Z: Q: R: C: 3. Green Cauchy

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2 Z: Q: R: C: 3. Green Cauchy Taylor Heisenberg Riemann Jacobi Jacobi, Euler Jacobi snu, κ, cnu, κ, dnu, κ sn u sn u Jacobi

3 A 7 B

4 . Green Green fx, y R D D x, y f x x, y, f y x, y.. D x D {x, y R ψx < y < φx, a < x < b} ψx, φx a x b a < x < b ψx < φx D C f y x, y dxdy fx, y dx D C D y f x x, y dxdy fx, y dy D C [ ] D f y x, y dxdy b a b a b a φx f y x, y dy dx ψx [ fx, y ] φx ψx dx fx, φx fx, ψx dx. C C + C + C 3 + C 4 C {x, ψx a x b}, C {b, t ψb t φb}, C 3 { t, φ t b t a}, C 4 {a, t φa t ψa} 3

5 fx, y dx fx, y dx, C C 4 b fx, y dx fx, ψx dx, C a a fx, y dx f t, φ t dt C 3 b fx, y dx C C fx, y dx b a b a fx, ψx dx b a b a fx, φx dx. fx, φx dx fx, φx fx, ψx dx D f y x, y dxdy. D y +. Green. D C,..., C m D D D x y D,..., D n P x, y, Qx, y D Q x P dxdy P dx + Q dy. y D [ ] D i. D ω P dx + Q dy dω Q dω x P dxdy y Green dω D D D ω 4

6 ..3. D C fz D z D fz z D fz fz lim z z z z f z fz z z D fz z D f z fz f z fz D f z D fz D z z fz z D α lim z z fz fz z z fz fz α, z z εz, z, z z, z z fz fz + αz z + z z εz, z,. lim z z εz, z. z x+iy, z x +iy, x, y x, y R fz ux, y + ivx, y, ux, y, vx, y x, y α a + ib, εz, z ε x, y, x, y + iε x, y, x, y.,. ux, y ux, y + a + ε x, y, x, y x x b + ε x, y, x, y y y, vx, y vx, y + b + ε x, y, x, y x x + a + ε x, y, x, y y y, lim ε j x, y, x, y, j,. x,y x,y 5

7 ux, y, vx, y x, y ux, y, vx, y x, y u x x, y a, u y x, y b, v x x, y b, u, v Cauchy-Riemann v y x, y a u x v y, u y v x.3 f z u x x, y + i v x x, y f z u x v y u y v x.3.4. D C fz D z x + iy, fz ux, y + ivx, y C D z zt xt + iyt, t ux, y + ivx, y dx + idy C uxt, yt dx dt + i vxt, ytdy dt uxt, yt dy + vxt, ytdx dt dt fz C fz dz C dt dt.4 Cauchy Cauchy-Riemann.3 Green. 6

8 .5 Cauchy. fz D D D D D D fz dz [ ] D Green fz dz u dx v dy + i u dy + v dx u y + v u dxdy + i x x v y dxdy. Cauchy-Riemann.6 Cauchy. D, fz.5 D z fz fζ πi D ζ z dζ [ ] r > r {ζ C ζ z r} D E D r E φζ fζ.5 ζ z fζ dζ. E ζ z E D r fζ D ζ z dζ fζ r ζ z dζ πifz r z x + iy ζ xθ + iyθ, xθ x + r cos θ, yθ y + r sin θ, θ π. r dζ π ζ z dx rcos θ + i sin θ dθ + idy dθ dθ π π r sin θ + ir cos θ dθ rcos θ + i sin θ i dθ πi. 7

9 r fζ fζ fz dζ πifz dζ ζ z r ζ z ε > r > ζ r fζ fz < ε π fζ fz dζ r ζ z fζ fz r ζ z dζ π ε r dθ ε. πr ε > r fζ fζ dζ πifz dζ πifz ζ z ζ z D fz πi r D fζ ζ z dζ.7. n, r > w, z C w r, z r, w z w z w + n + n + n + n+ z n+ z n+ w z. r n+3 [ ] w z w n+ z n+ w n+ z n+ w z w z zw n k w + n w n z + + wz + n z n w n + w n z + + wz n + z n w k+ z n+ k. n w z n + w z..4 w k+ z n+ k r n+ k 8

10 w z w + n + n+ z n+ z n+ n w k+ z + n + n+ k z n+ k n z n+ w k+ z n+ k k n k z k+ w k+ z n+ k..4 n + k + w z w n+ z n+ + n + z n+ n z k+ w k+ z n+ k n k + w z r k+ r n+ k k k n + n + w z. r n+3.8. D.5 φζ D z D φζ f n z dζ n,,,... ζ z n+ D f n z D f nz n + f n+ z [ ] z D r > r {ζ C ζ z r} D ζ D ζ / D ζ z > r < h < r, ζ D ζ z h ζ z h > r r r.7 z ζ z, w ζ z h h ζ z h + n + n+ ζ z n+ ζ z n+ n + n + h. r n+3 9

11 f n z + h f n z n + f n+ z h D h ζ z h n + φζ dζ n+ ζ z n+ ζ z n+ D h ζ z h n + n+ ζ z n+ ζ z n+ φζ d ζ n + n + h φζ dζ D r n+3 n + n + r n+3 ML h, M max ζ D φζ, L D dζ. f n z + h f n z lim n + f n+ z h h f n z D f nz n + f n+ z f n+ z f n+ z f nz f n z.9. D, fz.5, fz n f n z n! fζ dζ πi ζ z n+ [ ].6 D fz fζ πi D ζ z dζ. φζ fζ.8 f z fζ πi ζ z dζ. n D f n z n! fζ dζ. πi D ζ z n+.8 f n z f n+ n +! fζ z dζ. πi ζ z n+ D

12 .5 Taylor.. D.5 fz D c D R > R {z C z c R} D fz R fz a n z c n n a n n! f n c πi R fζ dζ. ζ c n+ [ ] < r < R z c < r fz fζ πi R ζ z dζ. ζ R, z c < r z c ζ c < r R < ζ z ζ c z c ζ c ζ c n n z c. ζ c ζ R fz n z c fζ dζ πi R ζ c ζ c n z c n fζ dζ πi n R ζ c n+ a n z c n, n a n πi R fζ ζ c dζ f n c. n+ n!.. fz D c D fz c c fz [ ]. r > z c < r fz a n z c n n

13 Taylor a fc fz a n, n,..., m, a m m gz a m + a m+ z c + a n z c n m z c < r gz fz z c m gz gc a m gz z c < δ < r z c < δ gz a m < a m / gz a m gz a m > a m / < z c < δ fz z c m gz.. D fz, gz D E fz gz fz gz [ ] F z fz gz c D E c n E, c n c, lim n c n c F c n fc n gc n F z..3 Liouville. fz fz [ ] z C fz M R >. z < R fz a n z n n nm R {ζ C ζ R} a n fζ dζ πi R ζn+ a n π fζ dζ M R ζ n+ πr n+ πr M R n. R > R a n, n fz a.6 D c D R D c R

14 .4. D {c} fz R a n πi fz r n a n z c n fζ dζ < r < R. ζ c n+ [ ] < z c < R z ε, ε >, r, r > < ε < ε z c r fζ fz < r < R gζ gζ ζ D {c, z} ζ z. ζ z Taylor fζ fz + b ζ z + b ζ z + gζ fζ fz ζ z b + b ζ z + ζ z ζ gζ ζ D {c} gζ {ζ D ε ζ c r}.5 r {ζ ζ c r}, ε {ζ ζ c ε} gζ dζ r ε gζ dζ gζ dζ r ε.6 dζ. πi r ζ z /ζ z ε.5 dζ. ε ζ z 3

15 gζ dζ fζ fz dζ πi r πi r ζ z fζ fz dζ πi r ζ z πi r ζ z dζ fζ dζ fz πi r ζ z gζ dζ fζ fz dζ πi ε πi ε ζ z πi πi fz πi r ε ε fζ fz dζ ζ z πi fζ ζ z dζ. fζ ζ z dζ πi ε ζ z z c n ζ c n+ n ε ζ z dζ fζ ζ z dζ. ζ r fζ πi ζ z dζ a n z c n, a n fζ dζ. πi ζ c n+ r n ζ z ζ c n z c n+ n ζ ε fζ πi ζ z dζ a n z c n, a n ζ c n fζ dζ. πi ε ε n r.5..4 fz z c Laurent c fz Laurent z c n a n z c n c fz 3 4

16 i fz z c D fz z c fc c fz fz a n z c n n a fc fz a, a,... a m fz z c m a m + a m+ z c + a m+ z c +, a m. m fz m fz z c m gz gz a m + a m+ z c + a m+ z c + c gz ii < z c < R fz a m z c + + a m z c + n a n z c n, a m c fz m gz z c m fz a m + a m+ z c + a m+ z c + z c < R gc a m c gz iii c fz a fz c Res zc [fz].4 r > c r r r fζ dζ πi Res zc [fz]..6. fz D c,..., c m D D fz dz πi m Res zcj [fz]. j 5

17 [ ] r > c j r r c j E D m j r c j fz Ē D m j r c j o.5 fz dz E D m j r c j D fz dz m j E r c j fz dz πi m Res zcj [fz]. j... fz C C c U c fz gz/hz, gz, hz U c hz fz.. fz C ω C fz + ω fz ω fz Ω fz Ω.3. C fz R ω, ω fz ω, ω ω, ω Ω Zω + Zω fz + ω fz, ω Ω. ω, ω C R Ω ω, ω Ω Zω + Zω C Ω C/Ω z, w C w z Ω w z mod Ω 6

18 C C/Ω z [z] C/Ω [z ] + [z ] [z + z ] C/Ω C/Ω p : C C/Ω, pz [z] U C/Ω p U C z C z, z z + ω, z z + ω, z 3 z + ω + ω z z z z 3 P [z ] z z 3, z z 3 P [z ] {z z + rω + sω r, s < } P [z ] z 3 z + ω + ω z z + ω P [z ] z z + ω z.4. z C z P [z ] z z mod Ω U P [z ] pu C/Ω p pu ω Ω ω + U p U : U pu.4 C/Ω Ū z z z +sω, s z z 3 z + ω + sω, s z z z + rω, r z z 3 z + rω + ω, r C/Ω 7

19 C/Ω q C/Ω q pz q z C z D z C pd z p Dz : D z pd z V q pd z q f p Dz : V q pd z D z C V q q, q C/Ω pz q, pz q z, z C f : V q D z, f : V q D z q V q V q pf q pf q q f q f q ω Ω f q ω + f q D z ω + D z D z, D z V q V q V q V q Ω, q f q f q f q f q + ω, q V q V q. f f : f V q V q f V q V q z z + ω C/Ω Riemann.5. α C, α Ω Zω + Zω z αz Ω αω αω Zαω + Zαω C/Ω C/αΩ α ω ω /ω / R Iω /ω ω /ω ω /ω Iω /ω > Iτ > τ τ Ω Z + Zτ C/Ω 8

20 .3 ω, ω C R Ω Zω + Zω Ω KΩ KΩ KΩ fz KΩ f z KΩ.6. P [z ] [ ] fz P [z ] c c U fz gz/hz, gz, hz U hz c hz a,..., a n fz P [z ] a i m i n fz.7. C fz i m i [ ] fz P [z ] M C ω Ω ω + P [z ] fz + ω fz M fz C.3 fz.8. fz P [z ] [ ] P [z ] fz P [z ] A + B + C + D z 3 C z B.6 D z A z πi 9 A+B+C+D fz dz.

