : LF = F T : T F. ( dφ : dφ = F T : ( T F. () dφ 1 k sin φ = dφ (3) 1 k sin φ 0 < k < 1 (1) δ < R r φ φ (3) k R, r, δ k R r 0 δ = 1 k 1 + k R, k = 1 k 4Rδ (R + δ) = k, AB δ CO O c O F F ACF = φ, ACF = φ (3) 6 T O F O = (R + δ) 1 k sin φ. (4) AOF = φ 1 CF : CO = sin φ 1 : sin(φ φ 1 ).

63 0. Landen 55 CF = R, CO = k 1 R ( k 1 = 1 ) k 1 + k sin(φ φ 1 ) = k sin φ 1 (5) φ 1 = φ + φ π dφ 1 = dφ + dφ, dφ 1 dφ = 1 + dφ dφ = 1 + OF F O = F F F O. (6) F O (4) F F F F = F O + OF F O OF OF OF = OC cos φ 1 = Rk 1 cos φ 1 OF OF = OA OB = R(1 + k 1 ) R(1 k 1 ) = R (1 k 1 ). OF + OF = (OF OF ) + 4 OF OF = 4R k 1 cos φ 1 + 4R (1 k 1 ) = R 1 k 1 sin φ 1. (4) (6) dφ 1 dφ = R 1 k1 sin φ 1 R + δ 1 k sin φ, R R + δ = 1 + k 1 dφ 1 k sin φ = 1 + k 1 dφ = 1 (7) 1 k1 sin φ 1 (7) k 1 = 1 k 1 + k k = 1 k k 1

64 56 Jacobi ( ) 1 k k1 4k 1 = 1 = 1 + k 1 (1 + k 1 ). ( ) k 4 = k 1 (1 + k 1 ) > 1, k 1 0 < k 1 < k (7) (5) (7) (5) Landen (5) k 1 = sin(φ φ 1) sin φ 1, k = 1 k k 1 = sin φ 1 sin(φ φ 1 ) sin φ 1 + sin(φ φ 1 ) = sin(φ 1 φ) cos φ sin φ cos(φ 1 φ). tan(φ 1 φ) = k tan φ. (8) (5) Landen sn (3) φ φ C F F f F T + F T = F f = R sin(φ φ), F T F T = ft = δ sin(φ + φ). 1 k sin φ + 1 k sin φ sin(φ φ) 1 k sin φ 1 k sin φ sin(φ + φ) = k sin α, sin α = 1 1 k sin α. sin α α φ = 0 φ sin φ sin φ sin φ cos φ 1 k sin φ + sin φ cos φ = sin α (9) 1 k sin φ φ, φ (3) α

65 1. 57 v = φ 0 dφ φ 1 k sin φ, u = dφ 1 k sin φ (3) u v = C 1 (9) sn u sn v sn u cn v dn v + sn v cn u dn u = C () sn(u v) = = 0 sn u sn v sn u cn v dn v + sn v cn u dn u sn u cn v dn v sn v cn u dn u 1 sn u sn. v 1. Landen 0 < k < 1 φ, φ 1 φ 0 tan(φ 1 φ) = k tan φ (1) dφ 1 k sin φ = 1 + k 1 φ1 0 dφ 1 1 k1 sin φ 1 () k 1 = 1 k 1 + k < k (3) φ φ 1 φ φ φ φ 3 φ 0 dφ 1 k sin φ = 1 + k k 1 + k n φn 0 dφ n 1 kn sin φ n, (4) tan(φ i φ i 1 ) = k (i) tan φ i 1 (i = 1,,, n; φ 0 = φ), k i = 1 k i k i 1 < k i 1 (i = 1,,, n; k 0 = k) k i i 0 (4) ( 9 ) k n = 0 φ 0 dφ 1 k sin φ = 1 + k k 1 + k n φ n (5)

66 58 Jacobi k k 1, k, (3) a, b (a > b) k = b a k 1 ( ) 1 k = k = 1 a 1 = a + b, b 1 = ab k 1 = b 1 a 1 ( ) a b = 4ab a + b (a + b). a = a 1 + b 1, b = a 1 b 1, k = b, a a 3 = a + b, b 3 = a b, k 3 = b 3, a 3 a n = a n 1 + b n 1, b n = a n 1 b n 1, k n = b n, a n. a, b a n, b n b n 1 < b n < a n < a n 1 a n a n 1 b n 1 a n b n < a n b n 1 = a n 1 b n 1. a n b n < a b lim a n = lim b n = M n n lim n k n = 1, lim k n = 0 n k 1, k, (4) (5) (1 + k 1 )(1 + k ) (1 + k n ) 1 + k 1 = k 1 = ab (a + b) = a a + b = a a 1.

67 k == a 1 a,, 1 + k n = a n 1 a n. (1 + k 1 )(1 + k ) (1 + k n ) = a a n a M (5) φ dφ 1 k sin φ = a φ n a n n (6) 0 φn n φ φ 1 54 φ 1 φ φ φ 0 π π φ 1 0 π π φ φ 1 < π, φ 1 φ 1 < π. φ 1, φ φ 1 φ < π, φ 1 φ π < 4 φ n 1 n 1 φ n π < n φ n lim n n = Φ n (6) n φ 0 dφ 1 k sin φ = a M Φ. Φ (1) φ 1, φ, k n 1 φn n Φ φ = π φ 1 = π, φ = π,, φ n = n 1 π,, K = 1 0 Φ = π. π dx (1 x )(1 k x ) = 0 dφ 1 k sin φ = a π M. K

68 60 Jacobi k = 4 5 K k = 7 5 a = 5, b = 7 a 1 = b 1 = a = b = a 3 = b 3 = a 4 = b 4 = M = a π M =

69 61 Weierstrass. sn, cn, dn u ω 1, ω 3 ω 1 ω 3 ω ω = ω 1 + ω ω 1 + ω + ω 3 = 0 0, ω 1, ω 3 ω 3 I ( ω 3 ) ω 1 ω ( ) ω 3 ω 1 > 0 0, ω 1, ω 3 1, 3 I 0, ω 1, ω, ω 3 (0, ω 1 ) (0, ω 3 ) 0 ω f(u) f(u + ω i ) = f(u), (i = 1, 3).

70 6 Weierstrass f (u + ω i ) = f (u), (i = 1, 3) f (u) 1.. f(u) g(u) f(u) ± g(u), f(u)g(u), f(u) g(u) f(u) 0, ω 1, ω, ω 3 ω 1, ω 3 sn u 0, 4K, 4(K + ik ), 4iK f(u), g(u) 3. f(u) f(u) f(u) Liouville f(u)

71 f(u) (P ) P (P ) a, a + ω 1, a ω, a ω 3 a f(u) du = (P ) a+ω1 f(u) a a ω a+ω3 a a+ω 1 a ω a+ω 3 a+ω1 a + a+ω3 a ω = 0, a ω a + a+ω 1 a+ω 3 = 0. (P ) f(u) du = 0. P f(u) P πi f(u) b 4 (P ) A u b (A = 0) f(u) du = πia = 0. f(u) f (u) f(u) 1 4 (P ) f (u) f(u) du = 0 P f(u) πi

72 64 Weierstrass 5. n n 9 f(u) c f(u) c f(u) c = f(u) = f(u) c = 0 f(u) = c u f (u) f(u) du (P ) f(u) (1), (), (3), (4) = (P ) (1) () (3) (4) (3) = = a+ω a ω a = = = u f (u) f(u) du (v + ω 3 ) f (v + ω 3 ) a+ω 1 f(v + ω 3 ) dv a+ω1 a a+ω1 a (1) (v + ω 3 ) f (v) f(v) dv v f (v) f(v) dv ω 3 a+ω1 a f (v) f(v) dv ω 3 {log f(a + ω 1 ) log f(a)}. f(a + ω 1 ) f(a) log f(u) u a (1) a + ω 1 f(u) k log f(a + ω 1 ) log f(a) = kπi (1) + = ω 3 kπi (3) + = ω 1 hπi (h ). () (4) (P ) = πi(hω 1 kω 3 ) = π ()

73 4. 65 P f(u) πi Liouville 4. sn 4K, ik k sn sn (ω 1, ( ω 3 ) ω, I 3 ω 1 f(u) = 1 (u w) m, (w = h 1 ω 1 + h 3 ω 3 ; h 1, h 3 = 0, ±1, ±, ). 30 h 1, h 3 w w w = 0 ω w G u G f(u) u m u G u < M M w w > M u w w u > w M. 1 ( w M) m (1) Weierstrass 1 (u w) m

74 66 Weierstrass m 31 (1) 1 w m () w w = 0 w = 0 0. () r, R s 1 8 R m < s 1 < 8 r m s 16 (R) m < s < 16 (r) m 8n (nr) m < s n < 8n, (n = 1,, 3, ) (nr) m 8 R m 1 n m 1 < 1 w m < 8 r m 1 n m 1. 1 n m 1 > 1 m > m m 1 3 f(u) = 1 (u w) 3 (3) w (3) u u + ω 1 f(u + ω 1 ) = 1 {u (w ω 1 )} 3 (4) w w (w ω 1 ) (4) (3) f(u + ω 1 ) = f(u).

