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In article <KAORU.96May22114944@ganymede.semi.ee.es.osaka-u.ac.jp>
kaoru@semi.ee.es.osaka-u.ac.jp (YAMASHITA Kaoru) writes:
>   $B$"$k(B$B4X?t$N86;O4X?t$,=iEy4X?t$GI=$5$l$k$+$I$&$+$H$$$&$3$H$r3N$+(B
> $B$a$kJ}K!$O$"$k$N$G$7$g$&$+!#(B

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>   $B$"$k7W;;$G!"(Br, c $B$r%Q%i%a!<%?$H$7$F(B
> 
>  I = \int_{c-r}^{c+r} ( \cosh x + \cos x ) \sqrt( r^2 - (x - c)^2 ) dx
>  ( r > 0 )
> 
> $B$H$$$&@QJ,$r5a$a$J$1$l$P$J$i$J$/$J$C$?$N$G$9$,!"(Bmathematica $B$G(B
> 
> Integrate[ Cos[x] Sqrt[ r^2 - (x - c)^2 ], { x, c-r, c+r } ]
> 
> $B$r5a$a$h$&$H$7$F$b$=$N$^$^JV$C(B$B$F$-$F$7$^$$$^$7$?!#(Bcosh $B$K$D$$$F(B
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$x = c + r \sin t$ $B$HCV$/$H(B,
$$
\eqalign{
 I &= \int_{-\pi/2}^{\pi/2}
         r^2 (\cosh(c + r \sin t) + \cos(c + r \sin t)) \cos^2 t \,dt\cr
   &= r^2 \cosh c \int_{-\pi/2}^{\pi/2}
                     \cosh(r \sin t) \cos^2 t \,dt\cr
   &\quad + r^2 \cos c \int_{-\pi/2}^{\pi/2}
                     \cos(r \sin t) \cos^2 t \,dt\cr
   &= r \cosh c \int_{-\pi/2}^{\pi/2}
                     \sinh(r \sin t) \sin t \,dt\cr
   &\quad + r \cos c \int_{-\pi/2}^{\pi/2}
                     \sin(r \sin t) \sin t \,dt\cr
   &= r \cosh c {d \over dr}
          \int_{-\pi/2}^{\pi/2} \cosh(r \sin t) \,dt\cr
   &\quad - r \cos c {d \over dr}
          \int_{-\pi/2}^{\pi/2} \cos(r \sin t) \,dt\cr
   &= \pi r (\cosh c {d \over dr} I_0(r) - \cos c {d \over dr} J_0(r))\cr
   &= \pi r (\cosh c I_1(r) + \cos c J_1(r))\cr
}
$$
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$B$3$3$G(B $J$ $B$O(B Bessel $BH!?t(B ($BBh0l<o1_ClH!?t(B),
$I$ $B$OBh0l<o$NJQ7A$5$l$?(B Bessel $BH!?t$G$9(B.
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$BE83+$7$F7W;;$5$l$F$$$l$P(B, $BNc$($P(B
$$
\eqalign{
\int_{-\pi/2}^{\pi/2}& \sin(r \sin t) \sin t \,dt\cr
 &= \sum_{k=0}^\infty {(-1)^k \over (2k+1)!} r^{2k+1}
                      \int_{-\pi/2}^{\pi/2} (\sin t)^{2(k+1)} \,dt\cr
 &= \sum_{k=0}^\infty
     {(-1)^k \over (2k+1)!} {(2k+1)!! \over (2k+2)!!} \pi r^{2k+1}\cr
 &= \pi \sum_{k=0}^\infty
     {(-1)^k \over k! (k+1)!} \left({r \over 2}\right)^{2k+1}\cr
 &= \pi J_1(r)\cr
}
$$
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> $B$G5M$^$C$F$7$^$$$^$7$?!#(B

$B=iEy(B$BH!?t$G$O$J$$$3$H$,CN$i$l$F$$$^$9$,(B,
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--
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Tsukamoto, C. : chiaki@ipc.kit.ac.jp