21 fz dz fz fz + ω dz, A+C A fz dz fz fz + ω dz. B+D D P [z ] fz.6 z z {P [z ] fz } {P [z ] fz }.9. [ ] fz P [z ].8.. fz r c C fz c P [z ] r [ ] gz f z gz b n fz c fz c fz c z b n hz, hz z b hz gz nz bn hz + z b n h z z b n hz n z b + h z hz h z hz z b Res zb gz n a m fz c fz c z a m hz, hz z a hz gz mz a m hz + z a m h z z a m hz m z a + h z hz h z hz z a Res za gz m

22 P [z ] fz c b,..., b l, n,..., n l P [z ] fz c a,..., a k, m,..., m k gz a,..., a k, b,..., b l m,..., m k, n,..., n l.8 gz P [z ] m m k + n + + n l. n + + n l m + + m k r r fz P [z ] a,..., a r a a. Abel. r fz P [z ] a,..., a r fz P [z ] b,..., b r a + + a r b + + b r mod Ω [ ] P [z ] fz φz z f z fz fz z ck hz, hz z c hz k φz z z c + z h z hz Res zc φz kc φz dz P [z ] kc z c + k + z h z hz a i k i b j l j P [z ].6 φz dz Res zai φz + Res zbj φz πi P [z ] k i a i + Res zbj l j b j r r b j a i. j i

23 φz + ω φz z + ω f z + ω fz + ω z f z fz z + ω f z fz z f z fz ω f z fz, A+C φz + ω φz ω f z fz φz dz φz φz + ω dz A f z ω fz dz, A B+D φz dz φz φz + ω dz D f z ω fz dz. D A f z fz dz πiz, x D f z fz dz πiz F x F, F A x f z + tω fz + tω ω dt F x f z + xω fz + xω ω f z fz dz Hx e F x fz +xω H x F xe F x fz + xω + e F x f z + xω ω f z + xω fz + xω ω e F x fz + xω + e F x f z + xω ω. Hx H H e F fz + ω e F fz

24 fz + ω fz e F f z f z F dz nπi, n Z dz mπi, A fz D fz m Z r b j j r i a i f z πi A+C fz dz + f z πi B+D fz dz nω + mω Ω. P [z ] fz z z P [z ] fz a i, b j mod Ω C/Z + Zτ, τ C, Iτ > H {τ C Iτ > } expπix ex expπix 3.. z, τ C H θz, τ n Z e n τ + nz θz, τ C H θz, τ C H [ ] Iz < r, Iτ > s > θz, τ α a + ib C eα expπia + ib expπai exp πb 3

25 eα exp πb exp πiα e n τ + nz exp πn Iτ πniz < exp πs n expπr n. 3. n r < n s exp πs n expπr n < exp πs n n n 3. a >, b n Z n n n > a n + b 3.3 y x 3., 3.3 e n τ + nz < exp πs a n +b C C n. 3.4 C exp πbs, C exp πas asπ < < C < e n τ + nz C C n C + C C n <. n Z n Z θz, τ Iz < r, Iτ > s > a n τ e n τ n 4

26 θz, τ n Z a n τenz 3.5 enz + enz θz +, τ θz, τ 3.6 z θz, τ θz, τ θz + τ, τ e n τ + nz + τ n Z e n + τ τ + nz n Z e n Z n + τ τ + n + z z e τ z e n Z e τ z θz, τ. n + τ + n + z θz + τ, τ e τ z θz, τ , 3.7 m, n Z θz + mτ + n, τ e m τ mz θz, τ a, b R θ a,b z, τ e a τ + az + b θz + aτ + b, τ θ a,b z, τ e a τ + az + b e n τ + nz + aτ + b n Z e n + a τ + n + az + b. 3.9 n Z 5

27 θ, z, τ θz, τ, 3. θ a,b z + b, τ θ a,b+b z, τ, 3. e a τ + a z θ a,b z + a τ, τ e a bθ a+a,bz, τ, 3. θ a+p,b+q z, τ eaqθ a,b z, τ. 3.3 z, τ C H, a, a, b, b R, p, q Z [3., 3.3 ] e a τ + a z θ a,b z + a τ, τ e n + a τ + n + az + a τ + b + a τ + a z n Z e n + a + a τ + n + a + a z + b a b n Z e a bθ a+a,bz, τ, θ a+p,b+q z, τ e n + a + p τ + n + a + pz + b + q n Z e m + a τ + m + az + b + mq + aq m Z τ H eaqθ a,b z, τ. V {fz fz, fz + mτ + n e m τ mzfz m, n Z} V C 3.. fz i fz V. ii fz + mτ e m τ mzfz, fz + n fz, m, n Z. iii fz + τ e τ zfz, fz + fz. [ ] i ii, ii iii ii i. fz + mτ + n fz + mτ e m τ mzfz. 6

28 fz V iii ii. fz + fz fz +n fz fz + τ e τ zfz m fz + τ + n fz + τ e τ zfz m fz + mτ e m τ mzfz fz + m + τ fz + mτ + τ m < e τ z + mτfz + mτ e τ z + mτe m τ mzfz e m + τ m + zfz M > fz fz + M fz fz c n enz, c n C n /MZ [ ] fz M z x + iy fz c n yenx, n /MZ c n y M c n yenx M M M M M enz M enz M fu + iye nu du fu + iye nu + nx du ft + x + iye nt dt M z+m z ft + ze nt + z dt fwe nw dw. z x + iy, z x + iy C z, z + M, z, z + M C Cauchy fwe nw dw. C 7

29 fwe nw dw z+m C z z z +M fwe nw dw + + fwe nw dw + z +M z+m z z +M z z+m z z fwe nw dw fwe nw dw + z +M z+m z z z fwe nw dw fwe nw dw z+m z z z z z +M z fwe nw dw fwe nw dw fwe nw dw fwe nw dw c n c n yenx c n enz fz c n enz. n /MZ fwe nw dw fwe nw dw 3.4. fz i fz V. ii fz n /Z c m c n c n e n τ + nz m n mod Z [ ] i ii. fz V fz + fz 3.3 fz c nenz n /Z n /Z c n c n e n τ fz c n e n τ enz 8

30 fz + τ e τ zfz fz + τ c n e n τ enz + τ n /Z n /Z e τ zfz e τ z n /Z m /Z c n e n τ + nτ enz, c n e n /Z c n e n τ enz n 4τ en z. n m n 4 m + 4m e τ zfz c m+ e m τ + mτ emz. n /Z c n+ c n ii fz + fz fz + τ c n e n τ + nz + τ n /Z c n e n + τ + n + z τ z n /Z e τ z e τ z n /Z n /Z e τ zfz. 3. fz V 3.5. dim C V 4. c n e n τ + nz c n e n τ + nz [ ] V fz fz c n e n τ + nz, c n+ c n n /Z c, c /, c, c 3/ c n+ c n dim C V 4 9

31 3.6. θ, z, τ, θ, z, τ, θ,z, τ, θ, z, τ V [ ] a,, b, θ a,bz, τ 3.9 θ a,b z, τ e n + a τ + n + az + b n Z c m e m τ + mz. m /Z c m { emb, m a mod Z,, m a mod Z m m mod Z m b mb mod Z c m c m 3.4 θ a,b z, τ V k a,b k, θ, z, τ + k, θ, z, τ + k,θ,z, τ + k, θ, z, τ m,,, 3 e m τ + mz k, k, k, + k,, k, + k, e, 4 k, k,, 3 k, + k, e. 4, k, k, 4 V 3.5 dim C V 4 4 V V {fz fz, fz +mτ +n e m τ mz fz m, n Z} V V 3. fz V fz + fz, fz + τ e τ z fz 3.4 fz V fz c n e n τ + nz n Z 3

32 fz + τ n Z c n e c n e n τ + nz + τ n + τ + n + z τ z n + τ + n + z n τ + nz, n Z e τ z c n e n Z e τ z c n e n Z e τ z fz e τ z c n e n τ + nz n Z fz V c n c n, n Z c n c, n Z dim C V θz, τ V V θz, τ. 3.3 Heisenberg a, b /Z fz V S b fz fz + b, T a fz e a, b Z S b fz + fz + + b fz + b S b fz, a τ + az fz + aτ S b fz + τ fz + τ + b e τ z bfz + b e τ zs b fz, T a fz + e a τ + az + fz + + aτ e a τ + az fz + aτ T a fz, T a fz + τ e a τ + az + τ fz + τ + aτ e a τ + az + τ e τ z + aτfz + aτ e a τ + az τ z fz + aτ e τ z T a fz. 3. S b f, T a f V S b S b S b +b, T a T a T a +a 3

33 S b T a, T a S b ] S b T a fz S b [T a fz] S b [e a τ + az fz + aτ e a τ + az + b fz + aτ + b, T a S b fz T a [S b fz] T a fz + b e a τ + az fz + aτ + b, S b T a eabt a S b C {c C c } ρ : C /Z/Z /Z/Z GLV ρc, a, b c T a S b ρ ρ GLV ρc, a, b ρc, a, b c T a S b c T a S b c c T a ea b T a S b S b c c ea b T a +a S b +b, ρc, a, b c S b T a c S b T a c eab T a S b ρ ρ C /Z/Z /Z/Z G Heisenberg z, τ V θ z, θ z, θ z, θ z θ, z, τ, θ, z, τ, θ,z, τ, θ, S / θ z θ z + θ z, S / θ z θ z + θ z + θ z, S / θ z θ z + e 8 τ + z + θ z, S / θ z θ z + e 8 τ + z e 8 τ + θ z + τ θ z + τ + z + θ z + τ + θ z, 3

34 T / θ z e 8 τ + z θ z + τ θ z, T / θ z e 8 τ + z θ z + τ e e 4 8 τ + z + θ z + τ + T / θ z e 8 τ + z θ z + τ e 8 τ + z e 8 τ + z + τ θ z + τ e 8 τ + z e 8 τ + z + τ e T / θ z e 8 τ + z θ z + τ e 8 τ + z e 8 τ + z + τ + θ z + τ + e τ + z + e 4 τ z θ z + e 4 e θ z, 4 τ z θ z θ z, θ z. ρ,, θ z, θ z, θ z, θ z θ z, θ z, θ z, θ zr S, ρ,, θ z, θ z, θ z, θ z θ z, θ z, θ z, θ zr T, R S, R i T. i a, b /Z, c C Rc, a, b c R T a R S b GL 4 C ρc, a, bθ z, θ z, θ z, θ z θ z, θ z, θ z, θ zrc, a, b G c, a, b Rc, a, b GL 4 C ρc, a, bθ ij z ce a τ + az θ ij z + aτ + b ce a τ + az θ z + aτ + b, θ z + aτ + b, θ z + aτ + b, θ z + aτ + b θ z, θ z, θ z, θ zrc, a, b

