$B9)A!Bg$NDMK\$G$9(B.

In article <20501ca5-e04e-4b1a-bc7a-52a199eb74a1@z7g2000prh.googlegroups.com>
KyokoYoshida <kyokoyoshida123@gmail.com> writes:
> $B$9$$$^$;$s!#$b$&0lEY;E@Z$j$J$*$7$5$;$F2<$5$$!#(B

$B$O$$$O$$(B.

> $B>ZL@$NJ}?K$O(B
> $B!X(B0$B!e(B|($B&#(B(s+h)-$B&#(B(s))/h-$B"i(B_0^$B!g(B (ln(x))x^{s-1}e^-x dx|
> $B!e(B|h|($B"i(B_0^1 (-ln(x))^2 x^{Re(s)-|h|-1} e^- dx + $B"i(B_1^$B!g(B(ln(x))^2 x^{Re(s)
> +|h|-1} e^-x dx)
> $B!e(B|h|($B"i(B_0^1 (-ln(x))^2 x^{Re(s)-|h|-1} e^- dx + $B"i(B_1^$B!g(B x^{Re(s)+|h|-1}
> e^-x dx)
> $B"*(B0 (as h$B"*(B0)
> $B$H$J$k(B($B"hMW>ZL@(B)$B$N$G(B

$B$H$3$m$I$3$mIT@53N$G$9$M(B.

 | (\Gamma(s+h) - \Gamma(s))/h - \int_0^\infty (\log x) x^{s-1} e^{-x} dx |
 = | \int_0^\infty ((x^h - 1)/h - \log x) x^{s-1} e^{-x} dx |
 \leq \int_0^\infty |(x^h - 1)/h - \log x| x^{Re(s)-1} e^{-x} dx

$B$K$*$$$F(B, 

 0 < x < 1 $B$G$O(B |(x^h - 1)/h - \log x| \leq |h| |\log x|^2 x^{-|h|},
 1 < x $B$G$O(B |(x^h - 1)/h - \log x| \leq |h| |\log x|^2 x^{|h|}
$B$G$9$+$i(B,

 | (\Gamma(s+h) - \Gamma(s))/h - \int_0^\infty (\log x) x^{s-1} e^{-x} dx |
 \leq |h| \int_0^1 |\log x|^2 x^{Re(s)-|h|-1} e^{-x} dx
      + |h| \int_1^\infty |\log x|^2 x^{Re(s)+|h|-1} e^{-x} dx

$B$H$J$j$^$9(B.

> $B%O%5%_%&%A$NDjM}$+$i(B
> lim_{h$B"*(B0}|($B&#(B(s+h)-$B&#(B(s))/h-$B"i(B_0^$B!g(B (ln(x))x^{s-1}e^-x dx|=0
> $B$,8@$((B,
> lim_{h$B"*(B0} ($B&#(B(s+h)-$B&#(B(s))/h=$B"i(B_0^$B!g(B(ln(x))x^{s-1}e^-x dx $B":(BC$B$H$J$k!Y(B
> $B$H$$$&Lu$G$9$h$M!#(B

$B$O$$(B.

> $B$3$l$N8eH>ItJ,(B
> $B!X(Blim_{h$B"*(B0}|($B&#(B(s+h)-$B&#(B(s))/h-$B"i(B_0^$B!g(B (ln(x))x^{s-1}e^-x dx|=0
> $B$,8@$((B,
> lim_{h$B"*(B0} ($B&#(B(s+h)-$B&#(B(s))/h=$B"i(B_0^$B!g(B(ln(x))x^{s-1}e^-x dx $B":(BC$B$H$J$k!Y(B
> $B$O!!(B
> $B!V(Blim_{z$B"*(B0}(f(z)-g(z))=0$B$+$D(Blim_{z$B"*(B0}g(z)$B$,<}B+$9$k$J$i(B
> lim_{z$B"*(B0}f(z)=lim_{z$B"*(B0}g(z)$B!W$H$$$&L?Bj$r;H$o$l$F$$$k$N$G$7$g$&$+(B
> ($B$3$N$h$&$JL?Bj$,$"$k$N$+CN$j$^$;$s$,(B)?

