Article View: fj.sci.math
Article #13816Re: pi
From: Tsukamoto Chiaki
Date: Fri, 23 Feb 1996 00:00
Date: Fri, 23 Feb 1996 00:00
172 lines
8380 bytes
8380 bytes
$B9)A!Bg$NDMK\$G$9(B.
# $BEv(B site $B$K$O$^$@3{$5$s$N5-;v$OFO$$$F$J$$$N$G$9$,(B
# $B$*NY$N(B sakura $B$+$i<h(B$B$C$F$-$F(B follow $B$7$^$9(B.
# References: $B$rIU$1$?$N$,$&$^$/5!G=$7$F$$$J$1$l$P(B
# $B$*OM$S$7$^$9(B.
$B!X86O@!Y$r(B$BFI$`;v$r4+$a$i$l$?$N$G(B Dover $BHG$r(B
$BGc$C$F$-$^$7$?(B. $B99$K0J2<$G=R$Y$k$3$H$O(B
$B!V?M4V@:?@$NL>M@$N$?$a$K(B $B!](B$B?t3X;>2N!]!W(B
$B#J!%%G%e%I%MCx(B $B9b66Ni;JLu(B $B4dGH=qE9(B, 1989.
ISBN4-00-005767-7
$B$rFI$_sv$C$F$N$b$N$G$"$k$3$H(B$B$r$*CG$j$7$F$*$-$^$9(B.
In article <WD.96Feb21144312@hokke.e.ics.nara-wu.ac.jp>
wd@e.ics.nara-wu.ac.jp (Hiroyasu Kamo) writes:
> $B%"%k%-%a%G%9$N5DO@$r8=BeE*$K=q$-49$($k$H!"<!$NFs$D$N1#$l$?2>Dj$,8=$l$k(B
> $B$3(B$B$H$,CN$i$l$F$$$^$9!#0l$D$O!"M-L>$J!V%"%k%-%a%G%9$N8xM}!W!"$b$&0l$D$O!"(B
> $B6J@~$ND9$5$NA4=g=x@-$dJ?LL?^(B$B7A$NLL@Q$NA4=g=x@-$J$I$G$9!#H`$N5DO@$NB?$/(B
> $B$O!"$^$:!"(Ba<x or a=x or a>x $B$G$"$k$3$H$+$i!J$3$3$G$R$=$+(B$B$KA4=g=x@-$r;H(B
> $BMQ!K!"(Ba<x $B$b(B a>x $B$b$"$j$($J$$$3$H$r<($;$P!J$3$3$G$R$=$+$K%"%k%-%a%G%9(B
> $B$N8xM}$r;HMQ(B$B!K!"(Ba=x $B$,=P$F$/$k$H$$$&7A$K$J$C$F$$$^$9!#(B
$B!V1#$l$?!W$H$+!V$R$=$+$K!W$H$+$H$$$&I=8=$NBEEv@-$O(B
$BFq$7$$(B$BLdBj$@$H;W$$$^$9(B. $B%"%k%-%a%G%9$O%f!<%/%j%C%I$N(B
$B!X86O@!Y$N>e$KN)$C$F5DO@$7$F$$$k$H$7$F$h$$$G$7$g$&$+(B.
$B!X86O@!Y$NBh(B 5$B4,Dj5A(B 5$B$,!T%"%k%-%a%G%9$N8xM}!U$G$9$,(B,
$B!VFs$D$NNL$,Hf$r;}$D$N$O0lJ}$r<+A3?tG\$7$F(B
$BB>(B$BJ}$h$jBg$-$/=PMh$k$H$-$G$"$k!W(B
$B$H$$$C$?7A$G$9(B. $BHf$r;}$D$H$9$l$P%"%k%-%a%G%9$N8xM}$r(B
$B;H$C$F$h$$$N$O$=$&(B$B$$$&Dj5A$J$N$@$+$iEv$?$jA0(B.
$B$G$O$3$N0UL#$G$NHf$r;}$D$3$H$r2>Dj$7$FNI$$$N$+(B, $B$,(B
$B3{$5$s$N8@$o$s$H$9$k=j(B$B$@$H;W$$$^$9(B.
$BNc$($P$"$k?M$,(B
$B<B?tBN$N9=@.K!$bCN$i$:(B,
$B<B?tBN$NB8:_$r<($9$3$H$,=PMh$F$b$$$J$$$N$K(B,
$B<B?t(B$BBN$N8xM}$rMQ$$$FHy@QJ,3X$r9=@.$7$F$$$?(B
$B$H$7$^$7$g$&(B.