35 3.4 Ωτ Z + Zτ, E τ C/Ωτ E τ P fz V fz Ωτ 4 [ ] Ωτ P [] τ C τ + D P [] B A fz P [] f z πi A+B+C+D fz dz. fz + fz, fz + τ e τ zfz f z + f z, f z + τ 4πie τ zfz + e τ zf z f z πi fz dz f z πi fz f z + dz fz + B+D D, f z πi A+C fz dz f z πi A fz f z + τ dz fz + τ f z πi A fz 4πifz + f z dz fz dz 4. A 3.8. z θ, z, τ θ, z, τ θ, z, τ. θ,, τ 34

36 [ ] 3.9 θ, z, τ e n + τ + n + z + n Z n n θ, z, τ e n Z n Z e n Z e n Z e n Z e n + τ + n + z + n + τ + n + z + n + τ + n + z n + τ + n + z + n θ, z, τ. n + τ + n + z θz, τ { p + τ + q + } p, q Z [ ] θ, z, τ e τ + z + θ z + τ +, τ θ τ +, τ 3.8 θ p + τ + q +, τ, p, q Z Ωτ P [] θ, z, τ 3.7 4,, τ, τ + 4 θ, z, τ z pτ + q, p, q Z θz, τ z p + τ + q +, p, q Z 35

37 3.. a, b {, /} θ a,b z, τ { a + p + τ + b + q + } p, q Z a, b a, b θ a,b z, τ θ a,b z, τ [ ] θ a,b z, τ e a τ + az + b θz + aτ + b, τ z C θ, z, τ, θ, z, τ, θ,z, τ, θ, z, τ,,, P 3 θ, z, τ : θ, z, τ : θ,z, τ : θ, z, τ Φz θ, z, τ : θ, Φ : C P 3 z, τ : θ,z, τ : θ, z, τ a, b {, /} 3.6 θ a,b z, τ V θ a,b z +, τ θ a,b z, τ, θ a,b z + τ, τ e τ 4zθ a,b z, τ Φ φ : E τ C/Ωτ P 3 a, b /Z 3.4 φ z + aτ + b θ z + aτ + b : θ z + aτ + b : θ z + aτ + b : θ z + aτ + b θ z : θ z : θ z : θ z R, a, b. θ i, j z, τ θ ij z 3.. a, b /Z φ z + aτ + b φzr, a, b. 36

38 3.. φ : E τ P 3 E τ C/Ωτ P 3 [ ] φ z, z E τ, z z φz φz z, z C z z mod Ωτ a, b /Z z z + aθ + b, z z + aθ + b z, z, z, z E τ 4 aτ + b, ±z z mod Ωτ w C 5 z, z, z, z, w E τ 5 fz V, fz fz fz fz fz fw c j C, j,,, 3 fz c θ z + c θ z + c θ z + c 3 θ z 4 c, c, c, c 3 3 fz c θ z + c θ z + c θ z + c 3 θ z, fz c θ z + c θ z + c θ z + c 3 θ z, fw c θ w + c θ w + c θ w + c 3 θ w c, c, c, c 3,,, fz c θ z + c θ z + c θ z + c 3 θ z fz V, fz fz fz fw φz φz c C fz θ ij z c θ ij z, i, j, cfz cc θ z + cc θ z + cc θ z + cc 3 θ z c θ z + c θ z + c θ z + c 3 θ z fz. 37

39 fz 3. φz φ z + aτ + b φz R, a, b cfz cc θ z + cc θ z + cc θ z + cc 3 θ z cθ z, θ z, θ z, θ z t c, c, c, c 3 cθ z, θ z, θ z, θ z R, a, b t c, c, c, c 3 θ z, θ z, θ z, θ z R, a, b t c, c, c, c 3 θ z, θ z, θ z, θ z t c, c, c, c 3 fz. fz fz V mod Ωτ 5 z, z, z, z, w 3.7 φ φ dφz : T z T φz dφz T z dφz a, b /Z aτ + b z C z z mod Ωτ, dφz φ z + aτ + b φzr, a, b dφ z + aτ + b dφz R, a, b. z z + aτ + b w C z, z, w mod Ωτ 3 c, c, c, c 3 C fz c θ z + c θ z + c θ z + c 3 θ z, fz fz fw dφz θ ijz z z fz dφz z z fz fz V, fz mod Ωτ c, c, c, c 3 C H {ξ : ξ : ξ : ξ 3 P 3 c ξ + c ξ + c ξ + c 3 ξ 3 } 38

40 3.3. H φe τ P 3 4 [ ] fz c θ z + c θ z + c θ z + c 3 θ z fz V, fz z C φz H fz φ 3.7 #H φe τ #{z mod Ωτ fz } 4. φe τ P 3 4 φe τ 3.9 θ z e n τ + nz, 3.5 n Z θ z e n τ + n z +, 3.6 n Z θ z n Z e θ z n Z e θ z θz, τ, θ z θ z +, τ, θ z e 8 τ + z θ z e 8 τ + n + τ + n + z, 3.7 n + τ + n + z θ z + τ, τ z + θ, z + τ +, τ. 3.9 θ z z p + τ + q +, p, q Z θ z z p + τ + q, p, q Z θ z z pτ + q +, p, q Z θ z z pτ + q, p, q Z 39

41 3.8 θ z θ z, θ z, θ z θ z θ z, θ z θ z, θ z θ z, θ z θ z. q e τ e πiτ, w e z e πiz θ z, τ n Z q n w n. 3.9 θ, τ + q n. 3. n θ z, τ n Z n q n w n, 3. θ, τ + n q n, 3. n θ z, τ q n+ w n+, n Z 3.3 θ, τ q n+, 3.4 n θ z, τ i n Z n q n+ w n+, 3.5 z θ z, τ π z n Z n+ n + q n+, Riemann Riemann A t AA A t A 4I 4 A 4

42 3.5. u, u, u 3, u 4, v, v, v 3, v 4 u u u 3 A u u u 3, v v v 3 A v v v 3 u 4 u 4 v 4 v 4 [ ] 4 u iv i i 4 u i v i i t u iv j t u i t A A v j t u i v j. θ x θ x θ x 3 θ x 4 θ x θ x θ x 3 θ x e m i + m,m,m 3,m 4 Z i m,m,m 3,m 4 Z i m i θ x θ x θ x 3 θ x 4 4 e m i + τ + m,m,m 3,m 4 Z m,m,m 3,m 4 Z i i τ + i e 4 i 4 m i τ + i 4 m i x i, i m i + x i, θ x θ x θ x 3 θ x 4 4 e m i m i + τ + θ x θ x θ x 3 θ x 4 + θ x θ x θ x 3 θ x 4 4 i 4 m i x i. +θ x θ x θ x 3 θ x 4 + θ x θ x θ x 3 θ x 4 4 m,m,m 3,m 4 Z e 4 m i τ + m i x i. i i i m i + x i. m, m, m 3, m 4 Z 4

43 i,, 3, 4 m i Z, m + m + m 3 + m 4 Z. i,, 3, 4 m i + Z, m + m + m 3 + m 4 Z. n n n 3 n 4 A m m m 3 m 4, y y y 3 y 4 A x x x 3 x 4 n m + m + m 3 + m 4, y x + x + x 3 + x 4, n m + m m 3 m 4, y x + x x 3 x 4, n 3 m m + m 3 m 4, y 3 x x + x 3 x 4, n 4 m m m 3 + m 4, y 4 x x x 3 + x m, m, m 3, m 4 i m, m, m 3, m 4 ii n, n, n 3, n 4 Z. [ ] i ii. m i Z, i,, 3, 4, 4 i m i Z n Z m i + Z, 4 i m i Z n Z n + n m + m Z, n + n 3 m + m 3 Z, n + n 4 m + m 4 Z n, n 3, n 4 Z ii i. n i Z, i,, 3, 4 A A m n + n + n 3 + n 4, m n + n n 3 n 4, m 3 n n + n 3 n 4, m 4 n n n 3 + n 4 m i Z, i,, 3, 4 m Z m + m n + n Z, m + m 3 n + n 3 Z, m + m 4 n + n 4 Z m, m 3, m 4 Z 4

44 m + Z m + m n + n Z, m + m 3 n + n 3 Z, m + m 4 n + n 4 Z m, m 3, m 4 + Z m i n i, m i x i n i y i i i i θ x θ x θ x 3 θ x 4 + θ x θ x θ x 3 θ x 4 +θ x θ x θ x 3 θ x 4 + θ x θ x θ x 3 θ x n i τ + n i y i n,n,n 3,n 4 Z e i θ y θ y θ y 3 θ y 4 Rimenann R θ x i + θ x i + θ x i + i i i i i 4 θ x i i 4 θ y i Heisenberg S θ z θ z, S 3.7. θ z θ z, S i θ z θ z, S θ z θ z θ z + θ z, θ z + θ z, θ z + θ z, θ z + θ z. R x x + y i y i + θ yi + θ y i 3.7 R 4 θ x i + i 4 θ x i i 4 θ x i i Heisenberg T θ z θ z, T θ z iθ z, T 4 θ x i i 4 θ y i. i θ z θ z, T θ z iθ z 3.8. e τ + z e τ + z θ z + τ θ z, θ z + τ θ z, e τ + z e τ + z θ z + τ θ z, θ z + τ θ z. 43

45 R x x + τ e τ + x yi y i + τ e 8 τ + y i θ yi + τ θ y i, 4 i y i x 3.8 R θ x i θ x i + θ x i θ x i θ y i. i i i i i 3.7, e τ + z e τ + z θ z + τ + θ z, θ z + τ + θ z, e τ + z e τ + z θ z + τ + θ z, θ z + τ + θ z. R x x + τ + e τ + x y i y i + τ + e τ + yi + θ yi + τ + 8 θ y i, 4 i y i x 3.8 R4 4 θ x i i 4 θ x i i 4 θ x i + θ ab x θ cd x θ ef x 3 θ gh x 4 i [ab, cd, ef, gh] θ ab y θ cd y θ ef y 3 θ gh y 4 [ab, cd, ef, gh] 4 θ x i R R4 i 4 θ y i. [,,, ] + [,,, ] + [,,, ] + [,,, ] [,,, ], [,,, ] + [,,, ] [,,, ] [,,, ] [,,, ], [,,, ] [,,, ] + [,,, ] [,,, ] [,,, ], [,,, ] [,,, ] [,,, ] + [,,, ] [,,, ]. R5 R [,,, ] + [,,, ] + [,,, ] + [,,, ] [,,, ], [,,, ] + [,,, ] [,,, ] [,,, ] [,,, ], [,,, ] [,,, ] + [,,, ] [,,, ] [,,, ], [,,, ] [,,, ] [,,, ] + [,,, ] [,,, ]. 44 i

