🚀 go-pugleaf

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$B!J$b$C$H$b:G=i$N(J "a^" $B$O$"$C$F$b0UL#$,$J$$$N$G$H$C$A$c$$$^$9!#!K(J

In article <DLAzAC.CFH@ulis.ac.jp> I wrote:
>$B<+J,$G$b>/$79M$($F$_$^$9!#(J

$B$G!">/$79M$($F$_$^$7$?!#(J
$B$^$@$^$@40A4$G$O$"$j$^$;$s$,!"0l1~$o$+$C$?HO0O$r=q$$$F$_$^$9!#(J
$B>/$7D9$/$J$j$^$9!"$9$_$^$;$s!#(J

======
$B;O$a$K5-9f$r>/$7!#(J
f(n) $B$r<+A3?t>e$N<B?tCM4X?t$H$7$^$9!#(J
$B$^$"!"(Ja_1 = f(1), a_2 = f(2), ..., a_n = f(n) $B$H$7$F!V?tNs!W$H(J
$B8@$C$F$7$^$C$F$$$$$N$G$9$,!"4X?t5-9f$NJ}$,2?$+$HJXMx!J$G$"$j$=$&!K$J$N$G!"(J
f(n) $B$NJ}$r;H$C$F$$$-$^$9!#(J

$B:#9M$($?$$$N$O7A<0E*$K(J
	T(f) === f(1)^f(2)^...^f(n)^...
             def
$B$HI=$5$l$k<0$G$9!J!V7A<0E*$K!W$H$$$&$N$O<}B+$,J]>Z$5$l$F$$$J$$$+$i$G$9!K!#(J
a^b^c^... $B$H$$$C$?%Y%-$NO":?$r(J tower $B$H8F$V$i$7$$$N$G!"(J
T(f) $B$r(J tower series $B$H8F$V$3$H$K$7$^$9!#(J
# power series $B$H$7$?$$$H$3$m$@$,!"$=$l$G$O!V$Y$-5i?t!W$K$J$C$F$7$^$&$N$G(J...

T(f) $B$NBh(J m $B9`$+$i(J n $B9`!"FC$KBh(J 1 $B9`$+$i(J n $B9`$^$G$r<h$j=P$7$?!VItJ,Ns!W$r!'(J
	T(f; m,n)             === f(m)^f(m+1)^...^f(n-1)^f(n)
	T(f; n) === T(f; 1,n) === f(1)^f(2)^...^f(n-1)^f(n)
$B$J$I$H=q$-$^$9!#(J
$B8@$&$^$G$b$J$/!"(JT(f) $B$O?tNs(J T(f;n)$B!'(J
	T(f;1) = f(1)
	T(f;2) = f(1)^f(2)
		...
	T(f;n) = f(1)^f(2)^...^f(n)
$B$G(J n->$B!g(J $B$H$7$?6K8B$H$7$FM?$($i$l$^$9!#(J

$B:#9M$($?$$$N$O!"$9$Y$F$N(J n $B$K$D$$$F(J 0<f(n)<1 $B$G$"$k>l9g$G$9!#(J
$B!J<B$O(J 0$B!e(Jf(n)$B!e(J1 $B$G$b$+$^$$$^$;$s$,!#!K(J
$BFC$KD>@\$NLdBj!'(J
	(1/2)^(1/3)^...^(1/n)^...
$B$N>l9g!"(Jf(n)=1/(n+1) $B$G$9(J
# $B$$$N$&$($5$s$N5-9f$G$O!"(Ja(m,n) === T(1/(n+1); m,m+n) $B$G$9!#(J

$B$3$l$K8B$i$:!"0lHL$K(J 0<f(n)<1 $B$G$"$k$J$i!"(JF(n)===T(f;n) $B$H=q$/$H!'(J
(A)	0 < F(1) < F(3) <...< F(2k+1) <...< F(2k) <...< F(4) < F(2) < 1
$B$,@.$jN)$D$3$H$,$o$+$j$^$9!#(J
$B$3$3$G(J F(1), F(3), ..., F(2k+1), ... $B$O>e$KM-3&$JC1D4A}2CNs!"(J
F(2), F(4), ..., F(2k), ... $B$O2<$KM-3&$JC1D48:>/Ns$G$9$+$i!"(J
$B$=$l$>$l$"$k6K8B&A!"&B$K<}B+$7$^$9!#(J
$B&A(J=$B&B$J$i(J T(f) $B$O<}B+!"&A(J<$B&B$J$i(J T(f) $B$OH/;6$G$9!#(J

