n( 3) p n n q n θ = π n p n = 2n sin θ, q n = 2n tan θ 1 sin θ tan θ O θ 1

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ぜんましんまつ
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1 n( 3) p n n q n θ π n p n 2n sin θ, q n 2n tan θ sin θ tan θ O θ

2 p 2n 4n sin θ, q 2 2n 4n tan θ 2 p n, q n, p 2n, q 2n q 2n 2p nq n p n + q n p 2n p n q 2n (p n q n ), () (p n q 2n ). (2) [ ] () 2(2n)2 sin θ tan θ 2n(sin θ + tan θ) 4n sin θ + cos θ 4n 2 sin θ 2 cos θ 2 2 cos 2 θ 2 4n tan θ 2 q 2n. (2) 8n 2 sin θ tan θ2 6n 2 sin θ 2 cos θ 2 tan θ 2 6n 2 sin 2 θ 2 4n sin θ 2 p 2n. n 6 p 6 6, q () (2) p 2 q (2 3). 6 24(2 3) p n < π < 2 q n n 2, < π < 3.253, n 24, < π < 3.596, n 48, < π < 3.460, n 96, 3.40 < π < 3.427, n 92, 3.44 < π < π

3 2 7 x tan θ π 2 < θ < π 2 θ arctan x < x < x tan θ π 2 < θ < π 2 θ arctan x x tan θ dx dθ cos 2 θ + tan2 θ + x 2 d dθ arctan x dx dx (3) + x 2 x < + x 2 x2 + x 4 x 6 + ( ) n x 2n (3) arctan x n0 arctan x x x3 3 + x5 5 x7 7 + n0 ( ) n x2n+ 2n + x < x tan π 4 arctan π 4 (4) x [ ] π ( ) n 2n + n k0 n0 ( ) k x 2k + ( )n x 2n+2 + x 2 0 n ( ) k x 2k dx + x dx k0 0 n k0 ( ) k 2k (4) ( ) n x 2n+2 + x 2 dx. (5)

4 arctan π 4 2 x 2n+2 0 < 0 + x dx < 2 0 n ( ) k 2k + π 4 k0 x 2n+2 dx 2n + 3 < 2n + 3 n (5) (5) tan x π 4 4 ( ) n (2n + )5 ( ) n. (6) 2n+ (2n + )2392n+ n0 n0 (5) n 4 π (5) 3. a n, n x, y f n (x, y) a x a 2 x a 3 x + + a n x + a nx + y, n, f n (x) f n (x, 0) x N n k n f N (x) x k f n (x) x k f(x) a x a 2 x a 3 x + a 4x +...

5 g(x) x k n f n (x, g(x)) x k f(x) x k [ ] A n, B n { A n A n + a n xa n 2, B n B n + a n xb n 2, n 2, 3,..., A 0, A + a x, B 0, B B n x n f n (x, y) A n y + A n B n y + B n (7) n A 0 + A y B 0 + B y + ( + a x)y + y + a x + y f (x, y). n k f k (x, y) f k (x, ) a k x + y A k 2 a k x + y + A k B k 2 a k x + y + B k A k y + A k + a k xa k 2 B k y + B k + a k xb k 2 A k y + A k B k y + B k n k n (7) y 0 f n (x) A n B n A k B k A k B k (A k + a k xa k 2 )B k A k (B k + a k xb k 2 ) a k x(a k B k 2 A k 2 B k )., k m, m,..., 2 A B 0 A 0 B a x A m B m A m B m ( ) m a a 2 a m x m (8) f m (x) f m (x) A m B m A m B m ( )m a a 2 a m x m B m B m. (9) B m, B m x (9) x x m f N (x) f n (x) N mn+ 5 (f m (x) f m (x))

6 x n+ f n (x, g(x)) f n (x) ( )n a a n x n g(x) B n (B n + B n g(x)) x n+ 2. a n, n 0 < δ a n δ <, n f(x) a x a 2 x a 3 x + a 4x x [0, ] A n [ ] f(x) n B n { A n A n + a n xa n 2, A 0, A + a x, B n B n + a n xb n 2, B 0 B, n 2, 3,..., B n x 0 x B n, n 0 B n B + B k B k a k xb k 2 a k x. n (B k B k ) + (a a n )x + (n )δ x. k2 3. a k, b k (b k 0), k λ b + b 2 + a a 2 a 3 b 3 + a 4 b

