🚀 go-pugleaf

$B9)A!Bg$NDMK\$H?=$7$^$9(B.

$B?t(B$B3X$N$b$N$+$i$b0l8@(B.

In article <OSHIKAWA.96Jan17134908@appi.mmm.t.u-tokyo.ac.jp>
oshikawa@mmm.t.u-tokyo.ac.jp (Masaki OSHIKAWA) writes:
> 1/(x-id) => i \pi theta(-y)
1/(x - i0) $B$N%U!<%j%(JQ49$O(B 2 \pi i \theta(- y) $B$G$9$M(B.
1/(x + i0) $B$N%U!<%j%(JQ49$O(B - 2 \pi i \thata y.

1/x $B$r(B generalized function $B$H9M(B$B$($k>l9g$O(B
$BIaDL%3!<%7!<$N<gCM$r;H$$$^$9(B.
$B$D$^$j(B test function f(x) $B$KBP$7$F(B
(1/x, f) = P.V. \int_{-\infty}^\infty f(x)/x dx
 = \int_0^\infty (f(x) - f(-x))/x dx,
$B$H$7$^$9(B.
$B$3$N0UL#$G$N(B 1/x $B$N%U!<%j%(JQ49(B$B$r5a$a$?$1$l$P(B
\lim_{c \to 0} P.V. \int_{-\infty}^\infty 1/x e^{- c^2 x^2} e^{-ixy} dx
$B$r7W;;$5$l$l$P(B$BNI$$$H;W$$$^$9(B.
 = \lim_{c \to 0} \int_0^\infty (- 2 i) (\sin x y / x) e^{- c^2 x^2} dx
 = - 2 i \sgn y \int_0^\infty (\sin x / x) dx
 = - \pi i \sgn y
$B$H$J$j$^$9(B.
 (\sgn $B$OId9f$r$H$k4X?t$G$9(B.)
 ($B6K8B$r(B$B@QJ,$NCf$KF~$l$kItJ,$O@5Ev2=$G$-$^$9(B.)

$B<B$O(B
1/x = 1/(x + i0) + \pi i \delta
    = 1/(x - i0) - \pi i \delta
$B$H$J$C$F$$$^$9(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