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Next: $B$3$NJ8=q$K$D$$$F(B... (PDF ե: riemann.pdf)

$BJ?@.(B 13 $BG/(B 6 $B7n(B 6 $BF|(B
$B%j!<%^%sOB(B ($B6hJ,5a@QK!(B) $B$N8m:9$K$D$$$F(B
$B?73c9)2JBg3X(B $B>pJsEE;R9)3X2J(B $BC]LnLP<#(B

$B%j!<%^%sOB$N6K8B$,LL@Q$H$J$k!"$H$$$&6hJ,5a@QK!$O!"87L)$K$O$=$N8m:9$,(B 0 $B$K<}B+$9$k$H$$$&$3$H$r>ZL@$9$kI,MW$,$"$k$,!"$=$l$O652J=q$K$O=q$+$l$F$$$J$$!#(B

$B$3$3$G$O!"(B$f(x)$ $B$,HyJ,2DG=$G!"$=$NF34X?t(B $f'(x)$ $B$,O"B3$G$"$k$h$&$J(B $B>l9g$K8B$C$F$=$N@bL@$r9T$&!#(B

$BJ,3d(B

\begin{displaymath}
\Delta : a=x_0<x_1<x_2<\ldots<x_{n-1}<x_n=b
\end{displaymath}

$B$KBP$7$F!"%j!<%^%sOB(B $S(f,\Delta)$ $B$O(B

\begin{displaymath}
S(f,\Delta)=\sum_{i=1}^n f(\xi_i)\Delta x_i
\end{displaymath}

$B$GDj5A$5$l$k!#(B $B$3$3$G!"(B$\xi_i$ $B$O(B $x_{i-1}\leq\xi_i\leq x_i$ $B$H$J$kE@(B ($BJ,E@4V$NG$0U$NE@(B)$B!"(B $\Delta x_i = x_i - x_{i-1}$ ($BJ,3dI}(B) $B$G$"$k(B ($B652J=q(B §12)$B!#(B $B$^$?!":GBg$NJ,3dI}$r(B $m(\Delta)$ $B$H$9$k!#(B

\begin{displaymath}
m(\Delta) = \max\{\Delta x_i \ ; \ 1\leq i \leq n\}
\end{displaymath}

$\xi_i$ $B$N.6h4V(B $[x_{i-1},x_i]$ $B$G(B $f(\xi_i)$ $B$,:GBgCM$H$J$k$h$&$Kl9g$G$"$k!#$=$N$h$&$J(B $\xi_i$ $B$r(B $\alpha_i$ $B$H=q$/$3$H$K$9$k!#(B

\begin{displaymath}
f(\alpha_i) = [x_{i-1},x_i] \mbox{ $B$G$N(B } f(x) \mbox{ $B$N:GBgCM(B}
\end{displaymath}

$B$^$?!"%j!<%^%sOB$,0lHV>.$5$/$J$k$N$O!"3F>.6h4V(B $[x_{i-1},x_i]$ $B$G(B $f(\xi_i)$ $B$,:G>.CM$H$J$k$h$&$Kl9g$G$"$k!#$=$N$h$&$J(B $\xi_i$ $B$r(B $\beta_i$ $B$H$9$k!#(B

\begin{displaymath}
f(\beta_i) = [x_{i-1},x_i] \mbox{ $B$G$N(B } f(x) \mbox{ $B$N:G>.CM(B}
\end{displaymath}

$B$h$C$F!"$=$l$>$l$N%j!<%^%sOB$r(B

\begin{displaymath}
S_M(f,\Delta)= \sum_{i=1}^n f(\alpha_i)\Delta x_i,\hspace{1zw}
S_m(f,\Delta)= \sum_{i=1}^n f(\beta_i)\Delta x_i
\end{displaymath}

$B$H$9$k$H!"$b$A$m$s(B
\begin{displaymath}
S_m(f,\Delta)\leq S(f,\Delta) \leq S_M(f,\Delta)
\end{displaymath} (1)

$B$H$J$k!#$=$7$F!"(B $\displaystyle \int_a^b f(x)dx$ $B$OLL@Q$J$N$G$b$A$m$s(B
\begin{displaymath}
S_m(f,\Delta)\leq \int_a^b f(x)dx \leq S_M(f,\Delta)
\end{displaymath} (2)