7 k b k a k + λ [ ] A k, B k { A k b k A k + a k A k 2, B k b k B k + a k B k 2, k 2, 3,..., A 0 0, A a, B 0, B b λ k λ k b + b 2 + a a 2 a a k b k + a k b k A k B k B k b k B k a k B k 2 b k B k ( b k ) B k 2, B k B k ( b k )( B k B k 2 ) ( b k ) ( b 2 )( B B 0 ) ( b k ) ( b 2 )( b ) a a 2 a k 0. 0 < b B B 2 B 3. A k B k A k B k (b k A k + a k A k 2 )B k A k (b k B k + a k B k 2 ) A k B k A B + a k (A k B k 2 A k 2 B k ) ( ) k a a 2 a k. ( A2 B 2 A B ) + + ( Ak A ) k B k B k A B a a 2 B B ( ) k a a 2 a k B k B k. (0) 7

8 a a 2 a k B k B k B k B k B k B k A + A k A k B k B k2 B k A B A B B k B k. () + ( B k ) B k k2 + B a +. (2) b A k λ lim k λ B k 4. c k, k 0 b + b 2 + a a 2 a 3 b 3 + a 4 b c a. c c 2 a 2 c b + c 2 c 3 a 3 c 2 b 2 + c 3 b 3 + c 3c 4 a 4 c 4 b A k [ ] k, k B k B k { A k b k A k + a k A k 2, B k b k B k + a k B k 2, k 2, 3,..., A 0 0, A a, B 0, B b, { Ãk c k b k Ã k + c k c k a k Ã k 2, B k c k b k Bk + c k c k a k Bk 2, k 2, 3,..., Ã 0 0, Ã c a, B 0, B c b A k c c 2 c k A k, B k c c 2 c k B k, k Ãk, Bk Ã k c c 2 c k A k, Bk c c 2 c k B k, k A k B k B k Ã k Ã k 4 (83 ) α n (α; n) α(α + ) (α + n ) 8

9 (α; 0) (; n) n! α, β, γ 0,, 2,... (α; n)(β; n) F (α, β, γ; x) x n (3) (γ; n)n! n0 x < (α; n)(β; n) [ ] a n (γ; n)n! a n a n+ (α; n)(β; n)(γ; n + )(n + )! (α; n + )(β; n + )(γ; n)n! (γ + n)(n + ) (α + n)(β + n) a n lim n a n+ F (α, β, γ; x) α, β, γ xf F (α, 0, α; x) F (α, β, β; x) xf (,, 2; x) ( 2,, 3 ) 2 ; x2 n0 n0 n0 n0 (0; n) x n, n! (α; n) x n n! ( α n n0 n!n! (n + )!n! xn+ n0 2n + ( )n x 2n+ arctan x. ) ( x) n ( x) α, n + xn+ log( x), 5 ( ). F (α, β, γ; x) F (α, β +, γ + ; x) F (α, β, γ; x) F (α +, β, γ + ; x) α(γ β) xf (α +, β +, γ + 2; x), (4) γ(γ + ) β(γ α) xf (α +, β +, γ + 2; x). (5) γ(γ + ) [ ] (α +; n) α (α; n+), (β +; n) β (β +n)(β; n) β (β; n+), (γ + ; n) γ (γ + n)(γ; n), (γ + 2; n) (γ + ) (γ + n + )(γ + ; n) γ (γ + ) (γ + n + )(γ; n + ) 9

10 (4) (α; n)(β + ; n) x n α(γ β) (γ + ; n)n! γ(γ + ) x n0 n0 γ(β + n)(α; n)(β; n) x n β(γ + n)(γ; n)n! n0 n0 γ(β + n)(α; n)(β; n) x n (γ β)n(α; n)(β; n) β(γ + n)(γ; n)n! β(γ + n)(γ; n)n! n0 n ( ) γ(β + n) (γ β)n (α; n)(β; n) + x n β(γ + n) (γ; n)n! n (α; n)(β; n) + x n F (α, β, γ; x). (γ; n)n! n (α + ; n)(β + ; n) x n (γ + 2; n)n! (γ β)(α; n + )(β; n + ) x n+ β(γ + + n)(γ; n + )n! F (α, β, γ; x) F (β, α, γ; x) (4) α β (5) (4) F (α, β +, γ + ; x) F (α, β, γ; x) F (α, β +, γ + ; x) α(γ β) γ(γ + ) x F (α, β +, γ + ; x) F (α +, β +, γ + 2; x) (5) (α, β, γ) (α, β +, γ + ) F (α, β+, γ+; x) F (α+, β+, γ+2; x) F (α +, β +, γ + 2; x) F (α, β +, γ + ; x) F (α +, β +, γ + 2; x) F (α, β, γ; x) F (α, β +, γ + ; x) x n (β + )(γ + α) xf (α+, β+2, γ+3; x). (γ + )(γ + 2) (β + )(γ + α) (γ + )(γ + 2) x α(γ β) x γ(γ+) (β+)(γ+ α) x (γ+)(γ+2) F (α+,β+,γ+2;x) F (α+,β+2,γ+3;x) F (α+,β+,γ+2;x) F (α+,β+2,γ+3;x) (α, β, γ) (α +, β +, γ + 2) F (α, β, γ; x) F (α, β +, γ + ; x) a x a 2 x + a 2n x F (α+n,β+n,γ+2n;x) F (α+n,β+n+,γ+2n+;x).... (6)

