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1 2 IT 1 *1 1 2 3 π i 1i 2i 3i πi i 2 1 *2 2 + 3 + 4i π ei 3 4 4 2 2 *1 *2 x 2 + 1 = x 2 + x + 1 = 2 3 1

2 2 2 2 *3 i 9 (1,) i (i,) (1,) 9 (i,) i i 2 1 ( 1, ) (1,) 18 2 i, 2 i 1 2 1 2 2 1 i r 3r + 4i 1 i 1 i *4 1 i 9 i 1 1 i 2 3 4 1 1 1 1 1 2 3 + 4i 3 9 + 12i 3 + 4i 3i 12 + 9i 3 i 9 *5 i 9 i 1 9 1 + 2i 9 2 + i 1 2i 2 i 3 *3 *4 3 3 1 n n 1 *5 (2 + 3i)(4 + 5i) = 8 15 + 12i + 1i = 7 + 22i r 1 (cos θ 1 + i sin θ 1 ) r 2 (cos θ 2 + i sin θ 2 ) = r 1 r 2 {cos(θ 1 + θ 2 ) + i sin(θ 1 + θ 2 )} 2

1 + 2i 1 i 1 4 1 i i 1 *6 1 *7 (1 + i)(a + bi) = a b + (a + b)i ( 1 + i)(a + bi) = a b + (a b)i ( 1 i)(a + bi) = a + b + ( a b)i (1 i)(a + bi) = a + b + ( a + b)i *8 *9 * 1 4 3 2 1 2 2 1 2 2 = (1 + i)(1 i) i 2 = 1 (1 + i)(1 i) = 1 2 i 2 = 1 2 ( 1) = 2 2 1 + i 1 i 9 1 + i 1 + i 1( 4 ) 1 + i 2 1 + i 2 2 1 + i 1 + i 1 + i a + bi a, b b -2-4 -4-2 2 4 a 2 1 + i * 11 3 4 5 4 1, 1, i, i * 12 *6 1 *7 a + bi a 2 + b 2 = 1 4 *8 *9. 2 1(mod 4). 4 1(mod 8) *1 *11 2 1+i 2 + 2 = 2 2 = 2 2 1 + i *12 *13 n n = 1 n = 2, 3, 7, 11, 19, 43, 67, 163 3

* 13 2 (a + bi)(a bi) (1 + 2i)(1 2i) = 5 (1 + 4i)(1 4i) = 17 (1 + 6i)(1 6i) = 37 (2 + 3i)(2 3i) = 13 (2 + 5i)(2 5i) = 29 (2 + 7i)(2 7i) = 53 (4 + 5i)(4 5i) = 41 4 1 4 3 * 14 1 + i * 15 1) 4 3 2) 4 3 2) a + bi a 2 +b 2 * 16 2 * 17 b 1 8 6 4 2 2 2) 2 2 2) a 2 + b 2 2 a, b 5 2 = 1 2 + 1 2 5 = 1 2 + 2 2 = 2 2 + 1 2 13 = 2 2 + 3 2 = 3 2 + 2 2 17 = 1 2 + 4 2 = 4 2 + 1 2 29 = 2 2 + 5 2 = 5 2 + 2 2 37 = 1 2 + 6 2 = 6 2 + 1 2 41 = 4 2 + 5 2 = 5 2 + 4 2 2 2) 4 1 a * 18 x x log x 4 1 4 3 x x 4 3 2 log x 2ax log x x + 2ax log x a a x.46 2 4 a 6 8 1 *14 4 2 2 *15 *16 a + bi a bi a bi *17 *18 4

3 5 (a + bi)(c + di) = (ac bd) + (ad + bc)i a + bi a 2 + b 2 c + di c 2 + d 2 (ac bd) + (ad + bc)i (a 2 + b 2 )(c 2 + d 2 ) A A A A 6(= 1 + 2 + 3) 28(= 1 + 2 + 4 + 7 + 14) 496 8128 3355336 858986956 137438691328 2 n M(M + 1) 1(= M) 2 (2 n 1) * 19 GIMPS * 2 a, b < 4 1 1 2 3 3 1+i 1 1 1 1 2 + 2 1 1+i 2 1 2 + 1 2 1+i 3 a b a 2 + b 2 4 5 1 2 + 2 2 1+2i 2+i * 21 a 2 + b 2 3 4 n a, b n = a 2 + b 2 a + bi n n 1 + 2i 2 + i * 22 * 23 a + bi a a b *19 *2 4 1 3 3 (2 n 1) 1 (2 2n + 1) n 2 Bernoulli *21 9 *22 *23 5

a + bi a a 1 a + bi * 24 a * 25 n n n n n 1/2 n 1/1.99 n 1/2.1 2 2 j = ( 1) 1/1.99 1 a x x x x * 28 x e ix = cos x + i sin x 2 4 2 3 3 4.1 π 3 2 * 26 * 27 a + bi e a+bi = e a e bi e a+bi = e a (cos b + i sin b) *24 a 1 a + bi *25 *26 *27 *28 6

