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1 III ( , 0 deg (f(xg(x deg f(x + deg g(x 12 f(x, g(x ( g(x 0 f(x q(xg(x + r(x, r(x 0 deg r(x < deg g(x q(x, r(x q(x, r(x f(x g(x r(x 0 f(x g(x g(x f(x g(x f(x g(x f(x 13 f(x x a q(x, r f(x q(x(x a + r x a f(a r f(x x a f(a f(x x a f(a (polynomial 1

2 14 (x a x x a f(x x 3 + 2x 2 + 3x + 4 (x 1 1 y x 1 x y + 1 x 3 + 2x 2 + 3x + 4 (y (y (y y 3 + 3y 2 + 3y y 2 + 4y y y 3 + 5y y + 10 (x (x (x x 1 Taylor 3 f (4 (x f(x f(1 + f (1(x 1 + f (1 2 (x f (1 (x 1 3 3! f(1 1, f (1 10, f (1 10, f (1 6 3 x x 3 + 2x 2 + 3x + 4 (x (x (x x 3 + 2x 2 + 3x + 4 (x 2 + 3x + 6(x , x 2 + 3x + 6 (x + 4(x , x + 4 (x x 2 + 3x + 6 ((x 1 + 5(x (x (x , x 3 + 2x 2 + 3x + 4 (((x 1 + 5(x [ (x (x ] (x (x (x (x β β 3 2

3 15 a f(x (x a f(x f(a 0, a f(x 0 a f(x m (x a m f(x a f(x m (x a m f(x (x a m+1 f(x m a 11 a f(x m f(a f (a f (m 1 (a 0 a f(x m f(a f (a f (m 1 (a 0, f (m (a 0 ( a f(x m q(x f(x (x a m q(x Leibniz n ( ( j n d f (n (x (x a m q (n j (x j dx j0 ( d j n m 1 j m 1 dx (x a m (x a x a 0 f (n (a 0 (0 n m 1 ( m 1 m f(x f(a f (a f (m 1 (a f (m (a 0 (x a m f(x q(x f (m (x f(x (x a m q(x m j0 ( ( j m d (x a m q (m j (x j dx f (m (a m!q(a q(a 0 q(x st q(x (x a q(x f(x (x a m q(x (x a m+1 q(x f(x (x a m+1 m + 1 3

4 16 11 (, Gauss n N n 1 Johann Carl Friedrich Gauss ( ( : 1 ax + b 0 (a 0 x b/a 2 ax 2 + bx + c 0 (a 0 x b ± b 2 4ac 2a (a, b, c 2 Leibniz 4

5 λ α + iβ (α, β λ 0 z 2 λ (λ β 0 ( λ β 0 z x + iy (x, y R z 2 (x 2 y 2 + 2ixy z 2 λ x 2 y 2 α, 2xy β y β/(2x x 2 β2 4x 2 α 4x 4 4αx 2 β 2 0 x 2 2 x 2 2α ± 4α 2 + 4β 2 4 α ± α 2 + β 2 2 x 2 0 x 2 α + α 2 + β 2 2 β 0 α + α x ± 2 + β 2, 2 y β 2x ± β 1 2 α + ( α 2 + β n (Abel 3 -Galois 4 [3] 18 1 α + iβ (α, β β 0 α iβ Niels Henrik Abel ( , 4 Evariste Galois ( , 5

6 11 f(x λ f(λ f(λ λ f(x λ ( λ m λ m f(λ a 0 λ n + a 1 λ n a n 1 λ + a n (a j, a 0 0 a j a j f(λ a 0 λ n + a 1 λ n a n 1 λ + a n a 0 λ n + a 1 λ n a n 1 λ + a n a 0 λ n + a 1 λ n a n 1 λ + a n f ( λ λ f(x f(λ 0 f(λ 0 f(λ 0 λ f(x λ f(x m f(λ f (λ f (m 1 (λ 0, f (m (λ 0 f(λ f (λ f (m 1 (λ 0, f (m (λ 0 λ m λ α + iβ (α, β (x λ ( x λ [x (α + iβ] [x (α iβ] [(x α + iβ] [(x α iβ] (x α 2 + β 2 x 2 2αx + (α 2 + β 2 ( (x λ m ( x λ m [ (x α 2 + β 2] m 2 Euclid 21 ( 6