$B?oJ,$H4m$J$C$+$7$$$b$N$K$b;W$o$l$k$G$7$g$&$,(B
$B!TA0(B$BDs(B-$B1ieh!U$NBN7O$H$7$F$O2?$bLdBj$O$J$$H&$G$9(B.
$B$3$l$KBP$7$F<B?tBN$NB8:_$r!V$R$=$+$K!W;H$C$F$$$k(B
$B$H$$$&Hc(B$BI>$OEv$?$i$J$$$G$7$g$&(B.
# $BL^O@(B, $B$3$l$@$1$G$OK~B-$G$-$J$$$3$H$K$h$C$F(B
# $B?t3X$OH/E8$7$F$-$?Lu$G$9$,(B.
$B$3(B$B$NHfSH$OEv$?$C$F$*$j$^$;$s$G$7$g$&$+(B.
$BNL$K$D$$$FA4=g=x@-$,!V$R$=$+$K!W2>Dj$5$l$F$$$k$N$O(B
$B$=$NDL$j$N$h(B$B$&$G(B, $B!VNL$H$O0lJ}$,B>J}$KBP$7$F(B
$BBg$-$$(B, $B>.$5$$(B, $BEy$7$$(B, $B$NUY$l$+$G$"$k$3$H$,(B
$B7hDj$G$-$k$b$N$G$"$k!W$H(B$B$$$C$?Dj5A$,(B
$B2?=h$K$b$J$$$N$O%f!<%/%j%C%I$N<jMn$A$G$7$g$&(B.
$B$7$+$7$J$,$iBh(B 5$B4,$NDj5A(B 5$B$G$NNL#A(B, $B#B$N(B
$BHf#A!?#B$NAjEy#A!?#B!a#C!?#D$NDj5A$O(B
$B#A!?#B!c#C!?#D$G$b#A!?#B!d#C!?#D$G$b$J$$$3$H(B
$B$HFI$a$k7A$GDj5A$5$l(B$B$F$$$k$N$,K\<AE*$J$3$H$N$h$&$G$9(B.
($B#A!?#B!c#C!?#D$NDj5A$ODj5A(B 7$B$G$9$,(B.)
$BHf$K$D$$$F(B a=x $B$r<($9$N$K(B,
a<x $B$H$9$k$HL7=b$,=P$k(B,
a>x $B$H$9$k$HL7=b$,=P$k(B, $B$NN>J}$r<($9(B
($B4V@\5"I5K!$H$$$&$=$&$G$9(B)
$B$N$O(B a=x $B$NDj(B$B5A$=$N$^$^$H$b$$$($^$9(B.
> $B7k6I$O<}B+@-$N5D(B
> $BO@$r$7$F$$$k$N$G$9$,!"6K8B$r9M$($k$3$H$O9*L/$KHr$1$k7A$K(B$B$J$C$F$$$^$9!#(B
$B!T<h$j?T$/$7K!!U$H!V6K8B!W$H$N4X78$OHyL/$G$9$M(B.
> $B$H$3$m$G!"$3$NFs$D$N1#$l$?2>Dj$K$D(B$B$$$F!"%"%k%-%a%G%9$?$A$,$I$NDxEY0U<1(B
> $B$7$F$$$?$+$O!"?t3X;K$NLdBj$H$7$FLLGr$$$H;W$$$^$9!#%"%k%-%a%G%9(B$B$N8xM}$r(B
> $B%"%k%-%a%G%9$N;~Be$N?M$,0U<1$7$F$$$J$+$C$?$3$H$OL@$i$+$G$7$g$&!#A4=g=x(B
> $B@-$K$D$$$F$O$I$&$G(B$B$7$g$&$+!#%f!<%/%j%C%I$N!X86O@!YBh8^4,!JHfNcO@!K$N$4(B
> $B$?$4$?$7$?5DO@$r@0M}$9$k$H<+A3$KA4=g=x@-$K4X$9(B$B$k5DO@$,=P$F$/$k$O$:$@$+(B
> $B$i!"$=$&$$$&5DO@$r9M$($F$$$??M$,$$$k$O$:$@$H$b9M$($i$l$^$9!#%"%k%-%a%G(B
> $B%9(B$B$N;~Be$O$9$G$K8EE5%.%j%7%"?t3X$N?jB`4|$KF~$C$F$$$?$+$i!"!X86O@!Y$r$5(B
> $B$i$K@0M}$7$h$&$H$$$&H/A[$J$I$J(B$B$+$C$?$@$m$&$H$$$&5$$b$7$^$9!#?t3X;K$G$O!"(B
> $B$3$NJU$N8&5f$O$5$l$F$$$k$N$G$7$g$&$+!#(B
$B%"%k%-%a%G%9$,!V(B$B%"%k%-%a%G%9$N8xM}!W$r(B
$B$=$l$N8=Be$N<B?tO@$K$*$1$k$HF1$80LCV$G(B
$B0U<1$7$F$$$?H&$O$J$$$G$7$g$&$,(B,
$B$=$l$r;H(B$B$o$J$1$l$P>ZL@$,=PMh$J$$(B, $BB($A(B,
$B5DO@$N!VA0Ds!W$H$7$FI,MW$J$3$H$O(B
$B==J,$K0U<1$7$F$$$?$N$G$O$J$$$G$7$g$&(B$B$+(B.
