## 3 Exact and Approximate Solutions  
The kernel functions \(M(\alpha)\) and \(N(\alpha)\) given by (44) are factorized as [23; 24]  
\[\left. \begin{array}{l}M(\alpha)=M_{+}(\alpha)M_{-}(\alpha)=M_{+}(\alpha)M_{ +}(-\alpha),\\ N(\alpha)=N_{+}(\alpha)N_{-}(\alpha)=N_{+}(\alpha)N_{+}(-\alpha),\end{array} \right\} \tag{46}\]  
where  
\[ \begin{array}{l}M_{+}(\alpha)=(\cos kb)^{1/2}e^{i3\pi/4}(k+\alpha)^{1/2}\exp \left(\frac{i\gamma b}{\pi}\ln\frac{\alpha-\gamma}{k}\right)\\ \exp\left[\frac{i\alpha b}{\pi}\big{(}1-C+\ln\big{(}\frac{\pi}{2k\pi}\big{)}+i \frac{\pi}{2}\big{)}\right]\prod\limits_{n=1,\text{odd}}^{\infty}\left(1+\frac {\alpha}{i\gamma_{n}}\right)\,e^{2i\alpha b/n\pi},\end{array}  \tag{47}\]  
\[ \begin{array}{l}N_{+}(\alpha)=(k\sin kb)^{1/2}e^{i\pi/2}\exp\left(\frac{i \gamma b}{\pi}\ln\frac{\alpha-\gamma}{k}\right)\\ \exp\left[\frac{i\alpha b}{\pi}\pi\big{(}1-C+\ln\big{(}\frac{\pi}{2k\pi}\big{)} +i\frac{\pi}{2}\big{)}\right]\Big{(}1+\frac{\alpha}{i\gamma_{0}}\Big{)}\prod \limits_{n=2,\text{even}}^{\infty}\left(1+\frac{\alpha}{i\gamma_{n}}\right)\, e^{2i\alpha b/n\pi}.\end{array}  \tag{48}\]  
with \(C\) (\(=0.57721566\dots\) ) being [PERSON]'s constant. According to (46)-(48), \(M_{\pm}(\alpha)\) and \(N_{\pm}(\alpha)\) are regular and nonzero in \(\tau\gtrless\mp k_{2}\), and show the following asymptotic behavior:  
\[M_{\pm}(\alpha),N_{\pm}(\alpha)\sim-\left(\mp i\alpha/2\right)^{1/2}\quad\text {as $\alpha\to\infty$ with $\tau\gtrless\mp k_{2}$}. \tag{49}\]  
We multiply both sides of (38) by \(M_{-}(\alpha)\) and decompose the resultant equations. This leads to  
\[ \begin{array}{l}M_{-}(\alpha)I_{-}^{d}(\alpha)-\left(\frac{2}{\pi}\right)^{ 1/2}\frac{i\cos(kb\sin\theta_{0})}{M_{+}(k\cos\theta_{0})(\alpha-k\cos\theta_ {0})}\\ +\sum\limits_{n=1,\text{odd}}^{\infty}\frac{n\pi}{b^{2}}\frac{1}{ \alpha+i\gamma_{n}}\Bigg{[}\frac{M_{-}(\alpha)e^{-i\alpha L_{5}}C_{5n}^{+}( \alpha)}{\alpha-i\gamma_{n}}+\frac{M_{+}(i\gamma_{n})e^{-\gamma_{n}L_{5}}C_{5n }^{+}(-i\gamma_{n})}{2i\gamma_{n}}\Bigg{]}\\ +\sum\limits_{n=1,\text{odd}}^{\infty}\frac{n\pi}{b^{2}}\frac{1}{ \alpha+i\gamma_{n}}\Bigg{[}\frac{M_{-}(\alpha)e^{-i\alpha L_{1}}C_{1n}^{-}( \alpha)}{\alpha-i\Gamma_{1n}}+\frac{M_{+}(-i\Gamma_{1n})e^{-\Gamma_{1n}L_{1}} C_{1n}^{+}(-i\Gamma_{1n})}{2i\Gamma_{1n}}\Bigg{]}\\ +\sum\limits_{m=2}^{5}\frac{1}{\alpha+i\Gamma_{mn}}\Bigg{\{}\frac{M_{-}(\alpha )\Big{[}e^{-i\alpha L_{m}}C_{mn}^{-}(\alpha)-e^{-i\alpha L_{m-1}}C_{(m-1)n}^{+} (\alpha)\Big{]}}{\alpha-i\Gamma_{mn}}\\ +\frac{M_{+}(i\Gamma_{mn})\Big{[}e^{-i\alpha L_{m}}C_{mn}^{-}(-i \Gamma_{mn})-e^{-i\alpha L_{m-1}}C_{(m-1)n}^{+}(i\Gamma_{mn})\Big{]}}{2i\Gamma _{mn}}\Bigg{\}}\\ =-\frac{U_{(+)}(\alpha)}{M_{+}(\alpha)}+\left(\frac{2}{\pi}\right)^{1/2}\frac{ i\cos(kb\sin\theta_{0})}{M_{+}(k\cos\theta_{0})(\alpha-k\cos\theta_{0})}\\ +\sum\limits_{n=1,\text{odd}}^{\infty}\frac{n\pi}{b^{2}}\frac{M_{+}(i\gamma_{n} )e^{-\gamma_{n}L_{5}}C_{5n}^{+}(-i\gamma_{n})}{2i\gamma_{n}(\alpha+i\gamma_{n })}\\ -\sum\limits_{n=1,\text{odd}}^{\infty}\frac{n\pi}{b^{2}}\frac{M_{+}(-i\Gamma_{1n })e^{-\Gamma_{1n}L_{1}}C_{1n}^{+}(-i\Gamma_{1n})}{2i\Gamma_{1n}(\alpha+i \Gamma_{1n})}\\ -\sum\limits_{m=2,\text{odd}}^{5}\frac{n\pi}{b^{2}}\frac{M_{+}(-i\Gamma_{mn}) \Big{[}e^{-\Gamma_{mn}L_{m}}C_{mn}^{-}(-i\Gamma_{mn})-e^{-\Gamma_{mn}L_{m-1}} C_{(m-1)n}^{+}(-i\Gamma_{mn})\Big{]}}{2i\Gamma_{mn}(\alpha+i\Gamma_{mn})}.\end{array}  \tag{50}\]Based on [PERSON]'s edge conditions [24; 48], we deduce that  
\[\frac{\partial\phi(\pm b,z)}{\partial x}=\left\{ \begin{array}{ll}-ik\sin\theta_ {0}e^{-ik(\pm b\sin\theta_{0}+z\cos\theta_{0})}+O(z^{-1/2}),&\text{for $L_{5}>0$, as $z\to+0$,}\\ -ik\sin\theta_{0}e^{-ik(\pm b\sin\theta_{0}+z\cos\theta_{0})}+O(z^{-1+\
u}),& \text{for $L_{5}=0$, as $z\to+0$,}\end{array} \right. \tag{51}\]  
\[\phi^{t}(\pm b\pm 0,z)-\phi^{t}(\pm b\mp 0,z)=\left\{ \begin{array}{ll}O(z^{1/2}), \text{ for $L_{5}>0$, as $z\to-0$,}\\ O(z^{\
u}),\text{ for $L_{5}=0$, as $z\to-0$,}\end{array} \right.