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とらふみふじがわ
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1 1 3 2 ( )

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18 *3 R R ( x, y R d (x, y) = x y ) S T f f T S ε δ 4 Kuratowski Zorn 10 Zorn Kuratowski-Zorn Kuratowski *3 Hahn-Banach

19 ( ) ( ) ( ) ( ) Topology for Analysis (Albert Wilansky Dover Publications) ( ) ( )

20 I ( ) ( )

21 2.3 I 21 (Taylor ) ( ) Cauchy f (z) = 1 f (ζ) 2πi C ζ z dζ. f (z) C ( ) Taylor Cauchy

22 Cauchy ( )

23 ( 2 ) 2 ( n 2 n+1 1 ) ( )

24 I I 5 (Cauchy Ultrafilter in nonstandard method ) R I ( ) G 2 : G G G 3 G (group) 1. a, b, c G (a b) c = a (b c) 2. e G a G e a = a e = a ( e G ) 3. a G a G a a = a a = e Fermat Lagrange (Fermat *1 ) *1 p a {1, 2,..., p 1} modp {a, 2a,..., (p 1) a} = {1, 2,..., p 1}

25 3.1 I 25 p x, y Z x 2 + y 2 = p p 1 (mod 4) Z [ 1 ] ( ) 0 Web Lemma TA TA

26 I ( ) ( ) ( ) nonstandard analysis Applied Nonstandard Analysis (Martin Davis Dover Publications) ( )

27 3.2 I I 2 R 2

28 ( I ( ) ) ( ) Riemann II

29 3.2 I 29 ( )

30 IV 1 ( ) ( ) ( ) ( )

31 3.3 IV 31 ( ) ( ) 2 ( ) ( ) ( ) (50%) (30%) (20%)

32 IV( ) Lp TA 4 5 ( )

33 3.4 II II II I I I I II C( Ĉ) ( ) I

34 ( ) ( ) D Ĉ D 2 a D f : D (1) f (a) = 0, f (a) > 0 (a ), df z d z (0) > 0 (a = ) 1 ( ( ) ) C

35 3.4 II 35 ( ) ϕ (z) = z i z + i ϕ : H (1) ( ) H C n D f : H D < a 1 < < a n + b j = f (a j ) D α j π (0 < a j < 2) f (z) f (z) = C 1 z n j=1 z n 1 f (z) = C 1 j=1 C 1 ( 0) C 2 (z a j ) α j 1 dz + C 2 (a n + ) (z a j ) α j 1 dz + C 2 (a n = + ) ( ( ) ) ( )

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37 3.5 I TEX 6 Jacobi SOR Newton Newton-Cotes Gauss Euler Heum Runge-Kutta gnuplot TEX 4

38 UNIX GCC gnuplot TEX 2 ( )

39 3.6 I I TA TA PC Lebesgue PC PC 1. PC PC

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60 l, m, n l A A m 60 (m 2 A ) m m n C A C 60 B A, B, C l, m, n AB = AC, BAC = 60 ABC 2 N = p i q j (p < q) p i+1 1 p i+1 p i qj+1 1 q j+1 q j = 2 p 2 p 3, q 5 ( ) < p p 1 q q = 15 8 < 2

61 61 p = 2 2 i+1 1, qj+1 1 q 1 ( 2 i+1 1 ) q j+1 1 q 1 = 2 i+1 q j 2, q 2 i+1 1 = q j, q j+1 1 = 2 i+1 (q 1) i j = 1, q = 2 i+1 1 N = 2 i ( 2 i+1 1 ) 3 9 e πi 9 + π (e) i 9 i π(e!) 2 π ( x x ) 3 (e! = Γ (e + 1) = )

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