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

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

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) f (n) (a) (x a) n + n! ( 4, ) f(x) n n s n (x) := n k=0 f (k) (a) (x a) k ( x ) k! f(x) 2.2 e (Euler, Napier ) e 2

3 e e x = exp x x = exp x = x n n! (x R) x = e = exp = n! = ! j=0 j! = ! + + 0! = ( ) ( 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 (j ) n s n = n = 0 j! j=0 REM N=0 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 N ( ) 3

4 2.3 ( ) ( ) BASIC PRINT USING e 00 S 0 PRINT USING PRINT USING "#.#########": S (. # 0 ) OK BASIC OPTION ARITHMETIC DECIMAL HIGH PRINT ( 0 ) # 00 PRINT USING "#.#######( )#####": S (. # 00 ) # 00 ( &, REPEAT$()) FMT$="#."&REPEAT$("#",00) PRINT USING FMT$: S FMT$, REPEAT$() $ 2.4 8A ( ) e 00 ( ) ( ) 4

5 2.5 tan x (arctan x) = 4 arctan ( tan 4 = ), = 6 arctan 3 ( tan 6 = 3 ) arctan arctan (, ) 3 ( exp arctan ) arctan arctan x = ( ) n 2n x2n = lim n a j = ( )j 2j x2j. a n t j := ( ) j x 2j n j=0 ( ) j 2j x2j t j t = x, t j = x 2 t j a j a j = t j 2j 3 5

6 piarctan.bas REM piarctan.bas --- REM arctan n REM INPUT X 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 REM PI PRINT USING " =-%.###^^^^^^";4*S-PI END = 4 arctan = 4 ( ) n BASIC PI n 4s n = ( 4 ) 2.6 8B piarctan.bas 4 = tan = ( )n 2n s n 6

7 000 3 ( ( A.2.2) ) = tan 3 = ( )n ( )n 2n 2n x2n (x = / 3) ( ) ( 50 ) piarctan.bas ( ) arctan ( ) piarctan.bas AGM (AGM ) ( ) BASIC kadai8b.pdf Oh-o! Meiji 6 8 ( ) 8:00 A 2007 (2009, 200, 20 ) A. A.. = ( (?) ) 7

8 A..2 r 2r r r 2 5 ( 4r 2 4r 3 /3 ) ( 2, 3 ) (2 n ) A..3 (Johann Heinrich Lambert, , Mülhausen (Mulhouse, ) Berlin, ) (76 ) tan tan x = x x 2 x x x 0 tan x ( ) (transcendental number) ( ) (Carl Louis Ferdinand von Lindemann, , Hilbert 6 ) (822 ) 5 r 2 r

9 A..4 ( ) (3.4 < < 3.5) ( ) ( ) (0 ) : (00 ) ( ) 3.4 ( ) 3 m mm A.2 A.2. ( syori2-203/jouhousyori /node7.html ) < < ( ) 9

10 Ludolph van Ceulen (540 60, Hildesheim Leiden ) (596 ) A.2.2 () (2) arctan x = 7 4 = arctan = n= ( ) n 2n x2n ( x < ) (2) Taylor f(x) = f (n) (a) (x a) n n! 400 ( ) arctan arctan x = y x (L. Euler ) Newton ( y ) 3 5 y2 +, y = x2 + x 2 arcsin x = x + x x x arcsin 2 (arcsin x) 2 = 2 (n!2 n ) 2 (2n + 2)! x2n+2 Euler x = 0

11 () ( ) x arctan

12 Abraham Sharp (65 742) 6 = arctan. 3 John Machin ( , ) = 4 arctan 5 arctan William Shanks (82 882) 707 (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, ) 4 = 2 arctan arctan 57 5 arctan 239, 4 = 2 arctan arctan arctan + 24 arctan (863 ) 9 4 = 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 2

13 2007, (203 ) (982) 4 = 2 arctan arctan 57 5 arctan arctan 0443 ( bit ) A.2.3 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 (2 ), n 2 2 n+4 e 2n+. 8 ( , ) WWW ( 3

14 ( ) 9 (i) K(k) := 0 dx /2 ( x2 )( k 2 x 2 ) = dθ 0 k2 sin 2 θ (0 k < ) E(k) := 0 k2 x 2 x 2 dx = /2 0 k 2 sin 2 θ dθ (0 k ) Legendre (Legendre s relation) (5) K(k)E(k ) + K(k )E(k) K(k)K(k ) = 2 ( k := k 2 ) k = / 2 ( ) 2 ( ) ( ) (6) 2K 2 E 2 K 2 = 2. (ii) K(k), E(k) (arithmetic geometric mean, AGM) 2 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, b)m(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). 4

15 (7) (6) = 2M (, / 2 ) 2. 2 n c 2 n ( ) arctan 97 Strassen Schönhage ( Fourier 965 ) 2 n O(n 2 ) O(n log n) A.2.4 Ramanujan (Srinivasa Aiyangar Ramanujan, , Erode Kumbakonam ) 8 = (4n)!( n) 980 (n!) n ( = 2 2 p n ) (2k )(4k 3)(4k ) (26390n + 03) k k= k= Chudnovsky ( ) ( = 2 p n ) (2k )(6k 5)(6k ) ( ) n ( n ) /2 9k DRM AGM AGM DRM ( ) arctan Ramanujan arctan, AGM, Ramanujan O(n(log n) p ) (p = 2, 3) (DRM), ( , 3 3 ) pdf 2002 ( 307 ) arctan arctan 5

16 200 6 Ramanujan ( ), AGM (Gauss-Legendre ) (T2K 29 ) Fabrice Bellard, Chudnovsky (Intel CPU, 5 ) BBP ( ) Alexander J. Yee ( ),, Chudnovsky 5 ( 96GB, 32+6TB 90 ) BBP ? Alexander J. Yee ( ),, Chudnovsky 0 6

2 4 BASIC (4) WWW BASIC 1 2 ( ) ( ) 1.2 3B 5 14 ( ) ( ) 3 1 1

2 4 BASIC (4) WWW BASIC 1 2 ( ) ( ) 1.2 3B 5 14 ( ) ( ) 3 1 1 2 4 BASIC (4 2007 5 8 WWW http://www.math.meiji.ac.jp/~mk/syori2-2007/ BASIC 2 (. (5 5 3.2 3B 5 4 ( 5 5 5 2 ( 3 ( PAINT PAINT ( MAT PLOT AREA.3 : MAT PLOT AREA PAINT ( BASIC (.3. MAT PLOT AREA REM testmatplotarea.bas