% <PLAINTEXT>
% $BNL;RJ*M}3X(B $B!'(B $B1i=,LdBj(B 5-1 $B2rEz(B
% $B:n(B  $B@.(B  $B<T(B $B!'(B f9311196 $B?e8M(B $BOB9,(B
% $B=$!!@5!!<T(B $B!'(B 240509f $B;-Eg!!>;90(B
% $B:n@.G/7nF|(B $B!'(B /96/02/20
% $B=$@5G/7nF|(B $B!'(B /02/12/02
% $BEE;R%a!<%k(B $B!'(B f11196@ced.cas.uec.ac.jp
% $B=$@5<TEE;R%a!<%k(B $B!'(B g240509@mail.ecc.u-tokyo.ac.jp
%
\mondai
{f9311196 $B?e8M(B $BOB9,(B}
{96/02/20,/02/12/02 modified}
{f11196@ced.cas.uec.ac.jp}
{5-1}
{
%----$BLdBjJ8(B----%
$B%U!<%j%(JQ49(B $ a(k )$ $B$r5a$a$h!#$^$?5U%U!<%j%(JQ49$r$7$F!"(B
$B$b$H$N(B $ f(x) $ $B$K$J$C$F$$$k$3$H$r3NG'$;$h!#(B\\
\begin{eqnarray}
{\rm (a)} \hspace{2mm} f(x)=e^{-b|x|}, &
{\rm (b)} \hspace{2mm} f(x)=e^{-bx^{2}}, &
{\rm (c)} \hspace{2mm} 
f(x)= \left\{ 
\begin{array}{ll}
 1 & (|x| < a/2) \\
 0 & (|x| \geq a/2)
\end{array} \right. \nonumber
\end{eqnarray}
}
{
%----$B2rEzJ8(B---%
$ {\rm (a)} $ $B%U!<%j%(JQ49(B $ a(k) $ $B$O!"(B\\
\begin{equation}
\begin{array}{lcl}
a(k) & = & \displaystyle \int_{-\infty}^{\infty}e^{-b|\xi|}e^{-ik\xi}d\xi \\
     & = & \displaystyle \int_{-\infty}^{0}e^{(b-ik)\xi}d\xi
            +\int_{0}^{\infty}e^{-(b+ik)\xi}d\xi \\
     & = & \displaystyle \left[\frac{1}{b-ik}e^{(b-ik)\xi}\right]_{-\infty}^{0}
           +\left[-\frac{1}{b+ik}e^{-(b+ik)\xi}\right]_{0}^{\infty} \\
     & = & \displaystyle \frac{1}{b-ik}+\frac{1}{b+ik} \\
     & = & \displaystyle \frac{2b}{b^{2}+k^{2}} 
\end{array}
\end{equation}
\\
$B<!$K5U%U!<%j%(JQ49(B $ f(x) $ $B$r!"N1?t$NDjM}$rMQ$$$F9T$J$&!#(B\\
\begin{equation}
 a(k) = \frac{2b}{b^{2} + k^{2}} \label{eq:5-1-2} 
\end{equation}
$B$N6K$O(B $ z=ib $ $B$H(B $ -ib $ $B$K$"$k!#(B\\
$ x > 0 $ $B$N$H$-!"(B\\
\begin{equation}
 f(x) = 2 \pi i \frac{1}{2 \pi}
        \left[\frac{2b(z-ib)}{(z-ib)(z+ib)}e^{ixz}\right]_{z=ib}
      = e^{-bx}
\end{equation}
$ x < 0 $ $B$N$H$-!"(B \\
\begin{equation}
  f(x) = -2 \pi i \frac{1}{2 \pi}
        \left[\frac{2b(z+ib)}{(z-ib)(z+ib)}e^{ixz}\right]_{z=-ib}
      = e^{bx}
\end{equation}
$B$H$J$k!#$h$C$F!"(B\\
\begin{equation}
 f(x) = e^{-b|x|}
\end{equation}
$B$H$J$k!#(B\\

$ {\rm (b)} $ $B%U!<%j%(JQ49(B $ a(k) $ $B$rHyJ,$9$k$H!"(B\\
\begin{equation}
\begin{array}{lcl}
 \displaystyle \frac{d}{dk}a(k) 
& = & \displaystyle \int_{-\infty}^{\infty} f(\xi)(-i\xi)e^{-ik\xi}d\xi \\
& = & \displaystyle \frac{i}{2b}\int_{-\infty}^{\infty}
      \left(\frac{d}{d\xi}e^{-b\xi^{2}}\right)e^{-ik\xi}d\xi \\
& = & \displaystyle \frac{i}{2b}
    \left(\left[e^{-b\xi^{2}}e^{-ik\xi}\right]_{-\infty}^{\infty}
    +ik\int_{-\infty}^{\infty}e^{-b\xi^{2}}e^{-ik\xi}d\xi\right) \\
& = & \displaystyle -\frac{k}{2b}
      \int_{-\infty}^{\infty}e^{-k\xi^{2}}e^{-ik\xi}d\xi \\
& = & \displaystyle -\frac{k}{2b}a(k)
\end{array}
\end{equation} 
$B$H$J$k!#$3$l$h$j!"(B\\
\begin{equation}
 \displaystyle \frac{d}{dk}a(k)+\frac{k}{2b}a(k) = 0 
\end{equation}
$B$H$J$k!#$3$NHyJ,J}Dx<0$r2r$H!"(B
\begin{equation}
 a(k)=Ce^{-k^{2}/4b}
\end{equation}
$B$H$J$k(B( $ C $ $B$OG$0UDj?t(B)$B!#$3$3$G!"(B
\begin{equation}