46 [,,, ] + [,,, ] + [,,, ] + [,,, ] [,,, ], [,,, ] + [,,, ] [,,, ] [,,, ] [,,, ], [,,, ] [,,, ] + [,,, ] [,,, ] [,,, ], [,,, ] [,,, ] [,,, ] + [,,, ] [,,, ]. [,,, ] + [,,, ] + [,,, ] + [,,, ] [,,, ], [,,, ] + [,,, ] [,,, ] [,,, ] [,,, ], [,,, ] [,,, ] + [,,, ] [,,, ] [,,, ], [,,, ] [,,, ] [,,, ] + [,,, ] [,,, ]. [,,, ] + [,,, ] + [,,, ] + [,,, ] [,,, ], [,,, ] + [,,, ] [,,, ] [,,, ] [,,, ], [,,, ] [,,, ] + [,,, ] [,,, ] [,,, ], [,,, ] [,,, ] [,,, ] + [,,, ] [,,, ]. Riemann 3.6 θ Riemann R, R4 x x x, x 3 x 4 u y x + u, y x u, y 3, y 4 θ x θ u + θ x θ u + θ x θ u + θ x θ u θ x + uθ x uθ, θ x θ u θ x θ u θ x θ u + θ x θ u. A θ x + uθ x uθ θ x θ u + θ x θ u θ x θ u + θ x θ u. R, R4 θ x θ u + θ x θ u θ x θ u θ x θ u θ x + uθ x uθ, θ x θ u θ x θ u θ x θ u + θ x θ u. 45

47 A θ x + uθ x uθ θ x θ u θ x θ u θ x θ u θ x θ u. R3, R4 θ x θ u θ x θ u + θ x θ u θ x θ u θ x + uθ x uθ, θ x θ u θ x θ u θ x θ u + θ x θ u. A3 θ x + uθ x uθ θ x θ u θ x θ u θ x θ u θ x θ u. R7 x x 3 x, x x 4 u θ xθ xθ uθ u + θ xθ xθ uθ u θ x + uθ x uθ θ, A4 θ x + uθ x uθ θ θ xθ xθ uθ u + θ xθ xθ uθ u. R4, R5 x x x, x 3 x 4 u θ x θ u + θ x θ u θ x θ u θ x θ u, θ x θ u θ x θ u + θ x θ u θ x θ u θ x + uθ x uθ, A5 θ x + uθ x uθ θ x θ u θ x θ u θ x θ u θ x θ u. R6 x x 3 x, x x 4 u θ xθ xθ uθ u θ xθ xθ uθ u θ x + uθ x uθ θ, A6 θ x + uθ x uθ θ θ xθ xθ uθ u θ xθ xθ uθ u. R6 x x 3 x, x x 4 u θ xθ xθ uθ u θ xθ xθ uθ u θ x + uθ x uθ θ, 46

48 A7 θ x + uθ x uθ θ θ xθ xθ uθ u θ xθ xθ uθ u. A, A5 u θ x θ θ x θ + θ x θ, 3.7 θ x θ θ x θ θ x θ. 3.8 θ a,b z, τ z τ θ a,b, τ 3.7 x Jacobi θ 4 θ 4 + θ θ +3.8 θ 3.9 θ x θ + θ x θ θ θ x θ 4 + θ 4 θ x θ 4, θ x θ θ x θ + θ x θ φ : E τ C/Ωτ P 3 φz θ z, τ : θ z, τ : θ z, τ : θ z, τ 3.7, 3.8 θ θ z θ θ z θ θ z, θ θ z θ θ z + θ θ z P 3 a : a : a : a 3 θ z : θ z : θ z : θ z θ a θ a θ a, θ a 3 θ a + θ a a : a : a : a 3 P 3 P 3 θ a θ a θ a, θ a 3 θ a + θ a 47

49 f x, x, x, x 3 θ x θ x θ x, f x, x, x, x 3 θ x 3 θ x + θ x f, f C x, x, x, x 3 ] f f V f, f {x : x : x : x 3 P 3 f x, x, x, x 3 f x, x, x, x 3 } φ φe τ V f, f 3.3 φe τ V f, f 3.3 V f, f P φe τ V f, f P 3 C V f, f φe τ C 3.33 P 3 H {x : x : x : x 3 P 3 b x + b x + b x + b 3 x 3 } b, b, b, b 3 C H V f, f V f, f # H V f, f 4 b θ z + b θ z + b θ z + b 3 θ z Ωτ 4 # H φe τ # H C. φe τ V f, f 48

50 A4, A θ x + yθ x yθ θ θ xθ xθ yθ y + θ xθ xθ yθ y, θ x + yθ x yθ θ x θ y θ x θ y. θ x + y θ θ θ x + y θ θ xθ xθ yθ y + θ xθ xθ yθ y θ x θ y θ x θ y 3.34 A6, A θ x + yθ x yθ θ θ xθ xθ yθ y θ xθ xθ yθ y, θ x + yθ x yθ θ x θ y θ x θ y. θ x + y θ θ xθ xθ yθ y θ xθ xθ yθ y 3.35 θ x + y θ θ x θ y θ x θ y A7, A θ x + yθ x yθ θ θ xθ xθ yθ y θ xθ xθ yθ y, θ x + yθ x yθ θ x θ y θ x θ y. θ x + y θ θ xθ xθ yθ y θ xθ xθ yθ y 3.36 θ x + y θ θ x θ y θ x θ y θ ij θ ij u πθ x Jacobi sn u, cn u, dn u sn u θ θ θ x θ x, cn u θ θ θ x θ x, 3.37 dn u θ θ θ x θ x. ω π θ φ E τ 4ω, 4ωτ 3.34, 3.36, 3.35 κ θ, κ θ θ θ

51 3.. sn u cn v dn v + sn v cn u dn u snu + v, κ sn u sn v cn u cn v sn u dn u sn v dn v cnu + v, κ sn u sn v dnu + v dn u dn v κ sn u cn u sn v cn v. κ sn u sn v 3.7 E τ C/Ωτ P 3 4 V f, f E τ 4 V f, f φ : E τ P 3 σx : x : x : x 3 x : θ x : θ x : θ x 3 θ θ θ φ σ φ : E τ P 3 φz σx : x : x : x 3 X : X : X : X 3 f i x, i, X + X3 θ x + θ x 3 θ θ θ θ x + θx3 θx x X, X + κ X 3 θ θ x + θ θ θ 4 θ θ θ x + θ x + θ θ 4 θ4 + θ 4 x θ 4 x X X : X : X : X 3 φz θ x 3 θ θ θ θ x 4 + θ θ x θx θ θ x 4 3 X + X 3 X, X + κ X 3 X E, κ φ : E τ E, κ 3.. X : X : X : X 3 E, κ X i X j i < j φ θ z φz : dn u : cn u : sn u 5

52 u πθ z X : X : X : X 3 φz, Y : Y : Y : Y 3 φz 3. φz + z W : W : W : W 3 W X Y κ X 3Y 3, W X X Y Y κ X X 3 Y Y 3, W X X Y Y X X 3 Y Y 3, W 3 X X 3 Y Y + X X Y Y 3. Z X X 3 Y Y X X Y Y 3 Z W X X 3 Y Y X X Y Y 3 W, Z W X X 3 Y Y X X Y Y 3 X X Y Y κ X X 3 Y Y 3 XX X 3 Y Y Y κ X X X3Y Y Y 3 X XX Y Y Y 3 + κ X XX 3 Y Y Y3 X X 3 Y Y XY + κ XY 3 X X Y Y 3 XY + κ X3Y X X 3 Y Y X Y κ Y3 + κ X X3Y 3 X X Y Y 3 X κ X3Y + κ X3Y Y3 X X 3 Y Y X X Y Y 3 X Y κ X 3Y 3 X X 3 Y Y X X Y Y 3 W. Z W X X 3 Y Y X X Y Y 3 X X Y Y X X 3 Y Y 3 XX X 3 Y Y Y X X X3Y Y Y 3 X X XY Y Y 3 + XX X 3 Y Y Y3 X X 3 Y Y XY + XY 3 X X Y Y 3 X3Y + XY X X 3 Y Y X Y Y 3 + X κ X 3Y 3 X X Y Y 3 X 3Y κ Y 3 + X X 3Y X X 3 Y Y X X Y Y 3 X Y κ X 3Y 3 X X 3 Y Y X X Y Y 3 W. Z W 3 X X 3 Y Y X X Y Y 3 X X 3 Y Y + X X Y Y 3 XX 3Y Y XX Y Y3 XX 3Y κ Y3 Y Y3 X κ X3X X3Y Y3 XX 3Y 4 + κ Y Y3 + κ Y3 4 X 4 + κ XX 3 + κ X3Y 4 Y3 XY X3Y XY 3 κ X3Y 3 X3Y XY 3 X 3Y X Y 3 X Y κ X 3Y 3 X 3Y X Y 3 W. 5

53 Z X X 3 Y Y X X Y Y 3, Z X X 3 Y Y X X Y Y 3, Z X X 3 Y Y X X Y Y 3, Z 3 X3Y XY 3 Z W P 3 W : W : W : W 3 Z : Z : Z : Z 3 Z W i Z i W, i,,, 3 W W Z Z i < j 3 W i W j Z i W i W j W W W i Z j Z i W j W i Z j W W j Z i W Z i Z j W Z j W i Z W j W j Z W i. W i Z i W j Z j W W W W 3 Z Z Z Z Z i W i, i,,, 3 Z W 3 X X 3 Y Y X X Y Y 3 i X Z 3 X3Y 3. X X 3 Y W X X 3 Y Y 3 X Y 3. X X 3, Y Y 3 Y X Y ii X W X X Y Y X 3. 5

54 X X Y Y Y Y Z X X 3 Y Y X X X 3 Y Y Y Y X Y X Y iii X X 3 X X 3 Y Y Y W X X Y Y Y X Y X Y X Y X 3 Y 3 W XY 3.39 λ C Z, Z, Z, Z 3 λw, W, W, W 3 P 3 Z : Z : Z : Z 3 W : W : W : W 3 E, κ X X : X : X : X 3, Y Y : Y : Y : Y 3 { W : W : W : W 3, W, W, W, W 3,,,, X + Y Z : Z : Z : Z 3, Z, Z, Z, Z 3,,,, W X Y κ X 3Y 3, Z X X 3 Y Y X X Y Y 3, W X X Y Y κ X X 3 Y Y 3, Z X X 3 Y Y X X Y Y 3, W X X Y Y X X 3 Y Y 3, Z X X 3 Y Y X X Y Y 3, W 3 X X 3 Y Y + X X Y Y 3, Z 3 X 3Y X Y 3. E, κ O φ : : : M, N C, M N EM, N {x : x : x : x 3 P 3 x Mx 3 x, x Nx 3 x } X : X : X : X 3 x : x : x : Mx 3 EM, N E, N/M κ N M EM, N EM, N x x : x : x : x 3, y y : y : y : y 3 { w : w : w : w 3, w, w, w, w 3,,,, x + y z : z : z : z 3, z, z, z, z 3,,,, w x y MNx 3y 3, z x x 3 y y x x y y 3, w x x y y + Nx x 3 y y 3, z x x 3 y y x x y y 3, w x x y y + Mx x 3 y y 3, z x x 3 y y x x y y 3, w 3 x x 3 y y + x x y y 3, z 3 x 3y x y 3. 53

