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1 2 4 BASIC ( WWW BASIC 2 (. ( B 5 4 ( ( 3

2 ( PAINT PAINT ( MAT PLOT AREA.3 : MAT PLOT AREA PAINT ( BASIC (.3. MAT PLOT AREA REM testmatplotarea.bas --- mat plot area DIM x(3,y(3 LET x(=0 LET y(=0 LET x(2=2 LET y(2= LET x(3= LET y(3=2 SET WINDOW -,3,-,3 DRAW grid SET AREA COLOR "red" MAT PLOT AREA: x,y END.3.2 PAINT Windows BASIC PAINT PAINT x,y (x, y set area color ( (, 2

3 : REM testpaint.bas --- paint (Windows SET WINDOW -,3,-,3 DRAW grid PLOT LINES: 0,0;2,;,2;0,0 SET AREA COLOR "red" paint, END PAINT, PAINT 0, PAINT JIS ( Linux BASIC BASIC PAINT f a f(x = f (n (a n! (x a n = f(a+f (a(x a+ f (a 2 (x a f (n (a (x a n + n! ( 4, 2 VRAM (video RAM VRAM ( PAINT Windows 3

4 f(x n n n f (k (a s n (x := (x a k k! k=0 f(x 2.2 e (Euler, Napier e e e x = exp x x = x = e = exp = exp x = 0 0 j= j! = ! + + 0! x n n! n! = ! +. = ( ( 7 e : n j= a j s=0 for j= to n s=s+(a j next j a j = j! a 0 =, a j = a j j 4

5 s n = n j=0 j! n = 0 REM REM a0, s0 LET a= LET s=a for j= to n LET a=a/j LET s=s+a next j print s end 2.3 4A ( Subject ( 2 4A e 00 ( arctan x = 4 arctan = 6 arctan 3 (, ( arctan arctan x = ( n 2n x2n = lim n a j = ( j 2j x2j n j=0 ( j 2j x2j t j = ( j x 2j t j t = x, t j = x 2 t j 5

6 a j a j = t j 2j piarctan.bas rem piarctan.bas --- rem arctan x n input x input n f=-x*x t=x s=0 for j= to n a=t/(2*j- s=s+a t=f*t next j print "arctan(x ";s print " 4 ";4*s print using " =-%.###^^^^^^";4*s-pi end = 4 arctan = 4 ( n n 4s n = ( B Subject ( 2 4B TEX TEX 5 22 ( 3 4s n 6

7 6 25 TEX PDF 00 ( piarctan.bas arctan AGM A A. A.. = ( (? A..2 r 2r r r 2 4 ( 4r 2 4r 3 /3 ( 2, 3 (2 n 4 r 2 r 2 3 7

8 A..3 (Johann Heinrich Lambert, , Mülhausen (Mulhouse, Berlin, (76 tan x = x x 2 x x x 0 tan x ( ( (Carl Louis Ferdinand von Lindemann, , Hilbert 5 (822 A (3.4 < < 3.5 ( (0 : (00 ( 3.4 ( 3 m mm

9 A.2 A.2. 2C ( ~mk/syori2-2007/jouhousyori /node8.html < < ( Ludolph van Ceulen (540 60, Hildesheim Leiden (596 A.2.2 ( 4 = arctan = (2 arctan x = n= ( n 2n x2n ( x < 6 (2 Taylor f(x = f (n (a (x a n n! 6 x = 9

10 400 ( arctan arctan x = y x ( y y2 +, y = x2 + x 2 (L. Euler Newton arcsin 2 arcsin x = x + x (arcsin x 2 = 2 x (n!2 n 2 (2n + 2! x2n+2 x Euler ( ( x arctan 0

11 Abraham Sharp ( = arctan. 3 John Machin ( , = 4 arctan 5 arctan William Shanks ( (567 L.Euler ( , Basel Petersburg = arctan 2 + arctan 3, = 20 arctan arctan Charles Huttion ( = 2 arctan 3 + arctan 7 = 2 arctan 2 arctan 7 = arctan 2 + arctan 3 = 3 arctan 4 + arctan C.F.Gauss (Johann Carl Friedrich Gauss, = 2 arctan arctan 57 5 arctan 239, = 2 arctan + 20 arctan arctan + 24 arctan ( = 2805 arctan 398 arctan arctan arctan arctan arctan arctan arctan arctan ( 896 F.C.M. Störmer (2002 a 4 = 44 arctan arctan 2 arctan arctan a 400h, Störmer 57h

12 (982 4 = 2 arctan + 32 arctan arctan arctan 0443 ( bit 983 4, A.3 AGM ( ( 976 E.Salamin R.P.Brent Salamin-Brent ( Gauss Legendre a =, b = / 2 a 0 := a, b 0 := b, (3 a n+ := a n + b n, b n+ := a n b n (n = 0,, 2,, 2 c n := a 2 n b 2 n (n = 0,, 2, n := lim n = n 2a 2 n+ n 2 k c 2 k k=0 (. 2 (4 (4 n+ 2 (n+ ( 2 n 2 (, n 2 2 n+4 e 2n+. ( 9 7 ( , WWW ( 2

13 (i K(k := 0 dx /2 ( x2 ( k 2 x 2 = dθ 0 k2 sin 2 θ (0 k < k2 x E(k := 2 dx = x 2 0 /2 0 k 2 sin 2 θ dθ (0 k Legendre (Legendre s relation (5 K(kE(k + K(k E(k K(kK(k = 2 ( k := k 2 k = / 2 (6 2K ( 2 ( ( 2 E 2 K 2 = 2. (ii K(k, E(k (arithmetic geometric mean, AGM I(a, b, J(a, b I(a, b := /2 0 dθ /2 a2 cos 2 θ + b 2 sin 2 θ, J(a, b := I(a, bm(a, b = 2, J(a, b = (a 0 a 2 cos 2 θ + b 2 sin 2 θ dθ 2 n c 2 n I(a, b M(a, b a, b (3 {a n }, {b n } : M(a, b := lim n a n = lim n b n. {c n } c n := a 2 n b 2 n K(k = I (, k, E(k = J (, k, k = k 2 ( (7 K 2 = ( ( 2 M(, / 2, E 2 = 2 2 n c 2 n M(, / 2. 3

14 (7 (6 = 2M (, / n c 2 n ( arctan 97 Strassen Schönhage ( Fourier n O(n 2 O(n log n A.4 A.4. ( Chudonovsky ( = /2 ( p n k= (2k (6k 5(6k 9k ( n ( n = 2 ( 2 p n (2k (4k 3(4k (26390n k k= A.4.2 DRM AGM AGM DRM ( arctan Ramanujan arctan, AGM, Ramanujan O(n(log n p (p = 2, 3 (DRM, ( , pdf arctan arctan Ramanujan 4

2 8 BASIC (4) WWW Taylor BASIC 1 2 ( ) 2 ( ) ( ) ( ) (A.2.1 ) 1

2 8 BASIC (4) WWW   Taylor BASIC 1 2 ( ) 2 ( ) ( ) ( ) (A.2.1 ) 1 2 8 BASIC (4) 203 6 5 WWW http://www.math.meiji.ac.jp/~mk/syori2-203/ Taylor BASIC 2 ( ) 2 ( ) ( ) ( ) 2 (A.2. ) 6B 2 2. f a f(x) = f (n) (a) n! (x a) n = f(a)+f (a)(x a)+ f (a) 2 (x a) 2 + + f (n) (a)

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