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$B$$$-$J$j0lHL$N>l9g$r9M$($k$N$OBgJQ$J$N$G!"4JC1$J%1!<%9!"$D$^$j(J
f(n) $B$,Dj?t4X?t$G$"$k>l9g!'(J a = f(1) = f(2) = ... = f(n) = ...
$B$r<h$j>e$2$^$9!#(J
$B$D$^$j(J T(a) = a^a^... $B$N>l9g$G$9!#(J
$B$3$N>l9g$O$9$G$KOC$K$b$"$C$?$h$&$K!"(J
(B)	T(a; 1) = a
	T(a; n) = a^T(a; n-1)
$B$HA22=<0$H$7$F07$($k$N$G!"7W;;$=$NB>$,$@$$$V%i%/$K$J$j$^$9!#(J

$B07$$$?$$$N$O(J 0<a<1 $B$N>l9g$J$N$GC&@~$K$J$j$^$9$,!"@h$KB>$N>l9g$b?($l$F$*$-$^$9!#(J
$B!J$b$A$m$s(J a$B!f(J0$B!K(J
a=1 $B$N>l9g$O(J T(1) = 1^1^... $B$OL@$i$+$K(J 1 $B$K<}B+$7$^$9!#(J
a>1 $B$N>l9g!"$^$:(J T(a;n) $B$O(J n $B$K$D$$$FC1D4A}2C!"$^$?(J n $B$r8GDj$7$?>l9g!"(J
T(a;n) $B$O(J a $B$K$D$$$FC1D4A}2C$G$9!#(J
$BL@$i$+$K(J T(2) = 2^2^... $B$OH/;6$7$^$9$,!"$3$l$bOC$K$"$C$?$h$&$K!"(J
a=sqrt(2) $B$N>l9g!"(JT(a) $B$O!J(J2 $B$K!K<}B+$7$^$9!#(J

$B$G$O<}B+$9$k8B3&$O2?$+$H8@$($P!"(Je^(1/e) $B$G$9!J7W;;$O>JN,!K!#(J
$B$D$^$j(J T(a) $B$O!'(J1$B!e(Ja$B!e(Je^(1/e) $B$N$H$-<}B+!"(Ja>e^(1/e) $B$N$H$-!g$KH/;6$7$^$9!#(J
e^(1/e) = 1.44466... $B$G$9$+$i!"NcBj$K=P$F$$$?(J a=sqrt(2) = 1.4142...
$B$N>l9g$O$+$J$j$3$N<}B+8B3&$K6a$+$C$?$o$1$G$9!#(J

$B0lJ}(J a=0 $B$N$H$-$O!"?6F0$7$FH/;6$G$9!#(J
$B$D$^$j!'(J
	T(0;2k+1) = 0^...^0 $B!J(J0 $B$,4q?t8D!K(J = 0
	T(0;2k)   = 0^...^0 $B!J(J0 $B$,6v?t8D!K(J = 1
$B$?$@$7(J 0^0=1, 0^1=0 $B$H$7$^$7$?!#(J
$B!J:#$N>l9g!"(Jx^x (x->0) $B$N7A$r9M$($F$$$^$9$+$i!"(J0^0 = 1 $B$O@5Ev2=$5$l$^$9!#!K(J

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$B$=$3$GK\Bj!#(J
$B>e$+$i!"(J0<a<1 $B$G$N(J T(a) $B$O!"$"$kCM$r6-$K!"(J
$B$=$l$h$jBg$-$$B&$G$O<}B+!">.$5$$B&$G$O?6F0$7$FH/;6$,M=A[$5$l$^$9!#(J
$B<B:]!"(J(*) $B$G8+$?$h$&$K!"(JT(a;2k+1) -> $B&A!"(JT(a;2k) -> $B&B$H$9$k$H!"(J
$B&A(J=$B&B$J$i(J T(a) $B$O<}B+!"&A(J<$B&B$J$i(J T(a) $B$OH/;6$G!"&A!"&B$N4V$r?6F0$7$^$9!#(J

$B2r@OE*$K87L)$KO@$8$k$N$O$a$s$I$&$J$N$G!"0J2<$G$OD>4QE*$J@bL@$r$7$^$9!#(J
$B$^$:(J y=x $B$H(J y=a^x $B$N%0%i%U$r=q$-$^$9!#(J
$B$3$l$O(J 0<x<1 $B$NHO0O$GI,$:8r$o$j$^$9!#(J
$B$=$N8rE@$N(J x $B:BI8$r(J c $B$H$7$^$9!J2<?^;2>H!K!#(J
	|              $B!?(J
	*            $B!?(J
	|**        $B!?(J
	|  ***   $B!?(J
	|     *$B!?(J
	|    $B!?(J ****
	|  $B!?(J  |    ******
	|$B!?(J    |
	+------+------------
	0      c

