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% $BEE;R%a!<%k(B : g140828@mail.ecc.u-tokyo.ac.jp
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% \mondai{$B:n@.<T(B}{$B:n@.G/7nF|(B}{$BEE;R%a!<%k(B}{$BLdBjHV9f(B}{$BLdBjJ8(B}{$B2rEz(B}
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\mondai
{g240423 $B:4F#?.M5(B}
{02/11/20}
{g240423@mail.ecc.u-tokyo.ac.jp}
{4-4}
{$x=a$$B$K$*$$$FGHF04X?t$NCM(B${\varphi}(x)$$B$H$=$NHyJ,78?t(B${\varphi}'(x)$$B$,O"B3$G$"$k$H$7$FLdBj$r2r$$$?$,!"GHF04X?t$NBP?t$NHyJ,(B$(\log{\varphi}(x))'={\varphi}'(x)/{\varphi}(x)$$B$,O"B3$G$"$k$H$7$FLdBj$r2r$$$F$bF1$87k2L$rF@$k!#$3$NJ}K!$@$H!">r7o$N<0$,>/$J$/$J$k$h$&$G$"$k$,!"F1$8Ez$($rF@$k$3$H$rM-8B$N0f8M$N?<$5$NNL;R0f8M$N%]%F%s%7%c%k(B(4.1)$B$G<($;(B($BEz$($r:G8e$^$G5a$a$kI,MW$O$J$$(B)$B!#(B}
{
(4.5),(4,6),(4,8),(4,9)$B<0$+$i!"(B
\begin{eqnarray}
{\varphi}_1(x) &=& C_1e^({\kappa}x)\\
{\varphi}_2(x) &=& C_5e^(ikx)+C_6e^(-ikx)\\
{\varphi}_3(x) &=& C_4e^{-{\kappa}x}
\end{eqnarray}
$B$h$C$F!"(B
\begin{eqnarray}
{\varphi}'_1(x) &=& {\kappa}C^1e^{{\kappa}x}\\
{\varphi}'_2(x) &=& ik(C_5e^{ikx}-C_6e^{-ikx})\\
{\varphi}'_3(x) &=& -{\kappa}C_4e^{-{\kappa}x}
\end{eqnarray}
$B$3$3$G(B$(\log{\varphi}(x))'={\varphi}'(x)/{\varphi}(x)$$B$,O"B3$G$"$k$+$i!"(B
\begin{equation}
\frac{{\varphi}'_1(-a)}{{\varphi}_1(-a)}=\frac{{\varphi}'_2(-a)}{{\varphi}_2(-a)}, \qquad \frac{{\varphi}'_3(a)}{{\varphi}_3(a)}=\frac{{\varphi}'_2(a)}{{\varphi}_2(a)}
\end{equation}
$B$9$J$o$A!"(B
\begin{eqnarray}
\kappa = \frac{ik(C_5e^{-ika}-C_6e^{ika})}{C_5e^{-ika}+C_6e^{ika}}, \qquad %
-\kappa = \frac{ik(C_5e^{ika}-C_6e^{-ika})}{C_5e^{ika}+C_6e^{-ika}} \nonumber\\
\left\{
\begin{array}{l}
{\kappa}(C_5e^{-ika}+C_6e^{ika}) = ik(C_5e^{-ika}-C_6e^{ika})\\
-{\kappa}(C_5e^{ika}+C_6e^{-ika}) = ik(C_5e^{ika}-C_6e^{-ika})
\end{array}
\right.\label{eq:Q4-4-9}
\end{eqnarray}
$B$3$3$G(B$\displaystyle\frac{C_6}{C_5}=A$$B$H$*$/$H!"(B(\ref{eq:Q4-4-9})$B$O!"(B
\begin{eqnarray}
& & \left\{
\begin{array}{l}
{\kappa}(e^{-ika}+Ae^{ika}) = ik(e^{-ika}-Ae^{ika})\\
-{\kappa}(e^{ika}+Ae^{-ika}) = ik(e^{ika}-Ae^{-ika})
\end{array}
\right. \nonumber\\
& & \left\{
\begin{array}{l}
{\kappa}(e^{-2ika}+A) = ik(e^{-2ika}-A)\\
-{\kappa}(e^{2ika}+A) = ik(e^{2ika}-A)
\end{array}
\right. \nonumber\\
& & \left\{
\begin{array}{l}
({\kappa}+ik)A = -{\kappa}e^{-2ika}+ike^{-2ika}\\
(-{\kappa}+ik)A = {\kappa}e^{2ika}+ike^{2ika}
\end{array}
\right.\label{eq:4-9-12}
\end{eqnarray}
(\ref{eq:4-9-12})$B$N>e$N<0$N:8JU$H2<$N<0$N1&JU!">e$N<0$N1&JU$H2<$N<0$N:8JU$r$+$19g$o$;$k$3$H$K$h$C$F!"(B
\begin{equation}
({\kappa}+ik)({\kappa}e^{2ika}+ike^{-2ika}) = (-{\kappa}+ik)(-{\kappa}e^{(-2ika)}+ike^{-2ika})
\end{equation}
$B$3$l$rE83+!&@0M}$9$k$H!"(B
\begin{equation}
{\kappa}^2(e^{2ika}-e^{-2ika})+2ik{\kappa}(e^{2ika}+e^{-2ika})-k^2(e^{2ika}-e^{-2ika}) = 0 
\end{equation}
\begin{eqnarray}
{\kappa}^2i\sin{(ka)}+4ik{\kappa}\cos{(ka)}-k^2\sin{(ka)} = 0 \nonumber\\
({\kappa}^2-k^2)\tan{(2ka)}+2k{\kappa} = 0 \label{eq:4-4-16}
\end{eqnarray}
$B$3$3$G!"(B
\begin{eqnarray}
\tan{2{\theta}} &=& \frac{\sin{2\theta}}{\cos{2\theta}} \nonumber\\
&=& \frac{2\sin{\theta}\cos{\theta}}{{\cos{\theta}}^2-{\sin{\theta}}^2} \nonumber\\
&=& \frac{2}{(\tan{\theta})^{-1}-\tan{\theta}} \nonumber\\
&=& \frac{2\tan{\theta}}{1-{\tan{\theta}}^2}
\end{eqnarray}
$B$G$"$k$+$i(B(\ref{eq:4-4-16})$B$O!"(B
\begin{eqnarray}
({\kappa}^2-k^2)\frac{2\tan{(ka)}}{1-{\tan(ka)}^2}+2k\kappa &=& 0 \nonumber\\
{\tan{(ka)}}^2-\frac{{\kappa}^2-k^2}{k\kappa}-1 &=& 0 \nonumber\\
(\tan{ka}+\frac{\kappa}{k})(\tan{ka}-\frac{k}{\kappa}) &=& 0
\end{eqnarray}
$B$7$?$,$C$F!"(B
\begin{equation}
\tan{(ka)} = -\frac{k}{\kappa}$B!!$^$?$O!!(B+\frac{\kappa}{k}
\end{equation}
}