75 4. 67 f(u) ω 1 ω 3 (ω 1, ω 3 ) f(u) w w w w 0 1 f(u) = (u w 0 ) (u w) 3 w =w 0 u = w 0 u (3) G f(u) u = w 0 1 (u w 0 ) 3 w = h 1 w 1 + h w (ω 1, ω 3 ) w f(u) f(u) (3) w w f( u) = 1 ( u w) 3 = 1 (u + w) 3 = 1 (u w) 3 = f(u) f(u) 0 u w f(u) 0 0 f(u) f(u) 1 u 3 u {f(u) 1u } u { } 1 3 du = (u w) 3 du 0 0 = u 1 0 (u w) 3 du = 1 { 1 (u w) 1 w }. f 1 (u) = { 1 (u w) 1 } w w u u = w 1 (u w) w 0

76 68 Weierstrass w = 0 f (u) = 1 u + { 1 (u w) 1 } w f (u) f ( u) = 1 u + { 1 (u + w) 1 } w = 1 u + { 1 [u ( w)] 1 ( w) } = f (u) f ( u) = u 3 + { } (u w) { 3 1 = u 3 + } 1 (u w) 3 = 1 (u w) 3 = f(u) f(u) f(u + ω 1 ) = f(u), f (u + ω 1 ) = f (u). f (u + ω 1 ) = f (u) + C, C u = ω 1 f (u) f (ω 1 ) = f ( ω 1 ) + C = f (ω 1 ) + C. f (u) u = w f (ω 1 ) C = 0 f (u + ω 1 ) = f (u). f (u) ω 1 ω 3 f (u) f (u) Weierstrass (u) (u) = 1 u + { 1 (u w) 1 } w, (u) = 1 (u w) 3. ω 1, ω 3 (u ω 1, ω 3 ), (u ω 1, ω 3 )

77 4. 69 (u) (u) (I) m 0 (mu mω 1, mω 3 ) = m (u ω 1, ω 3 ), (mu mω 1, mω 3 ) = m 3 (u ω 1, ω 3 )., u, ω 1, ω 3 3 (II) (u) u (u) = 1 u + c 0 + c 1 u + c u c n u n + [ { 1 c 0 = (u w) 1 }] w n >0 u=0 = 0, c n = 1 [ d n { 1 (n)! du n (u w) 1 }] w = (n + 1) 1 w n+ c 1 c g = 0c 1 = 60 1 w 4, g 3 = 8c = w 6 (u) = 1 u + g 0 u + g 3 8 u4 +, (u) = u 3 + g 10 u + g 3 7 u3 +. (III) 4 (u) 3 (u) u=0 4 (u) 3 (u) = g u + g 3 + (u ) 1 u g (u) 4 (u) 3 (u) g (u) = g 3 + (u ) w w = u = 0 g 3 (u) (u) = 4 (u) 3 g (u) g 3.

78 70 Weierstrass (IV) (u) = z u = 0 z = dz du = 4z 3 g z g 3. u = z dz 4z3 g z g 3 (u) z dz sn u u = (1 z )(1 k z ). (V) 0 Weierstrass zdz 4z3 g z g 3, dz (z a) 4z 3 g z g 3 (u) 5 (u) (u) u = ω (u + ω i ) = (u) (i = 1,, 3) (ω i ) = ( ω i ) = (ω i ). ω i (ω i ) (ω i ) = 0. (i = 1,, 3) (u) ω 1, ω, ω 3 (u) (VI) (u) = 4 (u) 3 g (u) g 3. (u) (u) = 4{ (u) e 1 }{ (u) e }{ (u) e 3 } u = ω i (i = 1,, 3) (ω i ) = 0 (ω i ) e 1 = 0 (ω i ) e = 0 (ω i ) e 3 = 0 (ω i ) e 1 = 0 0 (ω i ) = 0 (u)

79 5. 71 ω i (ω i ) e 1 = 0 (ω i ) e = 0 (ω i ) e 3 = 0 (ω i )(i = 1,, 3) e 1, e, e 3 e (ω 1 ) = e 1, (ω ) = e, (ω 3 ) = e 3 e 1 + e + e 3 = 0, e e 3 + e 3 e 1 + e 1 e 3 = 1 4 g, e 1 e e 3 = 1 4 g 3. (VII) (u) = 4 (u) 3 g (u) g 3. u (u) = 6 (u) 1 g, (u) = 1 (u) (u), (u) = 10 (u) 3 18g (u) 1g 3, (n) (u) n (u) n (u) (u) (VIII) (u) = 1 (u) (u). (u) = 1 u + c 1u + c u c n u n + u n 3 (n )(n + 3)c n = 3(c 1 c n + c c n c n 3 c + c n c 1 ). c 1, c c 3, c 4, c n (n 3) g, g 3 c 3 = g 100, c 4 = 3g g , c 5 = 49g g 3, sn

80 7 Weierstrass a ω 1 = a, ω 3 = ai g 3 = 0 g 3 = w 6 = [a(m + ni)] 6 = 35 16a 6 1 (m + ni) 6 m, n 0 1 (m + ni) ( n + mi) 6 = 0 g 3 = 0 g 0 g = 60 1 w 4 = 15 4a 4 1 (m + ni) 4 1 (m + ni) (m ni) 4 = () g (VIII) c n (u) = 1 u + g 0 u + g 100 u6 + g u10 +. (1) a (0, a), (0, ai) (u) (1) m = i (iu ai, a) = (u a, ai). (ai, a) (a, ai) (iu) = (u) () (1) (u) (a u) = (u). u 0 a (u) u = a (V) (a) = 0 u = a (V) (u) 0 a a (a) 0 (0, a)

81 6. ζ 73 (u) () (0, ai) (u) (u) (0, a) (0, ai) (u) 3 4 (u) 3 g (u) = 0 e 1 = (a) = 1 g, e = (1 + ia) = 0, e 3 = (ia) = 1 g. g > 0 6. ζ f(u) f (u) 1 (u) (u) (u) = 1 u + { 1 (u w) 1 } w 1 u { (u) 1u } u { 1 0 du = 0 (u w) 1 } w du = u { 1 0 (u w) 1 } w du = { u w w + 1 w + u } w, 0 w 1 u w u = 0 1 u

82 74 Weierstrass ζ(u) = 1 u + { 1 u w + 1 w + u } w. (1) (u) ζ(u) = 1 u u 0 (u) = ζ(u). { (u) 1 } u du, () (1) ζ(u) u, ω 1, ω 3 1 (u) ζ(u) = 1 u g 60 u3 g u5 g 8400 u7 g g u9. ζ(u) u u + ω 1 ζ(u) ζ(u + ω 1 ) = ζ(u) + Z (3) Z u ω 1, ω 3 (3) u () (u + ω 1 ) = (u) + Z u, Z u = 0. Z u ω 1, ω 3 (3) u = ω 1 Z ζ(ω 1 ) = ζ( ω 1 ) + Z. ζ(u) ζ( ω 1 ) = ζ(ω 1 ) ω 1 w ζ(ω 1 ) η 1 η 1 = η 1 + Z, Z = η 1. (3) ζ(u + ω 1 ) = ζ(u) + η 1, η 1 = ζ(ω 1 ). ω 1 ζ(u + ω i ) = ζ(u) + η i, η i = ζ(ω i ) (i = 1,, 3) (4) ζ(u + h 1 ω 1 + h 3 ω 3 ) = ζ(u) + h 1 η 1 + h 3 η 3 h 1, h 3 h 1 = h 3 = 1 ζ(u ω 1 ω 3 ) = ζ(u + ω ) = ζ(u) η 1 η 3

83 7. σ 75 (4) η 1 + η + η 3 = 0 (5) ζ(u) ω 1, ω 3 ( w) 3 4 ζ(u) P a+ω1 a ω a+ω3 a ζ(u) du = + (P ) a = πi. a+ω 1 + a ω + a+ω 3 (6) πi P ζ(u) 1 (6) u = v + ω 3 a+ω3 a+ω1 a ζ(u) du = ζ(v + ω 3 ) dv a ω a+ω 1 a+ω1 } = {ζ(v) + η 3 dv a = a a+ω1 a ζ(v) dv 4η 3 ω 1. a+ω3 + = 4η 3 ω 1. a ω a ω a + a+ω 1 a+ω 3 = +4η 1 ω 3. (6) η 1 ω 3 η 3 ω 1 = πi Legendre (5) ω 1 + ω + ω 3 = 0 (7) η ω 1 η 1 ω = πi, η 3ω η ω 3 = πi (7) 7. σ (u) ζ(u) ζ(u) u { ζ(u) 1 } u { 1 du = u u w + 1 w + u } w du 0 0 = { log u w w + u w + u w }.