$B$G$b$"$k!#$h$C$F!"(B(1),(2) $B$K$h$j(B

\begin{displaymath}
S_m(f,\Delta)-S_M(f,\Delta)
\leq S(f,\Delta)-\int_a^b f(x)dx
\leq S_M(f,\Delta)-S_m(f,\Delta)
\end{displaymath}

$B$H$J$k(B ($A\leq x\leq B$ $B$+$D(B $A\leq y\leq B$ $B$J$i$P(B $A-B\leq x-y\leq B-A$ $B$O(B $BMF0W$K<($5$l$k(B)$B!#(B

$B$h$C$F!"$b$7(B $S_M(f,\Delta)-S_m(f,\Delta)$ $B$,!"(B $n\rightarrow\infty$, $m(\Delta)\rightarrow 0$ $B$N$H$-$K(B 0 $B$K<}B+$9$k$3$H$,(B $B<($5$l$l$P!"$O$5$_$&$A$N86M}$K$h$C$F!"%j!<%^%sOB$HLL@Q$H$N8m:9(B

\begin{displaymath}
S(f,\Delta)-\int_a^b f(x)dx
\end{displaymath}

$B$b(B 0 $B$K<}B+$9$k$3$H$H$J$k!#$h$C$F!"(B
\begin{displaymath}
S_M(f,\Delta)-S_m(f,\Delta)\rightarrow 0
\hspace{1zw}(n\rightarrow\infty,\ \ m(\Delta)\rightarrow 0 \mbox{ $B$N$H$-(B})
\end{displaymath} (3)

$B$r>ZL@$9$k!#(B
\begin{displaymath}
S_M(f,\Delta)-S_m(f,\Delta)
=\sum_{i=1}^n \{f(\alpha_i)-f(\beta_i)\}\Delta x_i
\end{displaymath} (4)

$B$G$"$k$3$H$K$^$:Cm0U$9$k!#(B

$B:#!"(B$f'(x)$ $B$,O"B3$G$"$k$H$$$&2>Dj$K$h$j!"(B

\begin{displaymath}
-M\leq f'(x)\leq M \hspace{1zw}(a\leq x\leq b)
\end{displaymath}

$B$H$J$k@5$NDj?t(B $M$ $B$, $B$N79$-$O(B $-M$ $B$+$i(B $M$ $B$N(B $BHO0O$KF~$k$N$G!"I}(B $\Delta x_i$ $B$N6h4VFb$G$O$I$&JQ2=$7$F$b!"$=$N:GBgCM$H(B $B:G>.CM$N:9$O(B $M\Delta x_i$ $B0J2<$H$J$k(B ($B$b$7!"$=$l$h$jBg$-$$$H$9$k$H!"(B $B$=$N:GBgCM$r.CM$r $B$rD6$($F(B $B$7$^$&(B)$B!#(B

$B$h$C$F!"(B

\begin{displaymath}
f(\alpha_i)-f(\beta_i)\leq M\Delta x_i\hspace{1zw}(i=1,2,\ldots ,n)
\end{displaymath}

$B$,$$$($k$3$H$K$J$k!#$3$l$r;H$&$H!"(B(4) $B$O(B

\begin{eqnarray*}
& & S_M(f,\Delta)-S_m(f,\Delta)
\leq \sum_{i=1}^n M \Delta ...
...= M m(\Delta) \sum_{i=1}^n \Delta x_i\\
& = & M m(\Delta)(b-a)
\end{eqnarray*}



$B$H$J$k!#:G8e$N<0(B $M m(\Delta)(b-a)$ $B$O$b$A$m$s(B $n\rightarrow\infty$, $m(\Delta)\rightarrow 0$ $B$N$H$-(B 0 $B$K<}B+$9$k!#8N$K(B (3) $B$,(B $B@.$jN)$D$3$H$K$J$j!"7k6I8m:9$,(B 0 $B$K<}B+$9$k$3$H$,<($5$l$?!#(B

$B$J$*!"8m:9$N@dBPCM$O(B $M m(\Delta)(b-a)$ $B0J2<$G$"$k$3$H$b$D$$$G$K$o$+$C$?(B $B$3$H$K$J$k!#(B




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Next: $B$3$NJ8=q$K$D$$$F(B...
Shigeharu TAKENO
2001$BG/(B 8$B7n(B 9$BF|(B