55 3.. M 5, N 5 E5, 5 {x : x : x : x 3 P 3 x 5x 3 x, x + 5x 3 x }. x 4 : 3 : 49 : E5, , x : : : : 379 : 478 : , Jacobi θ, τ θ, τ z θ z, τ z 3.3 Jacobi. τ H 3.4. a, b, θ, τ π θ, τθ, τθ, τ z θ abz, τ 4πi τ θ abz, τ. [ ] θ ab z, τ e n + a τ + n + a z + b n Z 54

56 z θ abz, τ πi n Z τ θ abz, τ πi n + a n Z z θ abz, τ 4πi τ θ abz, τ. n + a e n + a τ + n + a z + b, e n + a τ + n + a z + b, 3.5. [ 3.3 ] θ ab z, τ θ ab z θ ab z z Taylor Riemann R7 [,,, ] + [,,, ] + [,,, ] + [,,, ] [,,, ] θ + θ x + θ + θ x + θ + θ x 3 + θ x θ x θ + θ x + θ + θ x + θ x θ x θ + θ x θ + θ x + θ x + 6 θ x 3 + θ + θ x 3 + θ + θ x θ x + 6 θ x 3 + θ + θ x + θ + θ x 3 + θ + θ x 4 + θ y + 6 θ y 3 + θ + θ y + θ + θ y3 + θ + θ y

57 y x + x + x 3 + x 4, y x + x x 3 x 4, y 3 x x + x 3 x 4, y 4 x x x 3 + x 4 x x 3 x 4 y i x, i,, 3, 4 θ θ θ θ x + 6 θ x 3 + θ x + 6 θ 8 x3 + θ + θ 4 x + θ + θ 4 x + θ + θ 4 x +. x 3 6 θ θ θ θ 4 θ θ θ θ + 8 θ θ θ θ + 8 θ θ θ θ + 8 θ θ θ θ. θ θ θ θ θ θ θ θ + θ θ θ θ + θ θ θ θ. θ θ θ θ θ θ θ + θ + θ, θ θ θ 3.4 θ θ θ θ θ. θ θ θ θ ab 4πi τ θ ab, θ 4πi τ θ F θ θ θ θ F F τ θ θ θ θ θ. θ θ θ 56

58 θ F 3., 3., 3.4, 3.6 θ θ θ τ i q e πiτ θ, τ, θ, τ, e 8 τ θ, τ, F π e 8 τ θ, τ π K C u n z n u nz K + u n z n K [ ] n u nz K < ε < / N Nε u n z < ε z K nn u < log + u n n un B u / log + u B u n N u n z nn u nz < ε log + u n z B u n z log + u n z B nn n u n z < Bε. nn 57

59 n log + u nz K sz log + u n z K + u n z n e sz K z θz, τ θz, τ { m + τ + n + } m, n Z n 3.7. z C, m Z i e m + τ z. ii n Z, πi z m + τ n + πi. iii n Z, z pz, τ m m + τ + n +. {[ + e m + ] [ τ z + e m + ]} τ + z C H c, d > Iz c, Iτ d m e + τ ± z e πc e πd m+ 3.6 pz, τ Iz c, Iτ d c, d pz, τ C H 3.7 pz, τ { m + τ + θz, τ 3.8. pz, τ i pz +, τ pz, τ. ii pz + τ, τ e τ z pz, τ. n + } m, n Z 58

60 [ ] i pz, τ ii pz + τ, τ m m [ + e {[ + e m + ] [ τ z τ + e m + ]} τ + z + τ {[ + e m ] [ τ z + e m + 3 ]} τ + z τ z ] m [ + e m + 3 ] τ + z m e [ ] τ z + e τ + z m [ + e m + ] τ + z e e m τ z m τ z pz, τ. [ + e m ] τ z [ + e m + ] τ z [ + e m + ] τ z m [ + e m + ] τ + z 3.9. θz, τ θz, τ cτpz, τ, cτ emτ. m [ ] cτ Iτ > emτ m e miτ <. m emτ τ H cτ m 3.8 pz, τ V V θz, τ C- τ c τ θz, τ c τpz, τ

61 c τ τ cτ c τ 3.4 θ z, τ, θ z, τ, θ z, τ θ z, τ θ z +, τ c τ m c τ θ z, τ e m [ + e m + τ z ] [ + e m 8 τ + z e 8 τ + z m e 8 τ + z c τe c τe 8 τ θ z, τ e 8 τ + ie m + τ + z + ] [ e m + ] τ z θ z + τ, τ c τ m [ + e m + τ + z + ] τ c τ 8 τ + z [ e 8 τ + z [ + e m m [ + e m + τ z ] τ [ + e mτ z] m [ + e z] [ e m + ] τ + z. [ + e m + τ + z] m [ + e mτ z] m [ + e mτ + z] m z + e ] z [ + e mτ z] [ + e mτ + z]. m z + τ +, τ z + θ c τ m m + c τ ie 8 τ + z ic τe 8 τ + z ic τe 8 τ [ e [ + e m + τ z τ ] τ + z + τ + ] [ e mτ z] m [ e z] [ e m + τ + z] m [ e mτ z] m [ e mτ + z] m z e ] z [ e mτ z] [ e mτ + z]. m 6

62 z θ, τ c τ θ, τ c τ m m θ, τ c τe [ + e m + τ], [ e m + τ], 8 τ m [ + e mτ]. [ θ z, τ e z e ] z hz, hz ic τe 8 τ [ e mτ z] [ e mτ + z] m z z θ, τ πi h πc τe 8 τ [ e mτ]. m 3.3 Jacobi πc τe 8 τ [ e mτ] m πc τ 3 e 8 τ [ + e m + ] τ m [ e m + τ] [ + e mτ]. m m 6

63 c τ m [ + e m + τ [ e mτ] m ] [ e [ e m + τ] [ e mτ] m m [ e m + τ] [ + e mτ] m [ e mτ] cτ. m m m + ] τ [ + e mτ] m c τ cτ c τ cτ Iτ cτ pz, τ, θz, τ Iτ pz, τ, θz, τ c τ cτ 3. Jacobi, Euler q e τ, w e z θz, τ m q m w m q m m + q m+ w + q m+ w m q m + q m w + q m w. 3.4 m Jacobi w iq /4 q q 3/ m m q m3m+ m q 3m q 3m q 3m m q m. m Euler 5 m q m3m+ q m. 3.4 m 6

64 3.4 w, w i q m q m + q m, 3.43 m m q m m m m q m q m a b 3.3. c d SL Z Ωτ Z + Zτ Zaτ + b + Zcτ + d. C/Ωτ C/ Zaτ + b + Zcτ + d Iτ >, c, d Z cτ + d cτ + d C z cτ + d z C C/ Zaτ + b + Zcτ + d C/ Z + Z aτ + b cτ + d a b 3.3. τ H, SL Z c d i aτ + b cτ + d H. ii cτ + d C/Ωτ C/Ω aτ + b cτ + d [ ] i aτ + b cτ + d aτ + bc τ + d cτ + d ac τ + bd + adτ + bc τ cτ + d 63

65 I aτ + b cτ + d ad bciτ cτ + d ii 3.45, z θ cτ + d, aτ + b cτ + d Iτ cτ + d >. θz, τ a d, b, c 3.3 Jacobi. z θ τ, e τ z e θ z, τ. τ 8 τ τ σ τ, Iσ > σ H [ ] z τ τ θ z, τ θ τ z, τ m + τ + n +, τ z n + τ + m + m, n, m, n Z z mτ + n + τ +, z m τ n + τ θ z, τ θ τ z, τ θ τ z, τ ψz e z τ θ z, τ 64

66 z 3.7, 3.8 ψz + θ τ z +, τ e z + τ θ z +, τ θ τ z τ, τ e z + τ θ z, τ e τ + τ z θ τ z, τ e z + τ θ z, τ θ τ z, τ e z τ ψz. θ z, τ ψz + τ θ τ z + τ, τ e z + τ τ θ z + τ, τ θ τ z +, τ e z + τ τ e τ z θ z, τ θ τ z, τ e z τ ψz. θ z, τ ψz.7 ψz A ψz θ τ z, τ Ae z τ θ z, τ 3.47 A e τ 3.47 z z+ 8, z+ τ, z+ τ θ τ z z + τ, τ τ Ae θ z +, τ Ae z τ zτ τ θ z, τ, 8 θ τ z τ, τ θ τ z + τ, τ e τ 8 zτ θ τ z, τ. 65

67 θ τ z, τ Ae z τ θ z, τ. θ τ z + z +, τ Ae τ τ Ae z τ z τ zττ θ z, τ, θ z + τ, τ τ τ e 8 8 τ z θ z, τ Ae θ τ z +, τ θ τ z, τ θ τ z +, τ θ τ z, τ. θ τ z, τ Ae z τ θ z, τ. θ τ z τ + z +, τ Ae τ + τ θ z + τ +, τ Ae z τ zτ + τ τ + τ e 8 τ z + Ae z τ zτ 8 τ 8 θ z, τ, θ z, τ, θ τ z τ +, τ θ τ z + τ, τ θ τ z + τ +, τ e τ 8 zτ + θ τ z, τ. θ τ z, τ iae z τ θ z, τ. θ τ z, τ Ae z τ θ z, τ, 3.48 θ τ z, τ Ae z τ θ z, τ, 3.49 θ τ z, τ iae z τ θ z, τ

68 z z Jacobi 3.3 τ θ, τ iaθ, τ. 3.5 θ, τ πθ, τθ, τθ, τ, 3.5 θ, τ πθ, τ θ, τ θ, τ , 3.47, 3.48, 3.49, 3.5 τ θ, τ τ πθ, τ θ, τ θ, τ τ πa 3 θ, τθ, τθ, τ τ A 3 θ, τ. 3.5 A i iτ e τ 4 τ A ±e 8 τ 3.47 z θ, τ Aθ, τ. τ H τ A τ τ q e τ θ, τ n Z q n θ, τ, θ, τ A τ arg τ π 4 e τ < < e τ 8 8 A e 8 τ 3.48, 3.49, 3.5 A e 8 τ θ z, τ, θ z, τ z θ z, τ z z θ τ, e τ z e θ z, τ, 3.54 τ 8 τ z θ τ, e τ z e θ z, τ, 3.55 τ 8 τ z θ τ, ie τ z e θ z, τ τ 8 τ 67

69 θ z, τ + θ z, τ θ z, τ + e θ z, τ. 8 [ ] θ z, τ + θ z, τ z mτ + + n, z mτ + n m, n, m, n Z ψz θ z, τ + θ z, τ z 3.7, 3.8 ψz + θ z +, τ + θ z +, τ θ z, τ + θ z, τ ψz, ψz + τ θ z + τ, τ + θ z + τ, τ θ z + τ +, τ + θ z + τ, τ e τ + z θ z, τ + e τ z θ z, τ θ z, τ + ψz. θ z, τ ψz.7 ψz B ψz z z θ z, τ + Bθ z, τ θ, τ + Bθ, τ θ, τ πe 8 τ c τ [ emτ]. c τ cτ m [ emτ] θ, τ πe 8 τ [ emτ] m 68 m