84 76 Weierstrass log exp { log u w w + u } w + u w = {( 1 u ) } e u w + u w w. u 0 w u = 0 u σ(u) = u {( 1 u w ) e u w + u w } σ(u) ζ(u) [ u { σ(u) = u exp ζ(u) 1 } ] du, 0 u u { log σ(u) = log u + ζ(u) 1 } du, u 0 (1) ζ(u) = d du log σ(u) = σ (u) σ(u). () σ(u) (u) ζ(u) u = w w σ(u) σ(u) (1) σ(u) lim u 0 u = 1 σ(u) = u + c 3 u 3 + c 5 u 5 + () ζ(u) e 3 = 0, e 5 = g 40, e 7 = g 3 840, σ(u) ω 1, ω 3 w σ(u) ζ(u) ζ(u + ω 1 ) = ζ(u) + η 1 u () log σ(u + ω 1 ) = log σ(u) + η 1 u + c (c ) σ(u + ω 1 ) = Ce η1u σ(u) (C ) C u = ω 1 σ(ω 1 ) = Ce η1ω1 σ( ω 1 ) = Ce η1ω1 σ(ω 1 ),

85 8. 77 σ(ω 1 ) = 0 C = e η 1 σ(u + ω 1 ) = e η 1(u+ω 1 ) σ(u). (3) σ(u + ω 3 ) = e η1(u+ω1) σ(u). (4) (3) (4) σ(u + h 1 ω 1 + h 3 ω 3 ) = ( 1) h1+h3 e E σ(u), E = (h 1 η 1 + h 3 η 3 )(u + h 1 ω 1 + h 3 ω 3 ) + h 1 h 3 (η 1 ω 3 η 3 ω 1 ), Legendre h 1 h 3 πi 8. m (z b i ) i=1 f(z) = C n (C ) (z a i ) i=1 f(z) = C + n i=1 { ci,1 + c i, z a i (z a i ) + + c } i,k i (z a i ) k i (C, c ) (I) f(u) a 1, a,, a n b 1, b,, b n 3 6 a 1 + a + + a n = b 1 + b + + b n + w, w b n + w b n a 1 + a + + a n = b 1 + b + + b n (1) b n φ(u) = n σ(u b i ) i=1 n σ(u a i ) i=1

86 78 Weierstrass f(u) φ(u) ω 1 ω 3 n σ(u b i + ω 1 ) i=1 φ(u + ω 1 ) = n σ(u a i + ω 1 ) i=1 ( 1) n e η np 1 i=1 = ( 1) n e η 1 (u b i +ω 1 ) i=1 np (u a i +ω 1 ) i=1 n σ(u b i ). n σ(u a i ) i=1 (1) φ(u + ω 1 ) = φ(u). ω 3 φ(u) f(u) n σ(u b i ) i f(u) = C n. (C ) σ(u a i ) i (II) f(u) a 1, a,, a n k 1, k,, k m a i (i = 1,,, m) f(u) A i,1 u a i + A i, (u a i ) + + A i,k i (u a i ) k i () 3 4 m A i,1 = 0 (3) i=1 ζ(u) = 1 u +, (u) = 1 u +, (u) = u 3 +,, (k i ) (u) = ( 1) k i (k i 1)! u k i φ i (u) a i () φ i (u) = A i,1 ζ(u a i ) + A i, (u a i ) A i,3 (u a i )! + + ( 1) ki A i,ki (k i 1)! (ki ) (u a i ). (4) m φ i (u) (5) i=1

87 9., ζ σ 79 f(u) (5) φ i (u) (5) m A i,1 ζ(u a i ) (6) i=1 u u + ω 1 m m } A i,1 ζ(u a i + ω 1 ) = A i,1 {ζ(u a i + η 1 ) i=1 = i=1 m A i,1 ζ(u a i ) + η 1 i=1 m i=1 A i,1, (3) 0 (6) (5) ω 1 ω 3 (5) 3 3 3,(5) f(u) f(u) = C + (III) n { A i,1 ζ(u a i ) + A i, (u a i ) A i,3 (u a i )! } A + + ( 1) ki i,ki (k i 1)! (ki ) (u a i ). (C ) i=1 (II) f(u) C u ζ,,,, log σ, ζ,,,,, ( i ) ( ii ) log σ (iii) ζ (iv) 9., ζ σ (u) (v) u v 0, 0 v, v (I) (u) (v) = C σ(u v)σ(u + v) σ(u) u C (u) (v) = 1 u, σ(u v)σ(u + v) σ(u) = C σ(v) + (u + ) = C σ(v) u +.

88 80 Weierstrass C = 1 σ(v) σ(u v)σ(u + v) (u) (v) = σ(u) σ(v) (1) v (1) u, v (1) u v (u) = ζ(u v) + ζ(u + v) ζ(u), (u) (v) (v) = ζ(u v) + ζ(u + v) ζ(v). (u) (v) ζ(u + v) = ζ(u) + ζ(v) + 1 (u) (v) (u) (v) () ζ 16 ζ(u + v) ζ(u), ζ(v) (u) (u) = ζ (u) () ζ(u), ζ(v) ζ () u (u + v) = (u) 1 u (u) (v) (u) (v). (3) (3) (u + v) = { (u) (v) 1 4 g } { (u) + (v)} g3 (u) (v) { (u) (v)}. (u + v) + (u) + (v) = 1 { (u) } (v), (4) 4 (u) (v) (u) (u) 1 (v) (v) 1 = 0 u + v + w = (). (5) (w) (w) 1 1. (5) (5) w z z 0, 0, 0 u, v w 3 6 u+v+w = () (5)

89 (u + v), (u), (v), (u), (v) R{ (u + v), (u), (v)} = 0 (R ) σ A, B, C, D (A B)(C D) + (A C)(D B) + (A D)(B C) = 0 A = (u), B = (u 1 ), C = (u ), D = (u 3 ) u, u 1, u, u 3 (1) A B = (u) (u 1 ) = σ(u + u 1)σ(u u 1 ) σ(u) σ(u 1 ), C D, σ(u + u 1 )σ(u u 1 )σ(u + u 3 )σ(u u 3 ) σ (6) + σ(u + u )σ(u u )σ(u 3 + u 1 )σ(u 3 u 1 ) + σ(u + u 3 )σ(u u 3 )σ(u 1 + u )σ(u 1 u ) = 0, (6) u + u 1 = a, u u 1 = b, u + u = c, u u 3 = d σ(a)σ(b)σ(c)σ(d) + σ(a )σ(b )σ(c )σ(d ) +σ(a )σ(b )σ(c )σ(d ) = 0, a = 1 (a + b + c + d) b = 1 (a + b c d) c = 1 (a b + c d) d = 1 ( a + b + c d) a = 1 (a + b + c d) b = 1 (a + b c + d) c = 1 (a b + c + d) d = 1 (a b c d) 30. (u), (3u), (u) (u) 80 (4) v = u (u) (v) lim v u (u) (v) = lim (v) v u (v) = 6 (u) 1 g (u)

90 8 Weierstrass (u) = 1 { 6 (u) 1 g } 4 4 (u) 3 (u) g (u) g 3 = (u)4 + 1 g (u) + g 3 (u) g 4 (u) 3 g (u) g 3. (3u) 80 (1) u, v nu, u n σ(n + 1u)σ(n 1u) (nu) (u) = σ(nu) σ(n). (1) ψ n (u) = σ(nu) σ(u) n (1) (nu) = (u) ψ n+1(u)ψ n 1 (u) ψ n (u). () ψ n (u) u u ω 1 ψ n (u + ω 1 ) = σ(nu + nω 1) σ(u + ω 1 ) n = ( 1)n e nη 1(nu+nω 1 ) σ(nu) ( 1) n e nη 1(u+ω 1) σ(u) n = σ(nu) σ(u) n = ψ n (u). ψ n (u) n ) ψ n (u) n (u) n (u) (u) n ψ n (u) u = 0 n 1 ψ n (u) (u) (u) ψ n (u) (u) n 1 ψ n (u) = c 0 (u) n 1 + c 1 (u) n 3 +. u c 0 = n ψ n (u) = n (u) n 1 +. n ψ n (u) (u) ψ n (u) = (u) { n } (u) n 4 +

91 n ψ n (u) = P n, n () n ψ n (u) = (u)p n (nu) = (u) (u)p n+1 P n 1 P n, (3) n (nu) = (u) P n+1p n 1 (u) P n. (4) (nu) P n P n m, n (u) ) (mu) (u) = ψ m+1(u)ψ m 1 (u) ψ m (u), (nu) (u) = ψ n+1(u)ψ n 1 (u) ψ n (u) (mu) (nu) = ψ m+1ψ m 1 ψ n ψ n+1 ψ n 1 ψ m 8 (1) σ(m + nu)σ(m nu) (mu) (nu) = σ(mu) σ(nu), ψ m ψ n. (5) (mu) (nu) = ψ m+nψ m n ψ n ψ m. (6) (5) (6) m = n + 1 m, n n + 1, n 1 ψ m+n ψ m n = ψ m+1 ψ m 1 ψ n ψ n+1 ψ n 1 ψ m. ψ n+1 = ψ n+ ψ n 3 ψ n+1 3 ψ n 1. (u)ψ n = ψ n (ψ n+ ψ n 1 ψ n+1 ψ n ). P { P n+ P 3 n (u) 4 P 3 n+1 P n 1 (n ) P n+1 = (u) 4 P n+ P 3 n P 3 n+1 P n 1 (n ) P n = P n (P n+ P n 1 P n+1 P n ).