$BA4=g=x@-$K$D$$$F%f!<%/%j%C%I$OBh(B 1$B4,(B
$B6&DL35G0(B 4$B!V=E$J$j9g$&$b$N$O8_$$$KEy$7$$!W$d(B
$B6&DL35G0(B 5$B!VIt(B$BJ,$OA4BN$h$j>.$5$$!W$r(B
$B1gMQ$9$k$3$H$K$h$j(B, $BD9$5$d3Q$NBg$-$5$K(B
$B$D$$$F$O>ZL@$7$h$&$H$7$F$$$?$h$&$K$b;W$$(B$B$^$9(B.
$BEy@Q$J@5J}7A$r:n?^$G$-$k$h$&$J?^7A$H$+(B,
$B$=$l$r85$K$7$F!T<h$j?T$/$7K!!U$GLL@Q$,(B
$BDj$a$i$l$k?^7A$H(B$B$+$J$iLL@Q$O(B
$BHf3S2DG=$G$7$g$&(B.
$B$3$&$$$C$?0UL#$G$NG[N8$O$"$kDxEY$5$l$F(B
$B$$$k$h$&$K;W$$$^$9(B.
$BNL$+$i<B?t(B$B$X$N0\9T(B, $BHf$r9M$($k$3$H$r(B
$B1i;;$H$7$FB*$($k$3$H(B, $B$H$$$&$N$O(B
$BFq$7$$$3$H$@$C$?$s$@$m$&$H;W$$$^$9(B.
$B$=$l(B$B$K$7$F$bNI$$?t3X;K$NK\$O2?$+$"$j$^$;$s$+(B.
$B$5$F(B, $B;d$N855-;v$N(B
In article <CHIAKI.96Feb19061624@kitipc.ipc.kit.ac.jp>
chiaki@ipc.kit.ac.jp (Tsukamoto Chiaki) writes:
> $B!V8EBe?M!W(BArchimedes$B$O1_$KFb@\$*$h$S(B$B30@\$9$k@5(B96$B3Q7A$N<~$r7W;;$7$F(B
> 3 + 10/71 < \pi < 3 + 1/7 $B$rF@$?$=$&$G$9$,(B, $B3Q?t$rG\$K$9$l$P(B,
> $BFb@\(B$BB?3Q7A$N<~$,C1D4$KA}2C$7(B, $B30@\B?3Q7A$N<~$,C1D4$K8:>/$7(B,
> $B$=$NHf$,(B 1$B$K6aIU$/$3$H$O>5CN$7$F$$$?$N$G$O$J(B$B$$$G$7$g$&$+(B.
> ($B<}B+$O!V8EBe?M!W$K$H$C$F$b!V8=Be?M!W$K$H$C$F$b!V<+L@!W(B?)
>
> $B8EBe?M$,<+L@$H9M$((B(?), $B;d$,5$$K$9$k$N$O(B
> $B1_$N<~$h$j30@\B?3Q7A$N<~$NJ}$,D9$$$3$H$N>ZL@$G$9(B.
> $BL5O@(B, ($B7B?tIU$1$i$l$?(B) $B6J@~(B$B$ND9$5$H$O(B,
> ($B7B?t$NG$0U$NJ,3d$KBP1~$9$k(B) $BJ,E@$r7k$V@^$l@~$ND9$5$N(B
> $B>e8B$G$"$k(B, $B$J$I$H$$$&Dj5A$,$J(B$B$1$l$P;O$^$i$J$$OC$G$9$,(B.
$B$G$9$,(B,
$B%G%e%I%M$K$h$l$P%"%k%-%a%G%9$O(B
(1) $B6J@~$ND9$5$rLdBj$K$9$k$H$-$K?7$7(B$B$$8x=`$,I,MW$G$"$k$3$H$K(B
$B5$IU$$$F$$$?(B.
(2) $B!T<h$j?T$/$7K!!U$K$h$k>ZL@$K$O<j$rIU$1$F$$$J$$(B.
$B$=$&$G(B$B$9(B.
(1) $B$N8x=`$H$O<!$N$h$&$J$b$N$G$9(B.
