2000 8 3.4 p q θ = 80 B E a H F b θ/2 O θ/2 D A B E
BD = a, EA = b, BH = a, BF = b 3 EF B, EOA, BOD EF B EOA BF : AO = BE : AE, b : = BE : b, AF = BF = b BE = bb. () EF = b AF = b b. (2) EF B BOD EF : BO = BE : DB, EF : = BE : a, BE = a EF. (3) (), (2), (3) bb = a(b b ), b = ab a + b. (4) 2 F BH, ABD BF : BA = BH : BD, b : BA = a : a BA = 2a 2(a ) 2 = ab, ab a = 2. (5) BB = 2a p = 2a EE = 2b q = 2b 2 AB = 2a p 2 = (2)(2a ) 2BF = 2b q 2 = (2)(2b ) (4), (5) p, q, p 2, q 2 q 2 = 4 (2) 2 p q (2) (p + q ) = 2p q, (6) p + q p 2 = 4 2 (2) p (4) q 2 = p q 2. (7) = 6 p 6 = 6, q 6 = 4 3 (6) q 2 = 2 6 4 3 6 + 4 3 = 24(2 3). (7) p 2 = 6 24(2 3) = 2 2 3. 2
= 2, 3.058 < π < 3.253, = 24, 3.326 < π < 3.596, = 48, 3.393 < π < 3.460, = 96, 3.40 < π < 3.427, = 92, 3.44 < π < 3.48. π 96 3.4 92 3.4 2 263 E B G O θ D A G B E 6 a BB = a OAB 2 OA BD = ( ) 2 2 a = 4 a s 2 = (2)(a /4) = a /2 ( ABB ) = ( OAB ) 2 ( OBB ) 3
(5 OBGG B ) = ( OAB ) 2 + ( ABB ) = ( OAB ) 4 ( OBB ) ( ) < 4 ( OAB ) ( OBB ) = 2s 2 s, s 2 < ( ) < s 2 + (s 2 s ) p = a s 2 = p /2 2 p < ( ) < 2 p + 2 (p p /2 ) π π a 2 a OD = (a /2) 2, AD = OD AB = BD 2 + AD 2 = (a /2) 2 + ( (a /2) 2 ) 2 = 2 4 a 2. AB = a 2 a 2 = 2 4 a 2. (8) (8) = 6 a 6 = a 2 = 2 3 (8) a 24 = 2 2 + 3, a 48 = 2 2 + 2 + 3, a 96 = 2 2 + 2 + 2 + 3, a 92 = 2 2 + 2 + 2 + 2 + 3,. 4
s 24 = 2 2 3 2 2 + 3 = 3.326, s 48 = 2 3 3 2 2 + 2 + 3 = 3.393, s 96 = 2 4 3 2 2 + 2 + 2 + 3 = 3.40, s 92 = 2 5 3 2 2 + 2 + 2 + 2 + 3 = 3.45,. 3.45 < π < 3.422 5 3.45926 < π < 3.45927 22 7 = 3. 4285 7 355 = 3.45929 3 355 3 6 3 7 x = si θ π 2 θ π 2 θ = arcsi x x x = si θ π 2 θ π 2 θ arcsi x x = ta θ π 2 < θ < π 2 θ = arcta x < x < x = ta θ π 2 < θ < π 2 θ arcta x x = si θ dx dθ = cos θ = si 2 θ = x 2 d dθ arcsi x = dx dx = x 2 5 (9)
x = ta θ dx dθ = cos 2 θ = + ta2 θ = + x 2 d dθ arcta x = dx dx = (0) + x 2 x < + x 2 = x2 + x 4 x 6 + = ( ) x 2 (0) arcta x = C + x x3 3 + x5 5 x7 7 + = C + ( ) x2+ 2 + x = 0 C = arcta 0 = 0 arcta x arcta x = x x3 3 + x5 5 x7 7 + = ( ) x2+ 2 + () x < x = ta π = 4 arcta = π 4 () x = π 4 = 3 + 5 7 + = ( ) 2 + (2) ta x ta(α + β) = ta 2α = ta α + ta β ta α ta β 2 ta α ta 2 α α = arcta 5 ta 2α = 2 ta α ta 2 α = 5 2. 6