70 θ, τ + πe τ + [ emτ] m 3.58, 3.59, 3.6 B e z z +, z + τ, z + τ θ z, τ + θ z +, τ + e θ z + 8, τ e θ z, τ, 8 θ z, τ + e θ z, τ. 8 e 8 τ + z θ z, τ + 4 θ z + τ +, τ + e θ z τ, τ e e 8 8 τ z + θ z + 4, τ e e 8 8 τ z + θ z, τ, 4 θ z, τ + θ z, τ. e 8 τ + z + θ z, τ + 4 e 8 τ + z + θ z + 4, τ + θ z + τ + +, τ + e θ z τ, τ e e 8 8 τ z + θ z +, τ 4 e e 8 8 τ z + θ z, τ, 4 θ z, τ + θ z, τ. 69

71 3.35. θ z, τ + θ z, τ, θ z, τ + θ z, τ, θ z, τ + e θ z, τ. 8 { } a b SL Z a, b, c, d Z, ad bc c d SL Z γ γ γ γ SL Z Γ a b γ SL Z a γ Γ c d a bc γ ± γ γ ± d d ± d ±γ d Γ. γ γ γ d γ γ3 γ d γ Γ n a < n γ Γ a n c aq + r, q, r Z, r < a γγ q q a b q c d γ c d a b r d + bq. a b γγ q γ Γ, γ Γ a b SL Z, c c d z θ cτ + d, aτ + b εcτ + d c e cτ + d cτ + d z θ z, τ ε 8 cτ + d σ cτ + d, Iσ, Rσ 7

72 a b [ ], c d a b p q SL Z A, B c d r s c, r z θ cτ + d, aτ + b εcτ + d c e cτ + d cτ + d z θ z, τ, z θ rτ + s, pτ + q ε rτ + s r e rτ + s rτ + s z θ z, τ. a b AB c d τ pτ + q rτ + s z z rτ + s a pτ + q rτ + s + b c pτ + q rτ + s + d z rτ + s c pτ + q rτ + s + d c pτ + q c c pτ + q rτ + s + d z θ c τ + d, a τ + b c τ + d rτ + s + d z rτ + s ap + brτ + aq + bs cp + drτ + cq + ds a τ + b c τ + d, z cp + drτ + cq + ds cp + drτ + cq + ds ε c τ + d rτ + s ε c τ + d rτ + s ε rτ + s e rτ + s z c τ + d, c τ + d rτ + s c rτ + s {cp + drτ + cq + ds} z c rτ + sc τ + d z c z e rτ + sc τ + d z θ c e rτ + sc τ + d z r rτ + s z θ z, τ rτ + sc τ + d z θ z, τ c c τ + d z θ z, τ. εε c τ + d e c + rc τ + d εε c τ + d e 7, rτ + s, pτ + q rτ + s,

73 a b SL Z, ab, cd c c d z θ cτ + d, aτ + b εcτ + d c e cτ + d cτ + d z θ z, τ ε 8 cτ + d σ cτ + d, Iσ, Rσ ε 4 c a b b [ ] i, b 3.35 c d a b ii c d 3.3 c + d c + d c + d > d > c d c < d, d + c < d d c < d d ± c < d ± a c b d ± a c b ± a d ± c c + d ± c < c + d z aτ + b ± a θ, εcτ + d ± c c e cτ + d ± c cτ + d ± c cτ + d ± c z θ z, τ. ε 4 c τ τ z aτ + b ± a θ, cτ + d ± c cτ + d ± c εcτ + d ± c c e cτ + d ± c z θ z, τ, z θ ctau + d, aτ + b cτ + d εcτ + d c e cτ + d z θ z, τ, εcτ + d c e cτ + d z θ z, τ. 7

74 gcdc, d, cd d c d < c a b b a c d d c d + c c + d, d < c z θ dτ c, bτ a εdτ c d e dτ c dτ c z θ z, τ. ε 4 d τ τ z z τ z θ τ b a τ d c, d c τ τ ε dτ c d z e z d c θ τ τ,, τ τ z θ cτ + d, aτ + b cτ + d cτ + d d z ε e τ τ cτ + d z θ τ,, τ cτ + d d ε e τ τ cτ + d z e τ z e θ z, τ 8 τ ε e cτ + d c e 8 τ cτ + d z θ z, τ ε cτ + d c e τ cτ + d z θ z, τ. gcdc, d, cd c d + mod ε 4 ε 4 d d+ c. 4 Jacobi 4. snu, κ, cnu, κ, dnu, κ τ H θ z, τ, θ z, τ, θ z, τ, θ z, τ θ z, θ z, θ z, θ z θ, θ, θ, θ θ, θ, θ, θ 73

75 u πθ z Jacobi sn u, cn u, dn u sn u θ θ θ z θ z, cn u θ θ θ z θ z, dn u θ θ θ z θ z. κ θ, κ θ θ θ sn, cn, dn z θ z, τ, θ z, τ, θ z, τ, θ z, τ sn u cn u, dn u κ, κ 3.9 κ + κ θ4 + θ 4 θ , 3.33 τ τ θ, τ θ, τ iτθ, τ iτθ, τ θ, τ θ, τ κ τ H θ, τ, θ, τ κτ, τ H a C, a, κτ a τ H jτ 3.7 cn u + sn u, 4.3 dn u + κ sn u 4.4 sn u, cn u, dn u d dz A4 θ z θ z θ zθ z θ zθ z θ z θ x + uθ x uθ θ θ xθ xθ uθ u + θ xθ xθ uθ u. 74

76 u u θ u, θ u θ θ θ xθ x θ xθ x θ θ θ xθ xθ θ. Jacobi 3.3 θ πθ θ θ θ zθ z θ zθ z πθ zθ zθ. d dz θ z θ z θ z θ z πθ θ z θ z d dz d θ θ z sn u du du dz θ θ z θ πθ πθ θ θ z θ z θ z θ z θ θ z θ θ z θ θ z θ θ z cn u dn u. 4.3 cn u d d cn u + sn u sn u. du du d sn u cn u dn u du 4.4 d sn u cn u dn u du d cn u dn u sn u du dn u d du dn u + κ sn u d sn u. du d du dn u κ sn u cn u. sn u, cn u, dn u d sn u cn u dn u, du d cn u dn u sn u, du d du dn u κ sn u cn u

77 sn, cn, dn 4.3, 4.4, 4.6 d du sn u sn u κ sn u, d du cn u cn uκ + κ cn u, d du dn u dn uκ dn u sn u F v v dx x κ x κ, ± x fx x κ x x f v sn sn cn dn u sn u F sn u d du F sn u sn u κ sn u d du sn u sn u κ sn u df dv v κ v F sn u u + C, d du sn u sn u κ sn u sn u κ sn u. C u sn, F C F sn u u sn u F v v sn F v v 76

78 κ < κ < F v v Taylor v Taylor x < x n n x n n a n v n!! n!! n df dv v κ v a l v l a m κ m v m l n l+mn b n m a l a m κ m v n n a n m a m κ m m n!! x n. n!! b n v n. F v F v n n b n n + vn v S S n b n n + n n m nm m l b n n + n + n + a n m l + m + a l 77 n a n m a m κ m m a m κ m a m κ m.

79 x < v < v x m x l a l x l+m x m dx x l + m + a l v l+m+. l Stirling a n πn C > l l + m + a l l l + a l l C l + 3 <. Abel x m dx x l + m + a l. x sin t l x m π dx x π m I m [ cos t sin m t] π + m π π sin m t cos t cos t dt sin m t dt I m sin m t sin m t dt m I m m I m. m cos t sin m t dt I m m m I m I m m!! I m!! S π m!! π m!! π a m a mκ m m 78

80 a m /πm, < κ < S v < 4.7 K Abel K dx x x κ x lim F v S v v sn F v v v < sn F v v v sn K 4.3 sn u 3.7, 3.8, 3.9 θ z + θ z, θ z + θ z, θ z + τ kθ z, θ z + τ kθ z, θ z + τ + kθ z, θ z + τ + kθ z, θ z + θ z, θ z + θ z, θ z + τ kθ z, θ z + τ kθ z, θ z + τ + kθ z, θ z + τ + kθ z. k e τ z u πθz ω π θ, ω π θ τ

81 z z +, z + τ u u + ω, u + ω sn u, cn u, dn u snu + ω θ θ z + θ θ z + θ θ z sn u, θ θ z snu + ω θ θ z + τ θ θ z + τ θ kθ z sn u, θ kθ z cnu + ω θ θ z + θ θ z + θ θ z cn u, θ θ z cnu + ω θ θ θ z + τ θ z + τ θ θ kθ z kθ z cn u, dnu + ω θ θ z + θ θ z + θ θ z dn u, θ θ z dnu + ω θ θ z + τ θ θ z + τ θ kθ z dn u. θ kθ z snu + ω sn u, snu + ω sn u, cnu + ω cn u, cnu + ω cn u, dnu + ω dn u, dnu + ω dn u. sn u 4ω, ω cn u 4ω, ω + ω dn u ω, 4ω θ z, θ z, θ z, θ z sn u, cn u, dn u n, n Z sn u nω + n ω nω + n + ω cn u n + ω + n ω nω + n + ω dn u n + ω + n + ω nω + n + ω 4 C : y x κ x κ, ± C C C W C W C x, y W x, y W i,ii : i xx, ii y y x. 8

82 C W C {x, y W y x κ x } C {x, y W y x x κ } C C W W C C C C {w : w : w : w 3 P 3 w w w, w w w κ w w 3 }. C w : w : w : w 3 w, w 3 C, w, w w w, w 3 C, w w w cn u dn u sn u κ sn u, sn u cn u dn u C u sn u, cn u dn u sn u, sn u C φ : C C C C sn u, sn u cn u dn u 4ω, ω 4 Ω Z4ω + Zω φ φ : C/Ω C C C φc/ω C/Ω φc/ω φc/ω C φ φ u, u C, u u mod Ω φu φu sn u sn u, sn u sn u sn u sn u snu + ω sn u, sn u+ω sn u fu sn u sn u fu fu fu Ω fu modω u, u u u fu f u sn u snω u sn u sn u fω u i ω u u i mod Ω sn u sn u i sn ω u sn u sn u sn u φ φ : C/Ω C 8

83 4.4 Jacobi dx x κ x κ, ± Bx x κ x Bx α κ, β, γ, δ κ p : C P, p : C P pξ, η : ξ, p ξ, η ξ : p p C C p : C P P : a P p P a : ± Ba C { p, P : α, : β, : γ, : δ, P, P 4 P : p : C P p P {, κ,, κ } P P { p, P : α, : β, : γ, : δ, P, P 4 C α β γ δ P p : C P C P P : α, : β, : γ, : δ p P p C C C C : y ± Bx 8