92 84 Weierstrass n 4 P n n > 4 n 4 P n P 1 = 1, P = 1, P 3 = 3 (u) 4 3 g (u) 3g 3 (u) 1 16 g, P 4 = (u) g (u) g 3 (u) g (u) + 1 g g 3 (u) + g g 3. P n = C λ,µ,ν g λ g 3 µ (u) ν (C ) λ, µ, ν λ + 3µ + ν = n 1 n 4 (n ) (n ) ( i ) ζ(u a) ( ii ) (u a) (iii) (u a) (iv) (u a) (iv) (ii) (i) 4 (VII) (ii) (iii) (u), (u), (a), (a) (a), (a) (u), (u) (ii), (iii) (iv) (a), (a) (i) Aζ(u a) ζ ζ(u) A Aζ(a) + 1 A (u) + (a) (u) (a) A = 0 (u), (u) 1.

93 f(u) f(u) = A + B (u) C + D (u) (A, B, C, D (u) ) C D (u) f(u) = P + Q (u) (P, Q (u) ) f(u) P Q (u) sn u ω 1 = K, ω 3 = ik { (u) e} sn u = (u) sn u u f 1 (u), f (u) (u), (u) (u), (u) f 1 (u) = P 1 + Q 1 (u), f (u) = P + Q (u) (1) (u) = 4 (u) 3 g (u) g 3 () R{f 1 (u), f (u)} = 0 (3) R. f = f 1 (3) f 1 (u). 3. f 1(u) f 1(u) = S{f 1 (u)} S w = f 1 (u) dw du = S(w), u = dw S(w).

94 86 Weierstrass 4. f(u) (u) f(u) R 1 {f(u), (u)} = 0 (4) u v, u + v } R 1 {f(u), (u)} = 0, R 1 {f(u + v), (u + v)} = 0 (5) R { (u + v), (u), (v)} = 0 (6) (4), (5), (6) (u), (v), (u + v) R 3 {f(u + v), f(u), f(v)} = 0 R 3 R f(u) R{f(u), f (u)} = 0 (R ) f(u) f(u) f(u) (i) u (ii) e cu (C ) (iii) u Briot-Bouquet (1) f(u) (i) (ii) (iii) u (ii) (i) (1) Briot-Bouquet, Théorie ces fonctions doublement périodiques. (1859). Koebe

95 3. σ 1, σ, σ Weierstrass, Phragmén Koebe () (i) (ii) (iii) 5 3. σ 1, σ, σ 3 9 (1) v = ω i (i = 1,, 3) σ (u) e i = σ(u ω i)σ(u + ω i ) σ(u) σ(ω i ) σ(u + ω i ) = e ηi(u+ωi) σ(u) u u ω i σ(u + ω i ) = e ηiu σ(u ω i ). (1) (u) e i = eηiu σ(u ω i ) σ(u) σ(ω i ) = e ηiu σ(u + ω i ) σ(u) σ(ω i ). (u) ei = ± eη iu σ(u ω i ) σ(u)σ(ω i ) = ± e ηiu σ(u + ω i ). σ(u)σ(ω i ) σ i (i = 1,, 3) σ i (u) = eηiu σ(u ω i ), σ(ω i ) σ i (u) = e ηiu σ(u + ω i ). σ(ω i ) () (1) (u) e i = σ i(u) σ(u) (3) (u) ei = σ i(u) σ(u) (4) (u) = 4{ (u) e 1 }{ (u) e }{ (u) e 3 } () Phragmén, Acta Math., 7. (1885). Koebe, SchwarzFestschrift. (1914)

96 88 Weierstrass (u) = ± (u) e 1 (u) e (u) e3 = ± σ 1(u)σ (u)σ (u) σ(u)σ(u)σ(u). u (3) (u) = σ 1(u)σ (u)σ 3 (u) σ(u) 3. (5) (u) = σ i(u) σ(u) (5) d du σ i (u) σ(u). d σ i (u) du σ(u) = σ j(u)σ k (u) σ(u) (6) i, j, k 1,, 3 σ(u) σ i (u) = 1 σ i(u) σ(u), σ i (u) σ j (u) = σ i(u) σ(u) σ j(u) σ(u) e i e j = { (u) e j } { (u) e i } = σ j(u) σ i (u) σ(u) (7) d σ(u) du σ i (u) = σ j(u)σ k (u) σ i (u), d σ i (u) du σ j (u) = (e i e j ) σ(u)σ k(u) σ j (u). (8) S(u) = σ(u) σ 3 (u), C(u) = σ 1(u) σ 3 (u), D(u) = σ (u) σ 3 (u) (7) { C(u) = 1 (e 1 e 3 )S(u), (8) D(u) = 1 (e e 3 )S(u). d S(u) = C(u)D(u), du d du C(u) = (e 1 e 3 )S(u)D(u), d du D(u) = (e e 3 )S(u)C(u). 4 sn, cn, dn

97 3. σ 1, σ, σ 3 89 S(u) du = ds(u) C(u)D(u) = ds(u) {1 (e1 e 3 )S(u) }{1 (e e 3 )S(u) } e1 e 3 S(u) = z, e1 e 3 u = w, e e 3 e 1 e 3 = k (9) dw = dz (1 z )(1 k z ) w = 0 z = 0 z sn w ( ) sn w = σ(u) e 1 e 3 σ 3 (u) = σ w e1 e 3 e 1 e 3 ( ). (10) w σ 3 e1 e 3 sn σ e1 e 3 S(u) = w C(u) = 1 (e 1 e 3 )S(u) = 1 sn w = cn w, D(u) = 1 (e e 3 )S(u) = 1 k sn w = dn w, u 0 C(u) 1, D(u) 1 C(u) = cn w, D(u) = dn w cn w = σ 1(u) σ 3 (u), dn w = σ (u) σ 3 (u) (11) sn w 4K, ik σ(u) σ 3 (u) { σ(u + ω1 ) = e η 1(u+ω 1 ) σ(u), σ(u + ω 3 ) = e η3(u+ω3) σ(u), { σ3 (u + ω 1 ) = e η1(u+ω1) σ 3 (u), σ 3 (u + ω 3 ) = e η 3(u+ω 3 ) σ 3 (u), σ σ σ(u + ω 1 ) σ 3 (u + ω 1 ) = σ(u) σ 3 (u), σ(u + ω 3 ) σ 3 (u + ω 3 ) = σ(u) σ 3 (u).

98 90 Weierstrass σ(u) σ 3 (u) 4ω 1, ω 3 ω 1, ω 3 K = ω 1 e1 e 3, ik = ω 3 e1 e 3 (1) sn(u e 1 e 3 ), e1 e 3 σ(u) σ 3 (u) u 0 1 (10) sn (3) i = 3 (u) e 3 = σ 3(u) σ(u) = e 1 e 3 sn w. ( 1 (u) = (e 1 e 3 ) sn w + e ) ( 3 1 = (e 1 e 3 ) e 1 e 3 sn w 1 + ) k, 3 ( K (u) = ω 1 ) ( 1 sn w 1 + ) k 3 (13)

99 91 ϑ 33. ϑ Weierstrass u, ω 1, ω 3 (u) (u) g, g 3 (u) ϑ wi v, τ u ω 1 = v, I(τ) = I ( ω3 ω 3 ω 1 = τ ω 1 ) > 0 e vπi = z, e τπi = q I(τ) > 0 q = e πi(τ) < 1 ϑ ϑ 1 (v) = i ϑ (v) = ϑ 3 (v) = ϑ 0 (v) = n= n= n= n= ( 1) n q ( n 1 ) z n 1, q ( n 1 ) z n 1, q n z n, ( 1) n q n z n. (1) q < 1 0 < z < v v < ϑ

100 9 ϑ ϑ 3 (v) ϑ 3 (v) = + q 4 z 4 + qz qz + q 4 z 4 + = 1 + q(z + z ) + q 4 (z 4 + z 4 ) + = 1 + q(e vπi + e vπi ) + q 4 (e 4vπi + e 4vπi ) + = 1 + q cos πv + q 4 cos 4πv +. ϑ 1 (v) = (q 1 4 sin πv q 9 4 sin 3πv + q 5 4 sin 5πv ), ϑ (v) = (q 1 4 cos πv + q 9 4 cos 3πv + q 5 4 cos 5πv + ), ϑ 3 (v) = 1 + (q cos πv + q 4 cos 4πv + q 9 cos 6πv + ), ϑ 0 (v) = 1 (q cos πv q 4 cos 4πv + q 9 cos 6πv ), () ϑ 1 (v) v = 0 ϑ 1, ϑ, ϑ 1 (0), ϑ (0), ϑ 1 = 0 ϑ = (q q q ), ϑ 3 = 1 + (q + q 4 + q 9 + ), ϑ 0 = 1 (q q 4 + q 9 ), ϑ 1 ϑ 1(0) ϑ ϑ 1 = π(q 1 4 3q q 5 4 ). (3 ) (3) ϑ v = 4πi ϑ τ (4) () 34. ω 1, ω 3 ϑ ω 1 1, τ ϑ v = u ω 1 v 1 v v + 1 z iz (1) ( ϑ 1 v + 1 ) ( = ϑ (v), ϑ v + 1 ) = ϑ 1 (v), ( ϑ 3 v + 1 ) ( = ϑ 0 (v), ϑ 0 v + 1 ) = ϑ 3 (v).