e α e β = e α+β c + di e e c+di = e c (cos d + i sin d) e a+bi e c+di = e a+c (cos b + i sin b)(cos d + i sin d) = e a+c (cos(b + d) + i sin(b + d)) = e (a+bi)+(c+di) * 29 2π = m = y i < 2π 2 1 log i = log 1 + πi/2( ) 2 i i = e i(πi/2) = e π/2 ( ) e a x = e x log a A A x = e x log A log A a + bi = e x+yi log(a + bi) = x + yi log x a y b OK 2 n n (n 1) (n 2) 1 3 3 2 1 = 6 (1/2) π * 3 e x+yi = e x (cos y + i sin y) = a 2 + b 2 ( a/ a 2 + b 2 + ib/ a 2 + b 2) = a + bi Γ(n) = (n 1) = x n 1 e x dx e x a 2 + b 2 x = log ( a 2 + b 2) y cos y = a/ a 2 + b 2 sin y = b/ a 2 + b 2 y y y = y + 2mπ m e x+yi = e x (cos y + i sin y) = a + bi x, y ( ) x+yi = log(a+bi) = log a2 + b 2 +i(sin y +2mπ), m log(a + bi) m y m y Γ(n) = (n 1)Γ(n 1) 1 2 n * 31 n = 2.46 Γ(1/2) π Γ(π) 2.288 Γ(1 + i) =.498.155i *29 e *3 e ax dx = 1/a e ax xe ax dx = 1/a 2 *31 n n 7

Gamma(n) 5 4 3 2 1 a aπ/2 a a a sin(n) sin(n) sin(n) = ( 1) x n 2x+1 (2x + 1)! x= = n n3 6 + n5 12 n7 54 + n9 36288 +..5 1. 1.5 2. 2.5 3. n 1 n n x n x n 1 nx n 1 * 32.5 1.3i d 1 sin(n) dn = cos(n) = ( 1) x n 2x (2x)! x= = 1 n2 2 + n4 24 n6 72 + n8 432 + x n 1 nx n 1 n(n 1)x n 2 n(n 1) n 1 n 1 Γ.5.5π/2 sin(n + π/4) n(n 1)(n 2)x n 3 a a x n a = Γ(n + 1) Γ(n + 1 a) xn a sin cos * 33 sin(n) 1 cos(n) sin π/2 9 cos 1 π/2.5 π/4 d.5 sin(n) dn = ( 1) x n 2x+1.5 Γ(2x + 2.5) x= 1 * 34 n * 35 a n *32 *33 *34 *35 n n = 5 8

d.5 sin(n) 1..5. -.5 * 36 π 2 2nπ 1 n n -1. 2 4 n 6 8 1 1 19 dy/dx n n n n n n n 3 n n 3 π/3 π/2 n π 2π/n (π 2) (π 2) 4 1. 1 n n 1 9 π/2 2. 1:1 π/2 3. 1 π/2 1 4. 1 4 n i 9 (π/2 ) 9 e ix x i e 1 1 (.368 ) i sin(a + bi) cos(ix) sin(ix) *36 π 2π/ 9

* 37 cos(ix) = e x + e x 2 sin(ix) = e x e x 2i = cosh x = i sinh x a + bi a + bi 2π a + bi π π 2π/(a + bi) a + bi 1 * 38 ( ) π = 2 sin a + bi A + Bi exp (i(a + Bi)) A e B 4 mod 1 mod 3 1/3 1 mod 1 mod 3.3.1 2 2 2 2 2 2 mod 2 2π mod 2π sin(x) cos(x) mod * 39 x 1, x 2, x 3 cx 4 i s s = x 2 1 + x2 2 + x2 3 c2 x 2 4 x 4 i 2 = 1 x 4 s x 4 * 4 mod 1 3 2 2 5 *37 x ix *38 *39 *4 s s *41 1 1

2 ( ) * 41 ( ) 2 4 8 4 8 4i 8i 2 2 * 42 a a n n n 1 1 a L(a) L(a) = a 1 n n 1 1/3 4/3 L(1/3) = (1/3) 1 n = 4/3 n n = log 3 4 = 1.26186 a a c+bi c + bi 4 2 2 x 3 3 x 3 x 3 3 Z 3 R 3 C 3 1 3 n * 43 n NANIKA n n NANIKA 3 3 n n 3 3 3 n Γ x n n a c+bi a c+bi c b x 2i +3x i +5x i + + + 2x 2 +x 1 +6x = 2 3 a 1x 2i + a 2x 2 + a 3x 1.99i + a 4x 1.99 + 2 1.99 2 2 *42 *43 11

x i = x 1 = x = x i = x i = e i log x = e i e log x = e i e log x = x x i = 1 e log x = 1/e i log x = i x = a + bi log x = log(a + bi) = log( a 2 + b 2 ) + i(sin y + 2mπ) = i log ( a 2 + b 2) = a 2 + b 2 = 1 a, b cos( 1) = a sin( 1) = b x = cos( 1) + i sin( 1) x = e i x 1 = 1 x = 1 = e x i = n log x = i log n e i log n x i = n (x i n)(x i m)(x i l) = n, m, l 5 1.999 K mixi 26 8 3 211 3 1 PDF http://www.mikaka.org/~kana/ 277-8691 54 D 19 / ISBN4-8731-71-2 C741 ` c Copyright 26-211 Printed in Japan 12

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