ta 4α = 2 ta 2α ta 2 2α = 20 9. β = 4α π 4 ta β = ta 4α + ta 4α = 239. β = arcta 239 π 4 = 4α β = 4 arcta 5 arcta 239. () (704 ) π 4 = 4 ( ) (2 + )5 ( ) (3) 2+ (2 + )239 2+ (2) = 4 π = 3.459 arcta x arcsi x 2 a ( + x) a = a ( ) a x, ( ) a = a C = a(a ) (a + ),! ( ) ( ) a a = a a 0 a = /2 ( ) 2 = ( )( 3 2 ) ( ) 2 2 2 (2 )(2 3) 3 = ( ),! 2! ( /2 ) ( /2 ) x x < ( + x) /2 + x = + (2 )(2 3) 3 ( ) x. 2! = x x 2 = + x 2 = (2 )(2 3) 3 x 2. 2! 7
(9) arcsi x = x + = (2 )(2 3) 3 x 2+ (4) 2 (2 + )! si π = 6 2 arcsi = π 2 6 (4) x = /2 π 6 = 2 + = (2 )(2 3) 3 2 3+ (2 + )! (5) = 3 π = 3.4 (5) π 6 (665 ) (2) (5) (739 ) 49 π 4 (83 ) α (α; ) = α(α + ) (α + ) (α; 0) = α, β, γ 0,, 2,... x < F (α, β, γ; x) = (α; )(β; ) x (6) (γ; )! F (α, β, γ; x) α, β, γ (α; ) ( ) α F (α, β, β; x) = x = x = ( + x) α,! xf (,, 2; x) =!! ( + )!! x+ = 8 + x+ = log( x),
( xf 2,, 3 ) 2 ; x2 = 2 + ( ) x 2+ = arcta x. 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), (7) γ(γ + ) β(γ α) xf (α +, β +, γ + 2; x). (8) γ(γ + ) (7) 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) (8) (α, β, γ) (α, β +, γ + ) 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) = = (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)... F (α, β, γ; x) F (α, β +, γ + ; x) = + +... + a x a 2 x... a 2ν x F (α+ν,β+ν,γ+2ν;x) F (α+ν,β+ν+,γ+2ν+;x). (α + )(γ + β) a 2 = (γ + 2 2)(γ + 2 ), a (β + )(γ + α) 2 = (γ + 2 )(γ + 2),. 9
α = γ =, β = 0 2 F (/2, 0, /2; x2 ) =, xf (/2,, 3/2; x 2 ) = arcta x x arcta x = + a x 2 a 2 x 2 +...... + a 2ν x 2 F (ν+/2,ν,2ν+/2; x 2 ) F (ν+/2,ν+,2ν+3/2; x 2 ) (2 ) 2 a 2 = (4 3)(4 ), a (2) 2 2 = (4 )(4 + ),. a = 2 (2 )(2 + ), ν arcta x. arcta x = + + + x x 2 3 (2x) 2 3 5 (3x) 2 5 7...... + (x) 2 (2 )(2+)... = + 3 + 5 + x x 2 (2x) 2 (3x) 2...... (9) 2 + (x)2... 0
(9) x = arcta = π 4 π π = 4 + 3 + 5 + 2 2 2 3 2 7 + 42 9 +... (20) {A }, {B } A = (2 + )A + 2 A 2, = 2, 3,..., A = 3, A 2 = 9, B = (2 + )B + 2 B 2, = 2, 3,..., B =, B 2 = 6. A B = + 3 + 5 + 4 7 + 2 2 2 3 2...... + 2 2 + = 5 π A B A /B 3 3 3 2 9 6 9/6 3.66 3 60 5 60/5 3.37 4 744 555 744/555 3.42 5 2384 7380 644/205 3.44 A 2 /B 2 = 3059428896/973843685 = 3.45926546 8.
3 2 ( 96 ) 263 2 ( 92 ) 5 7 665 6 ( ) 737 ( ) 67 ( ) 674 ( ) 706 00 ( ) 739 49 ( ) 83 [],,, 999. [2],,, 994. 2