101 v τ z q 1 z ( ϑ 1 v + τ ) = i n= ( 1) n q ( n 1 ) + n 1 z n 1 = iq 1 4 z 1 ϑ 0 (v) = ie πi(v+ τ 4 ) ϑ0 (v). ( ϑ v + τ ) = e πi(v+ τ 4 ) ϑ3 (v), ( ϑ 3 v + τ ) = e πi(v+ τ 4 ) ϑ (v), ( ϑ 0 v + τ ) = ie πi(v+ τ 4 ) ϑ1 (v). ε = e πi(v+ τ 4 ) δ = e πi(v+τ). v + 1 v + τ v τ v + 1 v + τ v τ ϑ 1 ϑ iεϑ 0 εϑ 3 ϑ 1 δϑ 1 δϑ 1 ϑ ϑ 1 εϑ 3 iεϑ 0 ϑ δϑ δϑ ϑ 3 ϑ 0 εϑ iεϑ 1 ϑ 3 δϑ 3 δϑ 3 ϑ 0 ϑ 3 iεϑ 1 εϑ ϑ 0 δϑ 0 δϑ 0 ϑ 1 { ϑ1 (v + 1) = ϑ 1 (v), ϑ 1 (v + τ) = e πi(v+τ) ϑ 1 (v). (1) v { ϑ 1 (v + 1) = ϑ 1(v), ϑ 1(v + τ) = πie πi(v+τ) ϑ 1 (v) e πi(v+τ) ϑ 1(v). ϑ 1(v + 1) ϑ 1 (v + 1) = ϑ 1(v) ϑ 1 (v), ϑ 1(v + τ) ϑ 1 (v + τ) = ϑ 1(v) ϑ 1 (v) πi a a, a + 1, a τ, a + τ P ϑ 1 (v) ϑ a+1 a+1+τ 1(v) ϑ 1 (v) dv = + + (P ) a a+1 a+τ a+1+τ a +. a+τ a+1+τ a+1 = a+τ a ϑ 1(v) ϑ 1 (v) dv, a+1+τ a+1 a + = 0, a+τ

102 94 ϑ a+τ a+1+τ = a a+1 { } ϑ 1 (v) ϑ 1 (v) πi dv, a+1 a a+τ + = πi. a+1+τ 1 ϑ 1(v) dv = 1. πi (P ) ϑ 1 (v) ϑ 1 (v) ϑ 1 (v) v = 0 ( ) ( 1 1 ϑ = 0, ϑ 3 + τ ) ( τ ) = 0, ϑ 0 = 0. ϑ (1) { ϑ1 (v + 1) = ϑ 1 (v), ϑ 1 (v + τ) = e πi(v+τ) ϑ 1 (v). () ϑ 1 () ϑ 1 ϑ, ϑ 3, ϑ 0 C F (v) = C ϑ ϑ 0 (v) ϑ 3 ϑ 1 (v) ϑ (v) () 1 τ ϑ (v) = 0 v = 1 ϑ ϑ 0 ( 1 ) ( ) 1 ϑ 3 ϑ 1 = ϑ ϑ 3 ϑ 3 ϑ = 0 ( ) F (v) 1 C Cϑ ϑ 0 (v) Cϑ 3 ϑ 1 (v) = ϑ (v). v = 0 Cϑ ϑ 0 = ϑ C ϑ ϑ 0 (v) ϑ 3 ϑ 1 (v) = ϑ 0 ϑ (v). (3)

103 ϑ 3 ϑ 0 (v) ϑ ϑ 1 (v) = ϑ 0 ϑ 3 (v) (4) (4) v = 1 ϑ ϑ 4 = ϑ 3 4 (5) (3) (4) ϑ 0 (v), ϑ 1 (v) ϑ 0 (v) = ϑ 0(v) {ϑ 3 ϑ 3 (v) ϑ ϑ (v) } ϑ 3 4 ϑ 4, ϑ 1 (v) = ϑ 0(v) {ϑ ϑ 3 (v) ϑ 3 ϑ (v) } ϑ 3 4 ϑ 4 (5) ϑ 0 (v) 4 + ϑ (v) 4 = ϑ 1 (v) 4 + ϑ 3 (v) 4. (5) v = ϑ m F m (z) = (1 + q n 1 z )(1 + q n 1 z ) n=1 z z 1 F m (z) = A (m) 0 + A (m) 1 (z + z ) + A (m) (z 4 + z 4 ) + + A (m) m (z m + z m ) A m (m) = qq 3 q 5 q m 1 = q m F m (z) z qz F m (qz) = (1 + qm+1 z )(1 + q 1 z ) (1 + q m 1 z )(1 + qz ) F m(z). (q m + qz )F m (qz) = (1 + q m+1 z )F m (z) z z m A k 1 (m) = 1 q m+k q k 1 (1 q m k+ ) A k (m) A m (m) k m, m 1, A m 1 (m), A m (m), A k (m) = q k (1 qm+k+ )(1 q m+k+4 ) (1 q 4m ) (1 q )(1 q 4 ) (1 q m k )

104 96 ϑ m lim F m(z) = F (z) m 0 < z < q < 1 F (z) F (z) = A 0 + A 1 (z + z ) + A (z 4 + z 4 ) + A k = lim m A k (m) A (m) k m (1 q n ) n=1 1 q k A k = (1 q n ) n=1 F (z) (1 q n ) = 1 + q k (z k + z k ), n=1 k=1 ϑ 3 (v) ϑ 3 (v) = (1 q n ) (1 + q n 1 z )(1 + q n 1 z ). n=1 n=1 ϑ v v + 1, v + τ, v τ ϑ 0(v), ϑ (v), ϑ 1 (v) ϑ 1 (v) = iq 1 4 (z z 1 ) (1 q n ) (1 q n z )(1 q n z ) n=1 n=1 n=1 = q 1 4 sin πv (1 q n ) (1 q n cos πv + q 4n ), n=1 n=1 ϑ (v) = q 1 4 (z + z 1 ) (1 q n ) (1 + q n z )(1 + q n z ) n=1 n=1 = q 1 4 cos πv (1 q n ) (1 + q n cos πv + q 4n ), n=1

105 ϑ 3 (v) = (1 q n ) (1 + q n 1 z )(1 + q n 1 z ) n=1 n=1 n=1 = (1 q n ) (1 + q n 1 cos πv + q 4n ), n=1 n=1 ϑ 0 (v) = (1 q n ) (1 q n 1 z )(1 q n 1 z ) n=1 n=1 = (1 q n ) (1 q n 1 cos πv + q 4n ). n=1 v = 0 Q 0 = (1 q n ), Q 1 = (1 + q n ), n=1 n=1 (1) Q = (1 + q n 1 ), Q 3 = (1 q n 1 ). n=1 n=1 ϑ = q 1 4 Q0 Q 1, ϑ 3 = Q 0 Q, ϑ 0 = Q 0 Q 3 () ϑ 1 0 ϑ 1(0) ϑ 1 ϑ 1 = lim v 0 ϑ 1(v) = lim v 0 ϑ 1 (v) v ϑ 1 = πq 1 4 Q0 3. (3) (1) Q Q 0 Q 1 = (1 q 4n ), Q Q 3 = (1 q 4n ). n=1 4n 4n Q 0 Q 1 Q Q 3 = (1 q n ) = Q 0, n=1 Q 0 0 () (3) (4) n=1 Q 1 Q Q 3 = 1. (4) ϑ 1 = πϑ ϑ 3 ϑ 0. (5) v τ τ 1 ϑ 1 ϑ 1 τ = 1 ϑ ϑ τ + 1 ϑ 3 ϑ 3 τ + 1 ϑ 0 ϑ 0 τ 33 (4) ϑ 1 ϑ 1 ϑ 1 + ϑ 3 + ϑ 0, ϑ ϑ 3 ϑ 0 = ϑ ϑ 1 (v) v = 0