11 α(γ β) a γ(γ + ), a (β + )(γ + α) 2, (γ + )(γ + 2) (α, β, γ) (α + n, β + n, γ + 2n 2) n a 2n (α + n )(γ + n β) (γ + 2n 2)(γ + 2n ), a 2n (β + n)(γ + n α) (γ + 2n )(γ + 2n). (7) 6 (Gauss). F (α, β, γ; x)/f (α, β +, γ + ; x) x < F (α, β, γ; x) F (α, β +, γ + ; x) a x a 2 x a 3 x + a 4x +... a 2n, a 2n (7) [ ] c k 2, k a x 2a x + a 2 x 4a 2 x 2 + a 3 x 4a 3 x + + a 2 + 4x a 4x (8) 0 < r < lim n a 2n lim n a 2n 4 N 4a 2n x 4 a 2n r, 4a 2n x 4 a 2n r, n N, x r a k 4a 2N+k x, b k 2 b k 2 a k +, k 3 g N (x) 4a 2N x 2 + 4a 2N+x 2 + 4a 2N+2x 2 + b + a a 2 b 2 + a 3 b 3 + (9)

12 (9) k 3 A k B k B k B k ( b k ) ( b 2 )( b ) a a 2 a k B k k + m > n m kn+ A k A k B k B k m kn+ ( B k ) B k B n B m < n + x r B k 0 A k B k x < r g N (x) x < r B 2N 2, B 2N 0 x g N (0) 0 (8) A 2N 2g N (x) + A 2N B 2N 2 g N (x) + B 2N x < r x 0 0 < r < (8) x < (8) k A k B k f 2n (x, y) f 2n (x) A 2n y + A 2n B 2n y + B 2n A 2n B 2n y + F (α + n, β + n, γ + 2n; x) F (α + n, β + n +, γ + 2n + ; x) a a 2n x 2n y B 2n (B 2n + B 2n y). y x F (α, β, γ; x) F (α, β +, γ + ; x) f 2n(x) f 2n (x, y) f 2n (x) a a 2n x 2n B 2n (B 2n + B 2n y) y x 2n+ F (α, β, γ; x) (8) F (α, β +, γ + ; x) x 2n n x 0 x < F (α, β, γ; x) F (α, β +, γ + ; x) (8) α γ, β 0 2 F ( 2, 0, ) ( 2 ; x2, xf 2,, 3 ) 2 ; x2 arctan x 2

13 6 x arctan x + a x 2 a 2 x a 2nx x. arctan x + + x. a x 2 a 2 x 2 + a 2nx ( 2 a 2n + n ) ( ( + n ) 2 + 2n 2) ( + 2n ) (2n )2 (4n 3)(4n ), 2 2 n ( 2 a 2n + n ) ( 2 + 2n ) ( (2n)2 + 2n) (4n )(4n + ), n. 2 2 a n n 2 (2n )(2n + ), n x < arctan x 3

14 arctan x x x 2 3 (2x) (3x) (nx) 2 (2n )(2n+) x x 2 (2x) 2 (3x) 2 (20) 2n + (nx)2 [0, ] [0, ] (20) x 0 arctan π 4 π π (2) {A n }, {B n } A n (2n + )A n + n 2 A n 2, n 3, 4..., A 3, A 2 9, B n (2n + )B n + n 2 B n 2, n 3, 4..., B, B

15 A n B n n2 2n + n 5 π n A n B n A n /B n / / / / A 2 /B / [],,, 999. [2],,,

(iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y = 0., y x, y = x. (v) 1x = x. (vii) (α + β)x = αx + βx. (viii) (αβ)x = α(βx)., V, C.,,., (1)

1. 1.1...,. 1.1.1 V, V x, y, x y x + y x + y V,, V x α, αx αx V,, (i) (viii) : x, y, z V, α, β C, (i) x + y = y + x. (ii) (x + y) + z = x + (y + z). 1 (iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y