84 p : y ± x a y x a x a η x a η C η, η η, η x x y x η + a x x + a x x +, y x η + b x x + b x x + y x x a, y x x a y x y x x a Γ y x Γ : xt a + x arteθt t, r r, θ θ y x ȳ t y a + x arteθt y x x a, y x η ȳ t x arteθt, η x a. ȳ t ±η rte θt. ȳ t t ȳ t η rte θt ȳ η γ y x x x ỹ x ỹ x η ỹ x y x y x a x a + eθx a 83

85 ȳ θ y a + eθx a y x x a ȳ θ eθx a. ȳ y x η. η x a, ȳ θ eθη. ȳ θ ±e θ η ȳ θ θ ȳ θ e θ η ȳ η a y x x x ỹ x ỹ x η ỹ x y x y x y x a y x y x y Bx κ x αx βx γx δ x α, β, γ, δ x x y x, y x y i x κ x αx βx γx δ i, y x Γ x x ȳ x { y x, # Γ {α, β, γ, δ}, ȳ x y x, # Γ {α, β, γ, δ} C {ξ, η W C η Bξ} y y x {x, y x C x C} C C C C C Riemann α β γ δ Riemann R αβ λ + λ γδ µ + 84

86 µ α κ, β, γ, δ κ α β, γ δ Riemann R λ +, λ, µ +, µ α, β, γ, δ α, β, γ, δ λ + λ λ λ + µ + µ µ µ + C Riemann R f i x x κ x i, f x, f x 4. Jacobi. κ f x f x dx x κ x π θ, τ. κ θ, τ θ, τ fx x κ x Riemann R f 4.3. κ C {,, } H κ C {,, } dx x κ x κ κτ θ, τ τ H θ, τ dx x κτ x τ H dx x κτ x 85

87 τ H hτ : τ H κτ < κτ <, x κ x > hτ dx x κτ x x κτ x [ 4. ] π sn θ, τ sn u π sn θ, τ θ θ, τ θ θ 4.9, τ. θ z, τ θ z +, τ, 4. θ z, τ e 8 τ + z θ z + τ, τ, 4. θ z, τ e 8 τ + z + θ z + τ +, τ θ, τ θ, τ , 4.5 θ, τ e 8 τ + θ τ, τ. 4.4 θ, τ e 8 τ θ τ, τ. 4.5 θ, τ θ, τ , 4.3, 4.6 π sn θ, τ θ θ, τ θ θ, τ 86.

88 ω π θ cn u cn u sn u sn u + sn u π θ sn u sn u Z4ω + Zω. P [] sn u u π θ C sn u u π θ + 4nω + n ω n, n Z K dx x κ x sn K n, n Z K π θ + 4nω + n ω n n τ 3., 3., 3.4 < κ θ θ R, κ θ θ R. κκ κ + κ < κ < 4. < κ < K π θ + 4nω + n ω 4n + ω + n ω ω π θ R, ω π θ τ ir n s K 4n + ω F s s n + πθ dx x κ x K F > n n n π θ < F n + πθ 87

89 < a < π θ F a a dx x κ x 4. sn F a a π a sn F a sn θ n τ K dx x κ x π θ κ θ, τ θ, τ dx x κ x, π θ, τ τ H τ H ir τ H K dx x κ x π θ, τ κ θ τ /τ θ κ + κ, κ θ, τ θ, τ K dx x κ x x Riemann R R R Riemann /κ /κ Riemann 4. K dx x κ x π θ, τ 88

90 K τ H τ /τ K τ H 4.4. K /κ dx x κ x ds s κ s s i Riemann R Riemann R /κ µ + τ H κ < κ < s κ s s /κ µ + Riemann R H gτ τ H κ < κ < gτ /κ ds s κ s [ ] τ ir κ, κ κ > κ + κ < κ <, κ + κ dt /κ t κ t ds s κ s 4.7 /κ [, ], [, /κ] s κ t, < t < s κ t s κ t κ t, κ s κ t κ t. 89

91 s κ s κ 4 t t κ t ds dt ds s κ s κ t κ t 3 dt t κ t. κ, κ κ + κ 4.7 τ H 4.7 τ τ H K i π θ τ. [ ] 4. K κ θ, τ θ, θ, τ θ τ, τ ds s κ s π θ, τ. 3.3 θ, τ iτ θ, τ. K i π θ τ C : y x κ x C 4.3 C dx x κ x dx y 4.8 9

92 4., 4.4, 4.5 y x κ x ydy x κ x κ x x dx dx y dy x κ x + κ x x. 4.9 dx/y y x, y C 4.9 dx/y y {x, C x ±, x ±κ } dx/y C dx/y x x, y y x dx y x y dx x dx y 4. dx/y C dx /y dx /y C dx/y C C C dx/y C 4.3 φ : E C/Ω C, Ω Z4ω + Zω, E φ dx/y x sn u, y cn u dn u sn u dx φ sn udu y sn u du du φ dx y dx/y C α κ, β, γ, δ κ C Riemann 9

93 λ + µ + α β γ δ λ µ dx/y β Riemann γ γ Riemann β Γ β γ γ β Γ Γ dx y 4, 4ω. dx y + dx, y dx x κ x + dx x κ x dx x κ x 4π θ, τ dx x κ x dx y C dx/y Γ 4. φ sn, sn, φ : E C/Ω C C Γ E Γ E φ, Γ E E E 9

94 4ω Γ E φ dx du 4. y Γ E C 4ω Γ C E C/Ω Γ Γ Γ E 4ω 4ω du du du 4. Γ Γ 4., 4. du Γ E du. Γ 4.3 E - Γ Γ E P [] 4ω + ω Γ 3 ω Γ Γ 4 4ω Γ E C/Ω Γ Γ 3 Γ Γ 4 C E C/Ω Γ i Γ i i,, 3, 4 C - Γ Riemann γ µ + δ /κ µ γ Γ dx/y Γ dx y Γ 93

95 Γ dx 4.4, y 4.5 dx /κ Γ y dx, µ + /κ i i /κ y + dx /κ, µ y dx x κ x + /κ i K πθ τ. dx x κ x ds s κ s dx x κ x /κ dx x κ x x s i ω π θ τ Γ dx y ω φ : C/Ω C C Γ E Γ Γ a, b, c C c u u d f df + c a + b + u abf 4.4 du du n Z, n a n aa + a + n a + n, a F a, b, c; u n a n b n n!c n un fu F a, b, c; u u <

96 [ ] A n an b n n!c n A n A n+ n + c + n a + nb + n n 4.6 F a, b, c; u F a, b, c; u u < D u d du Dun nu n [ a + Db + D c + D + D ] F a, b, c; u 4.7 u [ a + Db + D c + D + D ] A n u n u a + Db + DA n u n n a + Db + DA n u n n a + nb + na n u n n a + nb + na n u n n n c + D + DA n u n n c + D + DA n+ u n n c + n + na n+ u n n c + n + na n+ u n n [a + nb + na n c + n + na n+ ] u n. n D f DDf u d u df u df du du du + d f u du a + Db + Df D + a + bd + abf u df du + d f df u + a + bu du du + abf. D u f u u f + df u du u f + df du c + D + D [ ] u f c + D u f + D u f c + D df du c df du + ud f du. 95

97 u u d f df + a + b + u c + abf. du du 4.6 F a, b, c; u 4.4 F a, b, c; u 4.4 Γ Γ dx y 4 dx y i /κ κ u u F u Gu Γ / u dx x κ x, ds s κ s dx x ux, ds s us 4.7. y x ux dx dx 4F u, y y igu u a b /, c Γ u u d f df + u du du 4 f 4.8 [ ] 4. n!! u n F u π n n!! n!! n!! n!!n!! n!!n!! n / n n / n n n! n n! /n / n n! n 96

98 Γ dx y 4F u πf,, ; u Gu /κ ds s κ s dx x κ x F κ κ u Gu F u u u u d F u + u df u du du 4 F u 4.9 κ + κ u + u Gu F u u u d G + udg du du 4 G 4.3 Γ dx y igu 4.8 [ 4.7 ] x y y x ux y x, u u y y u x x. y dx/y u dx y u y y u dx x x dx. y 3 y x ux u dx y dx u y dx x u y u ux x dx ux y. 4.3 dx y + x x 4 dx ux y + x 4 dx 4 ux y x 4 dx ux y ux u dx y

99 u dx y 4.3, 4.3 [u u + u u u 4 [ 3u ux 4 3 x 4 dx 4 ux y. 4.3 ] dx ux + 4 ux ux 4 y ] dx y 3u ux4 + ux ux ux dx 4 ux y ux4 + ux 4 ux dx y xy d ux ux4 + ux dx 4 ux y y x ux ydy x ux ux x dx ux 3 ux xdx, dy ux3 ux x y xy d ux dx. x xy ux dx + y ux4 + ux ux dx y. 4.3, 4.34 [u u + u u u ] dx 4 y d xy ux xy ux dy Γ [u u + u u u ] dx 4 Γ y xy d. Γ ux dx dx 4.8 Γ y Γ y 4.8. K π F,, ; κ

100 K κ Kκ dx x κ x dx π/ x κ x dθ κ sin θ K κ Kκ κ + κ κ x Eκ dx 4.37 x x sin θ Eκ K κ π/ κ sin θ dθ 4.38 E κ Eκ 4.8. Kκ, K κ [ ] F u Kκ F κ κ 3 κ d y dκ + 3κ dy + κy 4.39 dκ dx 4.7 y F u x ux u u d y + udy du du 4 y 4.4 dk dκ κdf du κ, d K dκ 4κ d F du κ + df du κ, 99

101 d F du κ d K 4κ dκ dk 4κ 3 dκ, df du κ dk κ dκ κ κ d F du κ + κ df du κ 4 F κ κ 3 κ d K dκ + 3κ dk dκ + κk K κ Gκ G F K K C : y x κ x κ C {,, } 4.5 dx x κ x dx y C C κ x x dx κ x dx y C 4K, E κ x dx y dx/y C κ x C ζ κ x dx/y C C C C C {x, y C y x κ x }, C {x, y C y x x κ } x, y C x, y C i xx, ii y y x C ζ κ x dx/y C x /x, y y /x dx/y dx /y ζ κ dx x y

102 dx /y C C ζ x ζ C x, y, κ,, κ P, κ x ζ y x P, κ y x κ + a x + a x + y x κ + b x + b x + x x κ y x x κ κ + b x + b x +. modx κb, b y x κ + b x + ζ κ dx x y κ x κ + b x + κ x + x dx. dx ζ P, κ, κ ζ φu sn u, cn u dn u φ : C/Ω C, Ω Z4ω + Zω C ζ κ x dx/y C/Ω φ dx/y du φ ζ κ sn udu dn u du

103 4.9. X Riemann ξ ξ X P ξ X ζ C C φ η φ ζ η dn u du dn u Ω Z4ω + Zω P [] ω, ω + ω C/Ω dnu + ω dn u, dnu + ω dn u dn u Ω Zω+Zω ω +Ω C/Ω dn u du C/Ω η ψ : C/Ω C/Ω η ψ η 4.. d C xy d κ x 4 x + dx κ x κ x y. [ ] y x κ x xy d κ x x [ dy κ x 3 κ x x y xy κ x y κ x + κ x y κ x dx. dx + xy dy y κ x ] x dx + κ x y + κ x κ x dx + x κ x κ dx κ x y x κ x + κ x κ x y κ x 4 x + dx κ x y. dy dx + x κ x κ κ x y dx