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* $B$^$:(J y=x $B>e$NE@(J P1(a,a) $B$r=PH/E@$H$9$k!J$3$N$H$-(J a<c $B$G$9!K!#(J
* x $B<4$K?bD>>eJ}$K!"(Jy=a^x $B$N%0%i%U$KE@(J P2(a,a^a) $B$G$V$D$+$k$^$G?J$`!#(J
* x $B<4$KJ?9T$K!"1&$X(J y=x $B$N%0%i%U$KE@(J P3(a^a,a^a)$B$G$V$D$+$k$^$G?J$`!#(J
$B!!!J$3$N$H$-(J c<a^a$B!K!#(J
* x $B<4$K?bD>2<J}$K!"(Jy=a^x $B$N%0%i%U$KE@(J P4(a^a,a^a^a) $B$G$V$D$+$k$^$G?J$`!#(J
* x $B<4$KJ?9T$K!":8$X(J y=x $B$N%0%i%U$KE@(J P5(a^a^a,a^a^a)$B$G$V$D$+$k$^$G?J$`!#(J

$B0J2<F1MM$K$7$F!"(JP1->P2->P3->P4->P5->... $B$H!"E@(J (c,c) $B$rCf?4$K!"(J
$BMf@{>u$K?J$s$G$$$-$^$9!#(J
$B$3$3$G(J Pn $B$,6K8B$K$*$$$F(J (c,c) $B$KE~C#$9$k$J$i(J T(a) $B$O<}B+$7$^$9!#(J
$B$H$3$m$,=PH/E@$r(J Q1($B&A(J,$B&A(J) $B$H$7$F>e$NA`:n$r9T$C$?$H$-!"(J
	Q1 -> Q2 -> Q3($B&B(J,$B&B(J) -> Q4 -> Q5 = Q1
$B$H$J$k$h$&$J&A(J,$B&B!J&A(J<$B&B!K$,$"$k$H!"$=$3$G$0$k$0$k2s$j$r$7$F$7$^$C$F!"(J
$B$=$3$+$iFbB&$K?J$a$^$;$s!J$=$NCf$K(J (c,c) $B$,$"$j$^$9!K!#(J
$B$3$N&A(J,$B&B$,>e$N(J T(a;2k+1) -> $B&A!"(JT(a;2k) -> $B&B$K$[$+$J$j$^$;$s!#(J
$B&A!"&B$OO"N)J}Dx<0!'(J
(C)	a^x = y 
	a^y = x
$B$rK~$?$7$^$9!#(J
(C) $B$O>e$G8+$?$h$&$K!"(Jx=y=c $B$H$$$&2r$rI,$:;}$A$^$9!#(J
$BLdBj$O$=$l0J30$N!J(Jx$B!b(Jy $B$G$"$k!K2r$,$"$k$+$I$&$+$G$9!#(J 

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(C) $B$G(J y $B$r>C5n$9$k$H!"(Jx=a^a^x $B$K$J$j$^$9!#(J
g(x) = a^a^x $B$O!"(J
  * $BC1D4A}2C(J
  * x-> -$B!g(J $B$G(J g(x) -> 0
  * x-> +$B!g(J $B$G(J g(x) -> 1
$B$G!"%0%i%U$r=q$/$H%m%8%9%F%#%C%/6J@~$N$h$&$J7A$r$7$F$*$j!"(J
$BJQ6JE@$O#1$D$G$9!#(J

$B$7$?$,$C$FJ}Dx<0(J x=g(x) $B$O!J(Ja $B$NCM$K1~$8$F!K2r$,#18D$"$k$+!"#38D$"$k$+$N(J
$B$$$:$l$+$G!"#18D$N$H$-$,(J T(a) $B$,<}B+!"#38D$N$H$-$,&A!"&B$,I=$o$l!"(J
T(a) $B$,H/;6$N>l9g$K$J$j$^$9!#(J
# $B$3$l$O87L)$K$O3NG'$7$F$$$^$;$s$,!"$?$V$s$^$A$,$$$J$$$O$:$G$9!#(J

$B$=$N6-3&$H$J$k$N$,!"(JA = 0.065988035845... $B$G$9!#(J
$B$D$^$j(J 0<a<A $B$J$i(J T(a) $B$OH/;6!"(JA$B!e(Ja(<1) $B$J$i<}B+$G$9!#(J
$B!J(JA $B$N2r@OE*I=8=$O$o$+$j$^$;$s$,!"(J-log(x) = x^(1/log(x)) $B$N2r$G$9!K!#(J
1/A = 15.15... $B$G$9$+$i!"$?$H$($P(J T(1/15) $B$O<}B+!"(JT(1/16) $B$OH/;6$G$9!#(J
$B<B:](J a=1/16 $B$N$H$-!"&A(J=1/4$B!"&B(J=1/2 $B$H$J$j$^$9$+$i!"(J
T(1/16;n) $B$O(J 1/4, 1/2 $B$K?6F0$7$J$,$i6a$E$$$F$$$-$^$9!#(J