 \int_0^\infty (\log x) x^{s-1} e^{-x} dx $B$O(B h $B$K$h$i$J$$Dj?t$G$9(B.
$B0lHL$K(B \lim_{h \to 0} f(h) = \alpha $B$r<($9$3$H$O(B,
 \lim_{h \to 0} |f(h) - \alpha| = 0 $B$r<($9$3$H$K(B
$BB>$J$j$^$;$s(B.

> $B$3$NL?Bj$O0l8+Ev$?$jA0$+$J$H;W$($=$&$G$9$,ZL@$G$-$^$;$s$G$7$?!#(B
> $B>ZL@$O$I$&$9$l$P$$$$$N$G$7$g$&$+(B?

$B8+Ev$O$:$l$G$9$M(B.

 f(h) = (\Gamma(s+h) - \Gamma(s))/h $B$KBP$7$F(B.
 \alpha = \int_0^\infty (\log x) x^{s-1} e^{-x} dx $B$H$9$k$H$-(B,
 \lim_{h \to 0} |f(h) - \alpha| = 0 $B$r<($7$F(B,
 \lim_{h \to 0} f(h) = \alpha $B$rF3$$$F$$$k$@$1$G$9(B.

$B0x$_$K(B, \lim_{z \to 0} (f(z) - g(z)) = 0 $B$+$D(B,
 \lim_{z \to 0} g(z) = \beta $B$,B8:_$9$k$J$i(B,
 \lim_{z \to 0} f(z) = \lim_{z \to 0} (f(z) - g(z) + g(z)) $B$bB8:_$7$F(B
 = \lim_{z \to 0} (f(z) - g(z)) + \lim_{z \to 0} g(z) = \beta,
$B$D$^$j(B, \lim_{z \to 0} f(z) = \lim_{z \to 0} g(z) (= \beta)
$B$H$J$k$3$H$b(B, $BEvA3CN$C$F$$$J$1$l$P$$$1$J$$$3$H$G$9(B.

> $B99$K(B
> $B"i(B_0^1 (-ln(x))^2 x^{Re(s)-|h|-1} e^- dx$B":(BR$B!""i(B_1^$B!g(B x^{Re(s)+|h|-1} e^-x
> dx$B":(BR$B$J$i(B
> |h|($B"i(B_0^1 (-ln(x))^2 x^{Re(s)-|h|-1} e^- dx + $B"i(B_1^$B!g(B x^{Re(s)+|h|-1} e^-
> x dx)
> $B"*(B0 (as h$B"*(B0)
> $B$H2A$N;EJ}$H$$$&$b$N$G$9(B.

> |h|$B"*(B0 (as h$B"*(B0)$B$G$"$k$,(B
> ($B"i(B_0^1 (-ln(x))^2 x^{Re(s)-|h|-1} e^- dx + $B"i(B_1^$B!g(B x^{Re(s)+|h|-1} e^-x
> dx)$B"*!g(B (as h$B"*(B0)
> $B$G$H$J$k>l9g$@$C$F$"$k$+$b$7$l$^$;$s$h$M(B?

$B$@$+$i$"$j$^$;$s(B.

> |h|($B"i(B_0^1 (-ln(x))^2 x^{Re(s)-|h|-1} e^- dx + $B"i(B_1^$B!g(B x^{Re(s)+|h|-1} e^-
> x dx)
> $B"*(B0 (as h$B"*(B0)
> $B$N>ZL@$K$O(B
> |(x^h-1)/h-ln(x)|$B!e(B|h|(-ln(x))^2 x^-|h| (if 0<x$B!e(B1)

$B$3$A$i$O$=$&$G$9$,(B,

> |(x^h-1)/h-ln(x)|\xE2      ㇽh|(ln(x))^2 x^-|h| (if 1<x)

$B$3$A$i$O0c$$$^$9(B.

 |(x^h - 1)/h - \log x| \leq |h| (\log x)^2 x^{|h|}  (for 1 < x)

$B$G$9(B.