7 22 Euclid p(x, q(x d(x GCD(p(x, q(x r(xp(x + s(xq(x d(x r(x, s(s 1: d(x deg p(x deg q(x f 0 (x : p(x, f 1 (x : q(x j 0, 1, 2, f j (x f j+1 (x q j+1 (x, f j+2 (x (1 f j (x q j+1 (xf j+1 (x + f j+2 (x, deg f j+2 (x < deg f j+1 (x f j+2 (x 0 deg f 0 (x deg f 1 (x > deg f 2 (x > k 0 st f k+1 (x 0, f k+2 (x 0 (2 f k+1 (x f k (x, f k+1 (x (1 f j (x, f j+1 (x f j+2 (x f j+1 (x, f j+2 (x f j+1 (x, f j+2 (x f j (x f j (x, f j+1 (x f j (x, f j+1 (x f j+1 (x, f j+2 (x (3 p(x, q(x f k (x, f k+1 (x (2 (3 f k+1 (x p(x, q(x 2: r(x, s(x ( f j+1 (x f j+2 (x ( q j+1 (x ( ( ( f k (x f k+1 (x 1 q k (x 1 q k 1 (x ( f j (x f j+1 (x ( q 1 (x ( f 0 (x f 1 (x f 0 (x p(x, f 1 (x q(x d(x f k+1 (x r(xp(x+s(xq(x r(x, s(x r(x, s(x 7

8 21 0 f(x, g(x d(x s(xf(x + t(xg(x d(x s(x, t(x f(x, g(x deg s(x < deg g(x, deg t(x < deg f(x r(x, s(x s(x q(xg(x + s 0 (x, deg s 0 (x < deg g(x, t(x r(xf(x + t 0 (x, deg t 0 (x < deg f(x d(x s(xf(x + t(xg(x (q(xg(x + s 0 (x f(x + (r(xf(x + t 0 (x g(x s 0 (xf(x + t 0 (xg(x + (q(x + r(xf(xg(x q(x + r(x 0 (q(x + r(xf(xg(x d(x s 0 (xf(x t 0 (xg(x q(x + r(x 0 deg(q(x + r(x 0 deg((q(x + r(xf(xg(x deg(q(x + r(x + deg(f(xg(x deg(f(xg(x deg (s 0 (xf(x deg s 0 (x + deg f(x < deg g(x + deg f(x deg (f(xg(x, deg (t 0 (xg(x deg t 0 (x + deg g(x < deg f(x + deg g(x deg (f(xg(x, deg d(x min{deg f(x, deg g(x} < deg f(x + deg g(x deg (f(xg(x q(x + r(x 0 d(x s 0 (xf(x + t 0 (xg(x, deg s 0 (x < deg g(x, deg t 0 (x < deg f(x 23 Euclid ( f, g, h, p, q, r x f deg f 0 f, g f g (f, g : (f, g : {h; f f 1 h, g g 1 h f 1, g 1 } 8

9 (f, g f g (greatest common divisor, GCD 22 0 f, g h (f + hg, g (f, g p (f + hg, g f + hg h 1 p, g g 1 p h 1, g 1 f (f + hg hg h 1 p h g 1 p (h 1 hg 1 p p f p (f, g p (f, g f f 1 p, g g 1 p f 1, g 1 f + hg f 1 p + h g 1 p (f 1 + hg 1 p p f + hg p (f + hg, g 21 f, g ( 0 f g r (f, g (g, r f g q f qg + r (f, g (qg + r, g (r, g (g, r 21 ( f, g ( 0 (0 (2 d, r, s d f g (0 f 0 f, f 1 g rf + sg d (1 j 0, 1, 2, f j f j+1 q j, f j+2 f j+2 0 j k (2 (2 d f k+1 ( ( ( p q 0 1 r s 1 q k q k 2 ( q 1 ( q 0 r, s ( (2, 1 r, (2, 2 s f j f j+1 q j, f j+2 f j q j f j+1 + f j+2 f j+2 0 f j+2 0 deg f j+2 < deg f j+1 21 (f j, f j+1 (q j f j+1 + f j+2, f j+1 (f j+2, f j+1 (f j+1, f j+2 9

10 0 deg f 1 > deg f 2 > k f i 0 (1 i k + 1, f k+2 0 (f, g (f 0, f 1 (f 1, f 2 (f k, f k+1 (f k+1, f k+2 (d, 0 d d f g f j q j f j+1 + f j+2 ( ( ( f j f j f j+2 1 q j f j+1 ( ( ( ( ( ( f k 0 1 f k f k+1 1 q k 1 f k 1 q k 1 1 q k 2 ( ( ( ( ( q k 1 1 q k 2 1 q 1 1 q 0 ( ( ( p q f pf + qg r s g rf + sg rf + sg f k+1 d 21 r, s deg r < deg g deg s < deg f r q 1 g + r 0, s q 2 f + s 0 r 0 f + s 0 g d ( f k 2 f k 1 f 0 f g(x/f(x 1 q(x r(x/f(x : g(x/f(x q(x + r(x/f(x 2 f(x : (4 f(x A k (x a j m j j1 k+l jk+1 [ ] (x αj 2 + βj 2 mj 10