$B&A!"&B$O(J a $B$K1~$8$F7h$^$k?t$G$9$+$i!"$3$l$r&A(J(a)$B!"&B(J(a) $B$H=q$/$H!"(J
0<a<b<A $B$N$H$-!"&A(J(a)<$B&A(J(b)$B!"&B(J(b)<$B&B(J(a) $B$,@.$jN)$A$^$9!J>ZL@N,!K!#(J
$B$D$^$j&A(J(a)$B!"&B(J(a)$B$N4V$N!V%.%c%C%W!W$O!"(Ja $B$,>.$5$$$[$IA}2C$7$^$9!#(J
$B!J(Ja->0 $B$N6K8B$G&A(J=0$B!"&B(J=1 $B$H$J$k$3$H$O$9$G$K=R$Y$?DL$j$G$9!#!K(J

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$B$=$3$GK\Bj$N!'(J
	(1/2)^(1/3)^...^(1/n)^...
$B$G$9!#(J

(A) $B$+$i0lHL$N(J f(n) $B$K$D$$$F$b>e$N&A(J(f)$B!"&B(J(f) $B$r9M$($k$3$H$,$G$-$^$9!#(J
$B>e$N:G8e$G=R$Y$?$3$H$+$i!"$9$Y$F$N(J n $B$K$D$$$F(J 0<f(n)<a<A $B$G$"$l$P!"(J
$B&A(J(f)<$B&A(J(a)$B!"&B(J(a)<$B&B(J(f) $B$,@.$jN)$A$^$9!#(J
# $B$3$l$O$A$c$s$H>ZL@$7$F$$$^$;$s$,!"Bg>fIW$N$O$:$G$9!#(J

$B$7$?$,$C$FFC$K(J 0<f(n)<A $B$N$H$-!"(JT(f) $B$OH/;6$7$^$9!#(J

$B$3$3$G$O!V$9$Y$F$N(J n $B$K$D$$$F!W(J0<f(n)<A $B$H$7$F$$$^$9$,!"(J
$B<B:]$K$O$"$k(J N $B$,B8:_$7$F!"(Jn>N $B$J$i(J 0<f(n)<A $B$G$"$l$P!"(J
T(f; N,n) $B$O(J n->$B!g$N$H$-H/;6$7!"(J
	T(f) = f(1)^...^f(N) ^f(N)^f(N+1)^...
$B$G(J f(N) $B0J9_$NItJ,$,H/;6$9$k$3$H$h$j!"(J
$B$=$NA0$NM-8B8D$N7W;;$r2C$($F$b(J T(f) $BA4BN$,H/;6$9$k$3$H$K$J$j$^$9!#(J

$B>e$G8+$?$h$&$K!"(JT(1/16) $B$OH/;6$7$^$9$+$i!"(J
	(1/2)^(1/3)^...^(1/n)^...
	= (1/2)^...^(1/15) ^(1/16)^(1/17)^...
$B$O(J 1/16 $B0J9_$NItJ,$,H/;6$9$k$?$a!"A4BN$bH/;6$7$^$9!#(J

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$B0J>e$+$i!"(J
	(1/2)^(1/3)^...^(1/n)^...
$B$N<}B+@-<+BN$O%+%?$,$D$/$N$G$9$,!"(J
f(n) = 1/(n+1) $B$H$7$F&A(J(f), $B&B(J(f) $B$NCM$K$D$$$F$O!"(J
$B?tCME*$K$OA0JX$G=R$Y$?$h$&$K(J
	$B&A(J = 0.658365599...
	$B&B(J = 0.690347126...
$B$G$"$k$3$H$O8+Ev$,$D$-$^$9$,!"$3$NCM$N2r@OE*I=8=$O$b$H$h$j!"(J
$B$3$l$i$,K~$?$9$Y$-2r@OE*$JJ}Dx<0$J$I$b%J%>$G$9!#(J
$B$=$&$$$C$?$b$N$,8+$D$+$k$3$H$K$D$$$F$OHa4QE*$G$9$,!#(J

$B$^$?F1MM$N5DO@$K$h$j!"(J
	T(1/(n+1)^2) = (1/4)^(1/9)^(1/16)^...
	T(1/2^n) = (1/2)^(1/4)^(1/8)^...
	T(1/sqrt(n+1)) = (1/sqrt(2))^(1/sqrt(3))^(1/sqrt(4))^...
$B$J$I$bH/;6$9$k$3$H$,$o$+$j$^$9!#(J

$B$H$j$"$($:$O0J>e!#(J

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