106 98 ϑ 36. σ σ ϑ ϑ(u) = e η 1 ω 1 u σ(u) σ { σ(u + ω1 ) = σ(u), σ(u + ω 3 ) = e πi ω 1 (u+ω 3 ) σ(u). u ω 1 = v, ω 3 ω 1 = τ σ(ω 1 v) = φ(v) { φ(u + 1) = φ(u), φ(u + τ) = e πi(v+τ) φ(u). ϑ 1 (v) φ(v) ϑ 1 (v) ϑ 1 (v) = 0 φ(v) = 0 C φ(v) = Cϑ 1 (v). e η 1 ω 1 u σ(u) = e η 1ω 1 v σ(ω 1 v) = Cϑ 1 (v). v 4η 1 ω 1 ve η 1ω 1 v σ(ω 1 v) + ω 1 e η 1ω 1 v σ (ω 1 v) = Cϑ 1(v). v = 0 σ (0) = 1 ω 1 = Cϑ 1 σ(u) = ω 1 e ϑ 1(v) η1ω1v ϑ. (1) 1 ϑ ϑ σ (1) σ η 1

107 σ ϑ (1) u u + 1 v v + 1 u = 0(v = 0) σ(u + ω 1 ) = ω 1 e η1ω1(v+ 1 ) ϑ (v) ϑ, 1 σ(ω 1 ) = ω 1 e 1 η 1ω 1 ϑ ϑ. 1 σ 1 (u) = e η1u σ(u + ω 1 ) σ(ω 1 ) = e η 1ω 1 v ϑ (v) ϑ. σ (u) = e η 1ω 1 v ϑ 3(v) ϑ 3, σ 3 (u) = e η 1ω 1 v ϑ 0(v) ϑ 0 () (4) (u) ei = σ i(u), (i = 1,, 3) σ(u) () (u) e1 = 1 ϑ 1 ϑ (v) ω 1 ϑ ϑ 1 (v), (u) e = 1 ϑ 1 ϑ 3 (v) ω 1 ϑ 3 ϑ 1 (v), (u) e3 = 1 ϑ 1 ϑ 0 (v) ω 1 ϑ 0 ϑ 1 (v). 3 (5) ( ) 3 1 ϑ 13 ϑ (v)ϑ 3 (v)ϑ 0 (v) (u) = ω 1 ϑ ϑ 3 ϑ 0 ϑ 0 (v) 3 = π 4ω 3 ϑ 1 ϑ (v)ϑ 3 (v)ϑ 0 (v) 1 ϑ 0 (v) 3. ( 35 (5) ) (1) (1) u = ω = ω 1 = ω 3, v = 1 τ e e 1 = 1 ( ϑ 1 ϑ 1 + ) τ ( ω 1 ϑ ϑ ) τ = 1 ϑ 1 iϵϑ 0 ( 34 ) ω 1 ϑ ϵϑ 3 = iϑ 1ϑ 0 = i π ϑ 0. ω 1 ϑ ϑ 3 ω 1

108 100 ϑ e e 1 = i π ϑ 0, ω 1 e1 e = π ϑ 0, ω 1 e1 e 3 = π ϑ 3, ω 1 e3 e 1 = i π ϑ 3, ω 1 e3 e = i π ϑ, ω 1 e e 3 = π ϑ. ω 1 () () e 1, e, e 3 ( π ) (ϑ ϑ 0 4 ), e 1 = 1 3 ω 1 ( ) π (ϑ 4 ϑ 4 0 ), e = 1 3 e 3 = 1 3 ω 1 ( π ω 1 ) (ϑ 4 + ϑ 3 4 ) (1) (u) (1) σ(u) = ω 1 π eη1ω1v σ(u) = ω 1 e η 1ω 1 v ϑ 1(v) ϑ. 1 sin πv (1 q n z )(1 q n z ). (1 q n ) n=1 n=1 ( ) u v = u ω 1 ζ(v) = η 1 v + π cot πv πi ω 1 ω 1 n=1 ( q n z 1 q n z qn z ) 1 q n z q < z < q (3) q n z < 1, q n z < 1 z z ζ(v) = η 1 v + ζ(u) π cot πv πi ω 1 ω 1 = η 1 v + π ω 1 cot πv + π ω 1 n=1 n=1 q n 1 q n (zn z n ) q n sin nπv. (4) 1 qn

109 v τ v τ (3) τ < v < τ v 0, 1, 1 + τ, τ (4) u ( ) 1 (u) = { 4η 1 v + π ω 1 sin πv 8π n=1 ( ) { 3 1 cos πv (u) = ω 1 sin 3 πv + 16 n=1 nq n 1 q n q n sin nπv 1 qn n cos nπv } }, (5) (4) (5) ζ, η 1 6 ζ(u) u ζ(u) = 1 ω 1 v g 60 (ω 1v) 3 g (ω 1v) 5 (7) (4) u ζ(u) = η 1 v + π ( 1 ω 1 πv 1 3 πv 1 45 π3 v 3 ) 945 π5 v 5 + π q n ( ω 1 1 q n nπv 4 3 n3 π 3 v ) 15 n5 π 5 v 5 n=1 (7) v, v, v 3 ( ) η 1 = π 1 1 nq n 1 q n, ω 1 ω 1 n=1 ( ) ( 4 π 1 g = ω n=1 ( ) ( 6 π 1 g 3 = n=1 n 3 q n 1 q n n 5 q n 1 q n η 1 (4) (5) (5) u = ω 1, v = 1 { e 1 = 1 4η 4ω 1 ω 1 + π 8π 1 n=1 ) ) ( 1) n 1 q n,. nqn e 1, g, g 3 e, e 3 η 3 η 1 6 (7) }. (6)

110 10 ϑ 38. sn 3 sn w = σ(u) e 1 e 3 σ 3 (u), cn w = σ 1(u) σ 3 (u), (w = u e 1 e 3 ) dn w = σ (u) σ 3 (u) () (1) () sn w = ϑ 3 ϑ 1 (v) ϑ ϑ 0 (v), cn w = ϑ 0 ϑ (v) ϑ ϑ 0 (v), dn w = ϑ 0 ϑ 3 ϑ 3 (v) ϑ 0 (v). k = e e 3 = ϑ 4 e 1 e 4 3 ϑ, 3 k = 1 k = ϑ 3 4 ϑ 4 ϑ 3 4 = ϑ ( 34 (5) ) ϑ 3. sn w = 1 ϑ 1 (v) k ϑ 0 (v), k ϑ cn w = (v) k ϑ 0 (v), dn w = k ϑ 3(v) ϑ 0 (v). sn K, ik τ = ω 3 = ik ω 1 K ( 3 (1)) q ϑ sn, cn, dn 39.

111 z Jacobi 0 z 0 1 k z 1 z dz z = sn(u, k) 1 k z u dn u 1 z dz = d(sn u) = 0 cn u E(u) = u 0 u 0 dn u du dn u du (1) 9 (4) v = ω 3.. (1) (u + ω 3 ) + (u) + e 3 = 1 (u) 4 { (u) e 3 } = { (u) e 1}{ (u) e } (u) e 3, (u + ω 3 ) e 3 = { (u) e 1}{ (u) e } (u) e 3 (u) e 3 = e 1e + e 3 (u) e 3 = (e 1 e 3 )(e e 3 ) (u) e 3, (u + ω 3 ) e 3 e 1 e 3 = e e 3 (u) e 3. e e 3 e 1 e 3 e 1 e 3 (u) e 3 = k sn w ( 3 (9) (11) ) dn w = 1 k sn w = 1 (u + ω 3) e 3 e 1 e 3 = e 1 (u + ω 3 ) e 1 e 3. w e 1 (u + ω 3 ) E(w) = dw (w = u e 1 e 3 ) 0 e 1 e 3 1 u { } = e 1 (u + ω 3 ) du e1 e 3 = 0 1 e1 e 3 { e 1 u + ζ(u + ω 3 ) η 3 }. ()

112 104 ϑ () w 0 E(w) dw = u Ω(w). Ω(w) = exp 0 {e 1 u ζ(u + ω 3 ) η 1 } du = 1 e 1u + log σ(u + ω 3) σ(ω 3 ) w 0 η 3 u. E(w) dw = e 1 e 1u σ 3 (u) (3) z 0 dz (1 + nz ). (n ) (1 z )(1 k z ) z = sn u u du 1 + n sn u 0 1 u n = k sn a (a ) 0 n sn u du 1 + n sn u, u k sn sn u du a 0 1 k sn a sn u Π(u, a) = k sn a cn a dn a u 0 sn u du 1 k sn a sn u (4) (4) u u Π(u, a) = sn u k sn a cn a dn a 1 k sn a sn u = 1 { } k sn a sn u sn(u + a) + sn(u a), (5)