$B"$#A#B#C$NFbIt$K$"$j(B, $BJU#B#C$r89$H$9$k(B
$BFL$J6J@~&#$ND9$5#L$K$D(B$B$$$F(B
$B#B#C!c#L!c#B#A!\#A#C$,@.N)$9$k(B.
$B$3$l$r!VA0Ds!W$KF~$l$k$J$iXT$+$K1_$N<~$h$j(B
$B30@\B?3Q7A$N<~$NJ}$,(B$BD9$$Lu$G$9(B.
$B$3$N!V8x=`!W$r8=Be$N?t3X$NOHAH$_$K$*$$$F(B
$B>ZL@$9$k$N$O7k9=$J1i=,LdBj$+$bCN$l$^$;$s(B.
(2) $B$N(B$B!T<h$j?T$/$7K!!U$G$9$,(B, $B@h$:(B,
$B1_$KFb@\(B, $B30@\$9$k@5(B2^{n+1}$B3Q7A$NLL@Q(B, $B<~D9$r(B
$B$=$l$>$l(B a_n, A_n, l_n, L_n $B$H$7$^$7$g$&(B.
$B%f!<%/%j%C%I$,!X86O@!Y$NBh(B12$B4,(B $BL?Bj(B 2$B$G(B
$B=R$Y$F$$$k$N$O(B
A_n - a_n \leq (1/2)(A_{n-1} - a_{n-1})
$B$,@.N)$9$k$3$H$G(B, $B%f!<%/%j%C%I$O$3$l$+$i(B
[a_1, A_1], [a_2, A_2], [a_3, A_3], ...
$B$K6&DL$9$k(B$B?t$,!T1_$NLL@Q!U$rM?$($k$H$7$F(B
$B5DO@$r?J$a$kLu$G$9(B. $B0lJ}(B
L_n - l_n \leq (1/2)(L_{n-1} - l_{n-1})
$B$N$h$&(B$B$JITEy<0$O%"%k%-%a%G%9$NCx:n$K$O$J$/(B,
$B%G%e%I%M$O$=$NM}M3$r(B, $B$3$l$r%f!<%/%j%C%I$N$h$&$K(B
$B4v2?3XE*$K>ZL@(B$B$9$k$N$OFq$7$+$C$?$+$i$@$m$&(B,
$B$H?dB,$7$F$$$^$9(B.
l_n / L_n $B$,(B 1 $B$K6aIU$/$N$@$+$i$H$$$&(B
$B;d$N?dO@$O!V8E(B$BBe?M!WC#$NL\$+$i$O(B
$B@u$O$+$J$b$N$G$"$k$+$bCN$l$^$;$s(B.
# $B!V%"%k%-%a%G%9$N8xM}!W$H(B
# $B!T<h$j?T$/$7K!!U$,(B$B0l$D$N?t$rDj$a$k$3$H$H$,(B
# $B!V<B?t$NO"B3@-$N8xM}!W$HF1Ey$JLu$G$9$,(B,
# $B$d$O$j(B2000$BG/3]$+$k$N$G$7$g$&$M(B.
# $BM1(B, $B;d$,<+J,$N!V@u$O$+$J?dO@!W$KCQ$8$J$/$F:Q$`$N$O(B
# $B@h?MC#$,!V6K8B!W$N35G0Ey$rL@3N$K$7$F$*$$$F(B
# $B$/$l$?$+$i$KB>$J$j$^$;$s(B.
--
$BDMK\@i=)(B@$B1~MQ?t3X(B.$B9bJ,;R3X2J(B.$BA!0]3XIt(B.$B5~ET9)7]A!0]Bg3X(B
Tsukamoto, C. : chiaki@ipc.kit.ac.jp
Technical Details
Message-ID:
<CHIAKI.96Feb24043950@kitipc.ipc.kit.ac.jp>Path:
arc1-ca.pugleaf.net!unknown!not-for-mailReferences:
<WAKAMATU.96Feb5234430@seri.kurims.kyoto-u.ac.jp>Show Full Headers
Complete Article Headers
X-Google-Language: Japanese,JIS X-Google-Thread: f4956,ea1ef931fcaca9b1,start X-Google-Attributes: gidf4956,public From: chiaki@ipc.kit.ac.jp (Tsukamoto Chiaki) Subject: Re: pi Date: 1996/02/23 Message-ID: <CHIAKI.96Feb24043950@kitipc.ipc.kit.ac.jp> X-Deja-AN: 140774995 distribution: fj sender: news@kitnews.kit.ac.jp (news) references: <WAKAMATU.96Feb5234430@seri.kurims.kyoto-u.ac.jp> organization: Dept. of Textile Science, Kyoto Inst. of Tech., Kyoto 606, Japan newsgroups: fj.sci.math