\int_{-\infty}^{\infty}e^{-b\xi^{2}}d\xi=\sqrt{\frac{\pi}{b}}
\end{equation}
$B$h$j!"(B$ k=0 $ $B$N$H$-!"(B$ \displaystyle C=\sqrt{\frac{\pi}{b}} $ $B$rF@$k!#(B
$B$f$($K!"(B
\begin{equation}
 a(k) = \int_{-\infty}^{\infty}e^{-b\xi^{2}}e^{-ik\xi}d\xi \\
      = \sqrt{\frac{\pi}{b}}e^{-k^{2}/4b}
\end{equation}
$B$H$J$k!#(B\\
\\
$B<!$K!"%U!<%j%(5UJQ49(B $ f(x) $ $B$r9T$J$&!#(B$ f(x) $ $B$rHyJ,$9$k$H!"(B\\
\begin{equation}
\begin{array}{lcl}
 \displaystyle \frac{d}{dx} f(x)
& = & \displaystyle \frac{1}{2 \pi}\int_{-\infty}^{\infty}a(k)(ik)e^{ikx}dk \\
& = & \displaystyle -i\sqrt{\frac{b}{\pi}}
      \int_{-\infty}^{\infty}\left(\frac{d}{dk}e^{-k^{2}/4b}\right)e^{ikx}dk \\
& = & \displaystyle -i\sqrt{\frac{b}{\pi}}    
      \left\{ \left[ e^{-k^{2}/4b} e^{ikx} \right]_{-\infty}^{\infty}
      -ix \int_{-\infty}^{\infty} e^{-k^{2}/4b} e^{ikx}dk \right\} \\
& = & \displaystyle -2bx \frac{1}{2 \pi}
      \sqrt{\frac{\pi}{b}}\int_{-\infty}^{\infty}e^{-k^{2}/4b}e^{ikx}dk \\
& = & \displaystyle -2bx f(x) \label{eq:5-1-11}
\end{array}  
\end{equation}
\\
$B$H$J$j!"$3$NHyJ,J}Dx<0$r2r$/$H!"(B\\
\begin{equation}
 f(x) = A e^{-bx^{2}}
\end{equation}
$B$rF@$k(B( $ A $ $B$OG$0UDj?t(B)$B!#$3$3$G!"(B \\
\begin{equation}
\begin{array}{lcl}
 \displaystyle f(0)
& = & \displaystyle \frac{1}{2 \pi}\sqrt{\frac{\pi}{b}}e^{-k^{2}/4b}dk \\
& = & \displaystyle \frac{1}{2 \pi}\sqrt{\frac{\pi}{b}}\sqrt{4 \pi b} \\
& = & \displaystyle 1
\end{array}
\end{equation}
$B$G$"$k$+$i!"(B$ x = 0 $ $B$N$H$-!"(B$ A = 1 $ $B$rF@$k!#$f$($K!"(B\\
\begin{equation}
 f(x) = e^{-bx^{2}}
\end{equation}
$B$H$J$k!#(B\\

$ {\rm (c)} $ $B%U!<%j%(JQ49(B $ a(k) $ $B$O!"(B\\
\begin{equation}
\begin{array}{lcl}
 \displaystyle a(k) 
& = & \displaystyle \int_{-\infty}^{\infty}f(\xi)e^{-ik\xi}d\xi \\
& = & \displaystyle \int_{-a/2}^{a/2}e^{-ik\xi}d\xi \\
& = & \displaystyle \left[-\frac{1}{ik}e^{-ik\xi}\right]_{-a/2}^{a/2} \\
& = & \displaystyle \frac{2}{k}\left(\frac{e^{ika/2}-e^{-ika/2}}{2i}\right) \\
& = & \displaystyle \frac{2}{k}\sin\left(\frac{ka}{2}\right)
\end{array} 
\end{equation}
$B$H$J$k!#<!$K!"5U%U!<%j%(JQ49(B $ f(x) $ $B$O!"(B\\
\begin{equation}
f(x) = \displaystyle \frac{1}{\pi}
       \int_{-\infty}^{\infty}\frac{1}{k}\sin\left( \frac{ka}{2} \right)
       (\cos kx +i \sin kx)dk \label{eq:5-1-16}
\end{equation}
$B$GI=$5$l$k!#$3$3$G!"(B$ a(k) $ $B$O6v4X?t$G$"$k$+$i!"(B$ (5-1-16) $$B<0$N(B $ \sin kx $
$B$N9`$K$*$1$k@QJ,$O!"(B $ 0 $ $B$K$J$k!#$f$($K!"(B\\
\begin{equation}
\begin{array}{lcl}
 \displaystyle f(x)
& = & \displaystyle \frac{1}{\pi}
      \int_{-\infty}^{\infty} \frac{1}{k}
      \sin \left( \frac{ka}{2} \right) \cos kx dk \\
& = & \displaystyle \frac{2}{\pi}\int_{-\infty}^{\infty} \frac{1}{k}
      \left\{ \sin \left( \frac{a}{2}+x \right)k
      + \sin \left( \frac{a}{2}-x \right)k \right \}dk \\
& = & \left\{
      \begin{array}{ll}
       1 & (|x| < a/2) \\
       0 & (|x| \geq a/2)
      \end{array} \right.
\end{array}
\end{equation}
$B$H$J$k!#(B\\
}