104 4.. Eκ, Kκ. de dκ E K, κ [ ] κ x dx κ x dk dκ E κ K κκ. κx x κ x dx κ x κ x κ x dx κ x κ x κ x dx κ x dx κ x x κ x dx ζ dx. κ y Γ d ζ d κ x dx dκ Γ dκ Γ x dx ζ. κ Γ Γ y dx y 4 Γ Γ Γ ζ κ x dx y 4 dx κ x κ x Γ κ x dx κ x dx 4Kκ x κ x de dκ E K κ κ x x dx 4Eκ κx x κ x κ x dx 3

105 dx κ x κ x κκ κ x dx x κ x κ x dx κx x κ x κ x dx κκ κx x κ x κ x dx κx κ x κ x κ κ κ x 4 x + κ κ x d κ xy. κ κ x κκ dx x κ x dx y κ κ x x κ x κ x x κ x 4. Γ d dx dκ Γ x x κ x dx Γ κ x x κ x κ x κ dx κκ Γ x Γ x κ x dx + d κ xy κ κ x Γ κκ Γ κ x x dx Γ κ x κ x dx. dx dx dk dκ κκ E κ K 4. Legendre. < κ < EκK κ + E κkκ KκK κ π 4

106 [ ] 4.8 Kκ, K κ κ 3 κ d y dκ + 3κ dy dκ + κy y κ κ z z κ + κ dκ dκ κ κ dy dκ 3 κ κ κ κ κ z + κ κ κ dz dκ, 3κ κ 3 κ 3 z + κ κ dz dκ, d y dκ d dκ 3κ κ 3 κ 3 z + 3κ κ 3 κ 3 dz dκ + κ κ d z dκ, 3 5κ 4 κ + κ 5 κ 5 z + 3κ κ 3 κ 3 dz 4 dκ + κ κ d z dκ κ 3 κ d y dκ + 3κ dy dκ + κy 3 κ 3 κ 5κ 4 κ + κ 5 κ 5 z + 3κ κ 3 κ 3 dz 4 dκ + κ κ d z dκ + 3κ 3κ κ 3 κ 3 z + κ κ dz + κ κ z dκ κ κ d z dκ 4 + κ κ 3 κ 3 z κ κ d z dκ κ κ κ 4 z. z d z dκ + + κ z κ κ H κ κ κ Kκ, H κ κ κ K κ z H i κ, i, 4.4 W dh dκ H dh H dκ dw dκ d H dκ H H d H dκ. 5

107 W W d κ κ Kκ κ K + κ κ dk dκ dκ κ κ K d dκ κ κ K κ κ K κ κ dk dκ κ κ K dk κκ dκ K K dk. dκ dk 4. κκ dκ E κ K dk κκ dκ κ + κ κκ dk dκ κ κ κ κ κ dk dk κ Eκ + k Kκ. W E κ KK K Eκ + κ Kκ EK + E K KK. W EK + E K KK lim κ W K π E π W E KK + E K π sin θ dθ cos θ dθ [ sin θ ] π. lim W lim E κ κ KK + EK lim E KK + π κ. lim E KK κ π π K EK dθ κ sin θ κ sin θ dθ Kκ κ π κk κk π π π κ sin π θ κ sin θ dθ dθ κ sin θ π κ dθ cos θ + κ sin θ κ dθ κ cos θ + κ sin θ κk π < K EK κk π dθ κ sin θ κ dθ κ sin θ κ. 6

108 5 5. a b a a, b b, a n+ a n + b n, b n+ a n b n n,,,... {a n }, {b n } b b b b n b n+ a n+ a n a a a. b n b n+ a n+ a n {a n } {b n } lim a n α, lim b n β n n α lim n a n+ lim n a n + b n α + β α β α a b AGMa, b 5.. AGM, n a n b n a n b n i AGMa, b AGMa, b AGMa, b. ii λ AGMλa, λb λ AGMa.b. Kκ dx x κ x 7

109 5.3. a b > Ia, b π dθ a cos θ + b sin θ κ, κ κ + κ cos θ sin θ 5.4. [ ] Ia, b I, κ π π b tan θ u I u π 5. I, κ Kκ 5. π dθ cos θ + κ sin θ dθ κ sin θ Kκ. I du dθ cos θ a + b, ab Ia, b. b cos θ, + tan θ dθ cos θ du cos θ b cos θ cos θ b π dθ a cos θ + b sin θ a + u b + u du a + b, ab v ab v a+b dθ κ sin θ b b + u, b + u. dθ cos θ a + b tan θ a + u b + u du. + u ab + u ab + u 4v ab + v, du ab + u dv v ab + v ab + v v v, a + b + u 4v a + v b + v 8 du.

110 v u a + b I, ab dv Ia, b. a + v b + v 5.5. Ia, b π AGMa, b. [ ] 5.4 Ia, b Ia, b Ia n, b n. M AGMa, b lim a n lim b n M n n n Ia, b IM, M M π dθ π M π AGMa, b Kκ I, κ π κ κ Kκ θ π θ 5.5 π dx x κ x Kκ π κ AGM, κ. 5.3 dθ κ cos θ + sin θ dθ κ sin θ Kκ I κ, π AGM κ, dx x 4 π AGM, 799 9

111 5.6. < κ <, κ κ i Kκ π AGM, κ. ii Kκ κ + κ K + κ κ iii Kκ + κ K + κ.. iv Eκ + κ κ E + κ + κ Kκ. κ v Eκ + κ E κ Kκ. + κ [ ] i ii κ κ + κ + κ i 5.ii + κ AGM K κ π + κ. κ AGM + κ, κ + κ, AGM κ, + κ AGM + κ, κ. + κ + κ 5.i, + κ κ κ + κ K κ + κ AGM + κ, κ AGM, κ. π + κ AGM π AGM, κ Kκ. π κ + κ, AGM + κ, κ

112 iii κ κ + κ + κ i κ K π. + κ AGM, κ + κ 5.ii, i + κ AGM, κ + κ iv + κ K ii + κ κ 4. + κ dk dκ κ fκ + κ AGM, κ AGM, κ. κ π + κ fκ κ + κ AGM, κ Kκ. + κkκ Kfκ. dk df κ + Kκ fκ κ. 5.4 dκ dκ dk dκ κκ E κ K, 5.5 dk Eκ κκ dκ κ + κ Kκ. Efκ fκgκ dk fκ + gκkfκ 5.6 dκ gκ fκ κ + κ

113 Efκ fκgκ dk dκ fκ df dκ κ df dκ κ + κ dk κ + Kκ dκ + κ dk κ + Kκ dκ. + gκkfκ ii Kfκ + κkκ + κ Efκ fκgκ df dκ κ κκ Eκ κ Kκ + Kκ + gκkfκ + κ fκgκ df dκ κ κκ Eκ κ Kκ + Kκ + gκ + κkκ. df dκ κ Efκ κ, fκgκ κ + κ df dκ κ κ κ + κ κ κ κ, gκ + κ + κ + κ + κ κκ Eκ κ κ Kκ + Kκ + + κ Kκ + κ Eκ κ Kκ + Eκ κkκ. + κ iv v µ κ + κ κ κ κ Kκ + + κ + κ Kκ + µ µ µ 4κ + κ, + κ, µ κ + µ + κ κ + κ κ

114 iv, iii Eµ + µ E κ E + κ µ + µ + µ κ Eκ + + κ + κ K Kµ, κ + κ κ Eκ + + κ + κ Kκ, κ Eκ + κ E κ Kκ. + κ 5.7. a > b > Ja, b π a cos θ + b sin θ dθ J, κ Eκ 5.8 cos θ sin θ J, κ π J π κ sin θ dθ κ sin θ dθ Eκ. a + b, ab Ja, b 5.8. [ ] J a + b, ab Ja, b abia, b. κ b a κ b a, κ a b + κ a + b Ja, b aj, b aj, κ Eκ a J a + b, ab a + b E 4ab a + b a + b E a b. a + b 3

115 Ja, b aeκ Ia, b Kκ 5.6 v a a + b J, a b ab Ja, b a + be aeκ a + b κ a + be aeκ + κ κ a + b Eκ + + κ + κ Kκ aeκ bkκ abia, b c n a n b n, n,,... n c n a n b n lim n n c n [ ] n c n a n b an + b n an b n n a n b n a n > b n b, lim a n lim b n lim c n n n n n n n < c n < b n > n < c n b a n b n b a n b n ba n + b n c n ba n + b n < 4 cn b. < c n +m < b ++ + m cn 4 b m < b m b. 4 m+ < n +m c n +m < b n +m m+ m. 5.. Ja, b a n c n Ia, b. n 4

116 [ ] A n n Ja n, b n a nia n, b n A n+ A n n Ja n+, b n+ Ja n, b n n+ a n+ia n+, b n+ + n a nia n, b n n a n b n Ia n, b n n+ a n+ia n, b n + n a nia n, b n n a n b n a n + b n + a n Ian, b n n a n b nia n, b n n c nia, b. n A n a nia n, b n Ja n, b n π π c n a π n a n cos θ + b n sin θ dθ π a n a n cos θ b n sin θ a n cos θ + b n sin θ dθ π sin θ a n cos θ + b n sin θ dθ c n a n cos θ + b n sin θ dθ π a n b n sin θ a n cos θ + b n sin θ dθ dθ a n cos θ + b n sin θ c nia n, b n. < A n n c nia n, b n n c nia, b. 5.9 lim n n c n n A n+ A n lim N N A n+ A n lim A N+ A A. N n Ja, b a Ia, b A A n+ A n n c nia, b Ja, b Ia, b a n n c n, n n c n Ia, b. n n 5

117 a, b c, a n+ a n + b n, b n+ a n b n, c n+ a n b n, n,,... AGM, π. n c n n s n n k k c k, p n a n s n p p p p p π p 3 9 p 4 p 5 4 π π p π 799 [ ] a, b κ κ 4. E K K π. 5. E n c n K n n c n K n 6 π..

118 5.5 K AGM π, n c n 4 AGM n π, π. A A.. fz D f z fz a < x, b < y, K {t + is a t x, b s y} D A.. K Cauchy fz dz [ ] SK K K fz dz a + iy x + iy K a + ib x + ib SK x a y ft+ib dt+ fx+is ids b x a y ft+iy dt fa+is ids A. b SK K 4 7

1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C

1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C 0 9 (1990 1999 ) 10 (2000 ) 1900 1994 1995 1999 2 SAT ACT 1 1990 IMO 1990/1/15 1:00-4:00 1 N 1990 9 N N 1, N 1 N 2, N 2 N 3 N 3 2 x 2 + 25x + 52 = 3 x 2 + 25x + 80 3 2, 3 0 4 A, B, C 3,, A B, C 2,,,, 7,

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