> $B$rMQ$$$k$h$&$K6D$C$F$$$^$9$,$=$b$=$b$3$l$r$I$N$h$&$KMxMQ$9$k$N$G$7$g$&$+(B?

$BMQ$$$k$H$3$m$,0c$$$^$9(B.

 \int_0^\infty |(x^h - 1)/h - \log x| x^{Re(s)-1} e^{-x} dx
 \leq |h| \int_0^1 (-\log x)^2 x^{Re(s)-|h|-1} e^{-x} dx
      + |h| \int_1^\infty (\log x)^2 x^{Re(s)+|h|-1} e^{-x} dx

$B$N=j$K;H$$$^$7$?(B.

$B9-5A@QJ,$N<}B+$O$A$c$s$H<($7$F2<$5$$$M(B.
 
> > In article <67ef746d-4563-4c78-a6f2-2fa530835d1e@k10g2000prh.googlegroups.com>
> > KyokoYoshida <kyokoyoshida123@gmail.com> writes:
> > > $B$3$l$O(BArchimedean principle$B$+$i8@$($k$N$G$9$M!#(B
> In article <110306165740.M0105944@ras1.kit.ac.jp>
> Tsukamoto Chiaki <chiaki@kit.ac.jp> writes:
> > $B$I$N$h$&$K8@$&$N$G$9$+(B.

$B$3$l$O2?$r0J$C$F(B Archimedean principle $B$H$*$C$7$c$C$F(B
$B$$$k$N$@$m$&$H$$$&5?Ld$+$i@8$8$?Ld$$3]$1$@$C$?$N$G$9$,(B,

> $B!V(B0$B!c(Bx,0$B!c(Ba$B$J$iI,$:(B-log(x)$B!e(BAx^-a$B$J$k(B0$B!c(BA$B$,B8:_$9$k!W$r<($;$P$$$$$N$G$9$+$i(B

$B$=$l$O$=$&$G$9(B.

> f(x):=Ax^-a+ln(x)$B$HCV$$$?;~$K(Bf(x)>0$B$r<($;$P$$$$$N$G(B
> f(x)=Ax^-a+ln(x)$B$N;~(B,
> f'(x)=-aAx^{-a-1}+1/x$B$J$N$G(B
> (f'(x)=)-aAx^{-a-1}+1/x=0$B$H$9$k$H(B
> x=(Aa)^{1/a},$BB($A(B,x^a=aA$B!D(B(*)

$B$=$N$H$-$K(B f'(x) = 0 $B$H$J$j$^$9$M(B.
 f'(x) = (x^a - aA)/x^{a+1} $B$G$9$+$i(B,
 0 < x < (Aa)^{1/a} $B$N$H$-(B f'(x) < 0,
 x > (Aa)^{1/a} $B$N$H$-(B f'(x) > 0,
$B$KCm0U$7$F$*$/$b$N$G$9(B.

> x=Ax^-a+ln(x)$B$r(Bf(x)$B$KBeF~$7$F$_$k$H(B

$B$=$N8@$$J}$OJQ$G$9$,(B,

> f((Aa)^(1/a))

$B$r7W;;$9$k$N$ONI$$(B.

> = A/x^a + logx
> = A/(aA) + log{(Aa)^(1/a)}$B!!!!!J"h(B(*)$B$h$j(Bx^a=aA$B!K(B
> = 1/a + (1/a)(logAa)
> = (1/a)(1+logAa)
> $B$3$3$G(BA:=e/a$B$H:N$l$P(B(f((Aa)^(1/a))>0$B$H$J$k$N$G(B)$B$$$$;v$,J,$+$j$^$9$M!#(B

$B$=$l$O$=$&$G$9$,(B, $B$=$l$H(B Archimedean principle $B$H$O(B
$B$I$&4X78$9$k$N$G$9$+(B.
-- 
$BDMK\@i=)(B@$B?tM}!&<+A3ItLg(B.$B4pHW2J3X7O(B.$B5~ET9)7]A!0]Bg3X(B
Tsukamoto, C. : chiaki@kit.ac.jp