113 { } u a Π(u, a) = k cn a dn a sn u sn(u + a) + sn(u a) { + k sn a sn u cn(u + a) dn(u + a) } cn(u a) dn(u a) { = k sn u sn(u + a) cn a dn a } + sn a cn(u + a) dn(u + a) { + k sn u sn(u a) cn a dn a } sn a cn(u a) dn(u a) { } = k sn u sn(u + a) 1 k sn (u + a) sn a { } + k sn u sn(u a) 1 k sn (u a) sn a. (6) sn(u + v) = sn u sn v sn u cn v dn v sn v cn u dn u dn a dn (u + a) = k { } sn (u + a) sn a { = k sn(u + a) sn(u + a) cn a dn a } sn a cn(u + a) dn(u + a) { } = k sn(u + a) sn u 1 k sn (u + a) sn a, { } dn a dn (u a) = k sn(u a) sn u 1 k sn (u a) sn a. (6) u a Π(u, a) = dn a dn (u + a) dn (u a). a u u Π(u, a) = E(a) 1 E(u + a) + 1 E(u a) + C, C a a = 0 (1) (5) C = 0 Π(u, a) = ue(a) 1 log Ω(u + a) + 1 log Ω(u a) + C, C u u =0 C = 0 (3) (3)

114 106 ϑ u w α = a e 1 e 3 Π(w, α) = we(α) 1 log Ω(w + α) + 1 log Ω(w α) = we(α) + 1 Ω(w α) log Ω(w + α). (7) E(w) = = d dw log Ω(w) = 1 e1 e 3 { 1 e 1 u + σ 1(u) e1 e 3 σ 3 (u) E(α) = d { du log e 1 e1u} }, { } 1 e 1 a + σ 3(a). e1 e 3 σ 3 (a) Ω(w α) Ω(w + α) = e 1 e 1(u a) σ 3 (u a) e 1 e1(u+a) σ 3 (u + a) = e e 1au σ 3(u a) σ 3 (u + a). (7) { } Π(w, α) = u e 1 a + σ 3(a) e 1 au + 1 σ 3 (a) log σ 3(u a) σ 3 (u + a) = u σ 3(a) σ 3 (a) + 1 log σ 3(u a) σ 3 (u + a). 40. x a + y b = 1 (0, b) (x, y) z 1 k z L = a 1 z dz, z = x a, k = a b a 0 L u = z 0 dz (1 z )(1 k z ) u u L = a u 0 = au a z = sn u, dn u du u 0 k sn u du

115 d dv log ϑ 0(v) = d { ( dv log ie πi(v+ τ 4 ) ϑ1 v + τ )} = πi + d ( dv log ϑ 1 v + τ ). (1) σ ϑ 1 ( ϑ 1 v + τ ) = σ(u + ω 3 )ϑ 1(0) 1 e η 1ω 1(v+ τ ), ω 1 d ( dv log ϑ 1 v + τ ) ( = ω 1 ζ(u + ω 3 ) 4η 1 ω 1 v + τ ). (1) d dv log ϑ 0(v) = πi + ω 1 ζ(u + ω 3 ) 4η 1 ω 1 (v + τ ) v, u, w v = u w, u = ω 1 e1 e 3 ϑ 0 (v) w Θ(w) Z(w) = d dw log Θ(w) = 1 d ω 1 e1 e 3 dv log ϑ 0(v) 1 ( Z(w) = {πi + ω 1 ζ(u + ω 3 ) 4η 1 ω 1 v + τ ) } ω 1 e1 e 3 } 1 = {πi + ω 1 ζ(u + ω 3 ) η 1 u η 1 ω 3 ω 1 e1 e 3 } 1 = {ω 1 ζ(u + ω 3 ) η 1 u η 3 ω 1 ω 1 e1 e 3 { 1 = ζ(n + ω 3 ) η } 1 u η 3 e1 e 3 ω 1 Z (w) = { d dw log Θ(w) = 1 (u + ω 3 ) η } 1. e 1 e 3 ω 1 (u + ω 3 ) e 3 e 1 e 3 = k sn w Z (w) = k sn 1 w e 1 e 3 ( e 3 + η ) 1. ω 1 k sn w = Z (0) Z (w) ()

116 108 ϑ () Z(w) = d dw log Θ(w) = Θ (w) Θ(w), Z (w) = Θ (w)θ(w) Θ (w) Θ(w), u 0 Z (0) = Θ (0) Θ(0). k sn w = Θ (0) Θ(0) d Θ (w) dw Θ(w), k sn u du = Θ (0) Θ(0) u Θ (u) Θ(u). { Θ (0) L = au a = a Θ (u) Θ(u) + au } Θ(0) u Θ (u) Θ(u) { } 1 Θ (0) Θ(0) 41. x a y b = 1 (a, 0) (x, y) L = 1 y (a + b )y + b 4 b y + b dy 0 y = b a + b tan θ L = b θ dθ a + b 0 cos θ 1 k sin θ, k = a a + b u = θ 0 dθ 1 k sin θ u sin θ = sn u, cos θ = cn u, dθ = dn u du, 1 k sin θ = dn u,

117 L = b u du a + b cn u cn u = ik k cn(u + K + ik ), k b = a + b, L = u a + b k cn (u + K + ik ) du 0 k cn u = k k sn u = k Θ (0) Θ(0) + d log Θ(u), du k cn (u + K + ik ) = k Θ (0) Θ(0) + d du log Θ(u + K + ik ) Θ ( ) u Θ(u) = ϑ 0 ω 1 e1 e 3 ( Θ(u + K + ik u ) = ϑ ω 1 e1 e 3 + τ ) ( ) u = εϑ, ω 1 e1 e 3 ε u ϑ u H 1 (u) k cn (u + K + ik ) = k Θ (0) Θ(0) + d du log H 1(u) L = a + b u 0 { k Θ (0) = a + b {( Θ (0) Θ(0) k Θ(0) + d ) u H 1 (u) H 1 (u) } du log H 1(u) Jacobi ϑ 0 (v), ϑ 1 (v), ϑ (v), ϑ 3 (v) w Θ(v), H(w), H 1 (w), Θ 1 (v) }. du

118

119 g g 3 g = 60 1 (h 1 ω 1 + h 3 ω 3 ) 4 = 60 (ω 1 ) 4 1 (h 1 + h 3 τ) 4, g 3 = (h 1 ω 1 + h 3 ω 3 ) 6 = 140 (ω 1 ) 6 1 (h 1 + h 3 τ) 6 (ω 1 ) 4 g (ω 1 ) 6 g 3 1, (m, n, α > ) (m + nτ) α τ τ = x + yi ε, k m + nτ ε m + ni = (m + nx) + n y ε (m + n ) = m (1 1k ) ( 1 ε + k m + knx { + n y (k 1)x ε }, ) k > 1 ϵ x N, y η N, η k, ε 1, 0 { } ε m + nτ > ε m + ni 1 (m + nτ) α < 1 ε α 1 (m + ni) α (m+nτ) α

120 11 τ g g 3 τ ω 1 g : g 3 τ 0 0 J(τ) = g 3 g 3 7g 3 (1) 4 3 g g 3 = 0 e 1, e, e J(τ) J(τ) q = e τπi 101 ( π g = 1 1 ω 1 g 3 = 1 ( π 16 ω 1 ) 4 { } (q + 9q 4 + ), ) 6 { } 1 504(q + 33q 4 + ) J (1) J(τ) = 1 ( ) q q + () g g 3 0 ω 1, ω 3 ω 1 = aω 1 + bω 3, ω 3 = cω 1 + dω 3 (a, b, c, d ad bc = 1) 35 6 ad bc = ±1 τ = ω 3 ω 1 1 J(τ) τ τ = ω 3 ω = cω 1 + dω 3 = c + dτ 1 aω 1 + bω 3 a + bτ ( ) c + dτ J = J(τ), a + bτ ad bc = 1 (3) J(τ) ( ) (Modular a b transfomation) (3) ad bc = 1 c d ( ) ( ) S =, T =

121 τ = τ + 1, τ = 1 τ (4) ( ) a b M = c d ( ) ( ) ( ) S ± 1 0 a b a b M = =, ±1 1 c d ±a + c ±b + d ( ) ( ) ( ) 0 1 a b c d T M = = 1 0 c d a b M b > d M T M b d α ( ) S α a b M =, αb + d < b αa + c αb + d ( ) S α a b M = c d d = 0 T ( ) T S α c d M = a b. β ( S β T S α c M = βc a ) d βd b, βd b < d T S β T S α M = ( a1 ) b 1 c 1 0 (5) a 1 b 1 c 1 0 = b 1c 1 = 1, b 1 = 1, c 1 = 1 b 1 = 1, c 1 = 1 ( ) ( ) a1 b 1 a1 1 =. c (5) T ( ) a1 b 1 = c 1 0 ( ) 1 0 = S a 1. a 1 1 M = S α T S β T T S a 1

122 (u) u = u + ω 1, u = u + ω 3 J(τ) (4) ( ) a b τ c d ( ) τ a b = τ = c + dτ c d a + bτ τ τ τ D (i) D (ii) D D J(τ) D J(τ) τ = τ + 1 τ = 1 τ D 1 D τ = x + yi 1 x < 1, x + y 1 D D D (i) D τ τ = c + dτ a + bτ τ = τ D (I) b = 0 ad bc = 1 a = d = 1a, b, c, d a = d = 1 ) τ = τ + c. c = 0 τ D (II) b = 1 c = ad 1 34 τ = d τ. τ 1 d = τ ( a).