11 3 r(x/f(x : r(x f(x k j1 r j (x (x a j + k+l m j jk+1 r j (x [ (x αj 2 + β 2 j ] mj r j (x 4 r j (x/(x a j m j, r j (x/ [ ] (x α j 2 + βj 2 mj : r j (x (x a j m j m j l1 a j,l (x a j, r j (x [ ] l (x αj 2 + βj 2 mj m j l1 b j,l x + c j,l [ (x αj 2 + β 2 j ] l g(x/f(x 3 4 (4 r(x f(x m k j j1 l1 a j,l (x a j + k+l l m j jk+1 l1 b j,l x + c j,l [ (x αj 2 + β 2 j ] l a j,l, b j,l, c j,l 32 f(x, g(x ( f(x 0 g(x f(x q(x, r(x g(x q(xf(x + r(x q(x + r(x f(x f(x f(x 2 r(x f(x ( 33 ( 31 ( f 1 (x, f 2 (x 0 r(x g 1 (x, g 2 (x r(x f 1 (xf 2 (x g 1(x f 1 (x + g 2(x f 2 (x g 1 (x, g 2 (x 1 f 1 (x, f 2 (x h 1 (x, h 2 (x st h 1 (xf 1 (x + h 2 (xf 2 (x 1 11

12 r(x f 1 (xf 2 (x r(xh 1 (x f 2 (x + r(xh 2(x f 1 (x r(x f 1 (xf 2 (x g 1 (x : r(xh 2 (x, g 2 (x : r(xh 1 (x r(x f 1 (xf 2 (x g 1(x f 1 (x + g 2(x f 2 (x g j (x f j (x q j (x, r j (x g j (x q j (xf j (x + r j (x, deg r j (x < deg f j (x g j (x f j (x q j(xf j (x + r j (x f j (x r j(x f j (x + q j(x r(x f 1 (xf 2 (x r 1(x f 1 (x + r 2(x f 2 (x + q 1(x + q 2 (x x q 1 (x + q 2 (x 0 r(x f 1 (xf 2 (x r 1(x f 1 (x + r 2(x f 2 (x 34 x a, (x α 2 + β 2 r(x, deg r(x < m (x a m r(x [(x α 2 + β 2 ] m, deg r(x < 2m 341 r(x/(x a m (14 m 1 r(x (x a m j0 r(x a j (x a j m 1 (x a m j0 m 1 j0 a j (x a j a j (x a m m j l1 ã l (x a l, ã l : a m l 12

13 342 r(x/ [(x α 2 + β 2 ] m Q(x : (x α 2 + β 2 2 2m 1 r(x r(x m 1 j0 a j (xq(x j, deg a j (x 1 ( m 1 r(x Q(x m j0 a j Q(x j Q(x m m 1 j0 a j Q(x m m j l1 ã l Q(x l, ã l : a m l : Leibniz (Gottfried Wilhelm Leibniz, , ( Arctan A (x a n, px + q [(x α 2 + β 2 ] m (A, a, p, q, α, β n, m dx (x a n 1 1 (n > 1 n 1 (x a n 1 log x a (n 1 px + q [(x α 2 + β 2 ] m dx x α u u 1 (u 2 + β 2 du n 13

14 41 (1 b, n x (x 2 + b 2 n dx 1 1 2(n 1 (x 2 + b 2 n 1 (n > log(x2 + b 2 (n 1 (2 b 0 1 I n dx (n (x 2 + b 2 n ( 1 x 2n 3 + b I n 2 (2n 2(x 2 + b 2 n 1 2n 2 I n 1 1 ( x b Arctan b (n > 1 (n 1 (1 u x 2 + b 2 (2 n 1 (x b tan θ n > 1 I n 1 I n 1 (x 2 + b 2 1 n dx (x (x 2 + b 2 1 n dx x(x 2 + b 2 1 n x (1 n(x 2 + b 2 n (2x dx x x 2 + 2(n 1 (x 2 + b 2 n 1 (x 2 + b 2 dx n x x 2 + b 2 b 2 + 2(n 1 (x 2 + b 2 n 1 (x 2 + b 2 dx n x (x 2 + b 2 + 2(n 1(I n 1 n 1 b 2 I n 2(n 1b 2 I n x (x 2 + b 2 n 1 + (2n 3I n 1 ( 1 x I n 2(n 1b 2 (x 2 + b 2 + (2n 3I n 1 n 1 14

15 5 2: Laplce ( 6 ( [3] ( Strum [1] [4] [1],, (1966 [2], I, (1980 [3],, (1965 [4],, (1992, 1992,

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