123 a D τ a τ ( a) 1 τ d < 1 τ D a = d = 0 τ τ D 1 x 0 D τ τ D ( ) 0 1 τ = i τ = τ = i, ( ) ( 0 1 τ = ρ τ = 1 + τ = ρ, ρ = 1 + ) 3i 1 1 τ ( ) 1 1 τ = ρ τ = τ = ρ i, ρ 1, 1 3 D (III) (IV) b = 1 (II) b > 1 b τ d b = 1 1 b τ + a. b a b τ D τ + a 3 b 3 τ + d b 1 b 1 4, τ + d b 1 3 < 3. τ d b 3 D D 34 (i) i, ρ D S : τ = τ + 1 D S S S, S 3, τ = τ 1 S 1, S, T : τ = 1 τ S

124 116 ST T S D τ y > h (0 < h < 1) D, S, T, T T D, T S, T, T, T S, D, 1 h, T, T S, D, S y > h h = 1 1 h h < h. x D, S, T, A, B τ D τ 1, τ τ = Aτ 1, τ = Bτ τ 1 = A 1 Bτ D σ, ρ D (ii) D D S, T,

125 44. J(τ) J(τ) D J(τ) D τ = i g 3 = 0 (7 ) J(i) = 1. τ = ρ g = 0 (1) τ = x + yi J(ρ) = 0. q = e τπi = e yπ e xπi. y q = e yπ 0 () J(τ) D τ a J(τ) = a τ x cc J(τ) = a τ cc D a = 0, 1 J(τ) = a τ D τ N = 1 J (τ) πi J(τ) a dτ, J( ) =. 37 { i ρ c c } ρ N = πi ρ i ρ c c J(τ) i ρ ρ + = 0, i ρ c c + = 0. (1) ρ N = 1 πi J (τ) J(τ) a dτ = 1 [ ] c log{j(τ) a} πi c (1) P 1 w 4 w = ω 1 (m + nρ), ω 1 ( n + m nρ), ω 1 ( m + n mρ) 0

126 118 y () J(τ) a = eyπ e xπi (1 + ε), y ε 0 [ ] c [ ] c log{j(τ) a} = πi + log(1 + ε) πi. c c N = 1 J(τ) a D J(τ) = a (a = 0, 1) τ D τ = ρ, i τ A A S A = SA B = T B (1) N = 1 J(τ) = 1 τ J(i) = 1 i τ N 39 cρ + B c c = B +πi T B = B B + B i B+B = B + =. B B B B + B J(τ) 1 = 0 i J(τ) 1 = 7g 3 g 3 7g 3 38 i J(τ) 1 = 0 g 3 = 0 ν = πi.ν. B+B 39 B = πi.ν,

127 44. J(τ) 119 N = 1 ( ) +πi = 1 ν. πi B N 0, ν 1 N = 0, ν = 1. J(τ) 1 D i τ = i J(τ) = 1 J(τ) = 0 ρ A c + + = + +πi B c A B A = T SA, A = S 1 T A, B = T B, B = S 1 T S 1 B, B = S 1 B C = A + B + A + B + A + B C ( ) = 3 +. A B 40 C J(τ) ρ J(τ) = 0 g 3 = 0 3 3ν A + = πi.ν, B N = 1 ν N = 0, ν = 1 J(τ) D τ = ρ 0 3 J(τ) = a τ a = 0, 1 D a = 1 τ = ia = 0 τ = ρ i 1 ρ 1 3 D J(τ) J(τ) = J(τ ) τ τ J τ τ τ J(τ ) = J(τ) J(τ ) = J(τ) τ τ J τ τ

128 10 τ J(τ) τ x = R(τ) = 0 τ ω 1 ω 3 ω 1 g, g 3 J(τ) R(τ) = ± 1 ω 1 41 τ = 1 ω 4 J(τ) r = 1 R(r) = τ D y J(τ) τ =, ρ, i τ J 43 J =, 0, 1 J τ J, 0, 1 Riemann

129 J(τ) J(τ) R{J(τ)} = f(τ) ( I ) τ f(τ) ( II ) τ D τ = f(τ) (III) τ D f(τ) ( ) (IV) f(τ) D (I) (II) (III) τ f(τ) J(τ) (IV) f(τ) J f(τ) = F (J) τ D J (, 1) F (J) (II) f(τ) (I) J F (J) F (J) J (III) F (J) J f(τ) (I) ( I ) τ f(τ) m (II) (III) f(τ) (Modular function) J(τ) m = 1 f(τ) τ f(τ), f 1 (τ), f (τ),, f m 1 (τ) (1) G f(τ) G 0 G = G 0 + H 1 G 0 + H G H m 1 G 0 = G 0 + G 0 H G 0 H G 0 H 1 m 1. H 1, H, f(τ) f 1 (τ), f (τ), f 1 (τ) H 1 G 0 H 1 1 G 0 f 1 (τ) τ m (1) f (τ) (1) m {x f(τ)}{x f 1 (τ)} {x f m 1 (τ)} = x m + R 1 x m R m = 0

130 1 R (1) R J(τ) J(τ) f(τ) τ J(τ) f(τ) f(τ) (I ) (II) (III) 43 G J(τ) G G 0 (i) (ii) G 0 f(τ) G 0 G D H 1 1, H 1, D 46. λ(τ) (u ω 1, ω 3 ) ω 1, ω 3 ( ) (ω 1, ω a b 3 ) = (ω c d 1, ω 3 ) ( a, b, c, d, ad bc = 1 ( ) a b (ω 1 ) = e 1 c d a 1, b 0 (mod. ) e 1 e, e 3 a + c 1, b + d 1 (mod. ) ) c 0, d 1 (mod. ) ( ) a b a, b, c, d c d ( ) 1 0, 0 1 ( ) 0 1, 1 0 ( ) 1 1, 1 0 ( ) 1 1, 0 1 ( ) 1 0, 1 1 ( ) ad bc = 1 e 1, e, e 3 λ(τ) = e e 3 e 1 e 3

131 46. λ(τ) 13 ( 1 ) 0 ( 0 ) 1 ( 1 ) 1 ( 1 ) 1 ( 1 ) 0 ( 0 ) e 1 e 1 e 3 e e e 1 e 3 e e e e 3 e 1 e 3 e 1 e 3 e 3 e 1 e 1 e 3 e e λ λ 1 λ 1 1 λ 1 λ λ λ 1 λ 1 λ λ(τ) ( ) 1 0 m = 6, G 0 = 0 1 λ J J = 4 7 (λ λ + 1) 3 λ. (λ 1) λ J λ J 89 (9) λ = k sn λ(τ) S, T ( ) 1 0 S =, T = 1 1 ( 0 ) ( ) ( ) ( ) T S = =, ( ) ( ) ( ) ( ) ST S = =, ( ) ( ) ( ) S T = =, , S, T, T S, ST S, S 1 T λ(τ) 35 1, S, T, T S, ST S, S 1 T 1, S 1, T, S 1 T, S 1 T S 1, T S D T T D

132 14 0 x = 0, x = 1, (x + 1) + y = 1, 1, S, T, T S, ST S, S 1 T λ(τ) D 1, S, T, T S, ST S, S 1 T 44 E λ(τ) E a λ(τ) = a τ 44 τ J(τ) = 4 (a a + 1) 3 7 a (a 1) (= A ) J(τ) = A τ D, S, T, E λ(τ) = a λ(τ) = a, 1 a, 1 1 a, 1 a, a a 1, a 1 a λ(τ) = a τ E (1) (1) a = 0, 1, ; 1,, 1 ; 1 ± 3i A = 0, 1, J(τ) τ E λ τ q = e τπi = e πy > 0, (τ = x + yi)

133 46. λ(τ) 15 q λ(τ) = e e 3 e 1 e 3 = = ( ϑ ( ϑ 3 λ(τ) ) 4 ( 37 ) q q q + y q 0, λ(τ) 0. τ = i 5 e 3 = e 1, e = 0 λ(i) = e 1 e 1 + e = 1. ) 4 > 0, τ i λ ( 0, 1 ) i T i 0 λ 13 1 λ τ (0, i) λ ( ) 1, 1 τ S 1 τ x = 1 λ (0, ) x = 1 x = 1 T τ ( 1, 0) λ (1, ) τ 46 λ

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