## 3 Exact and Approximate Solutions
Using (A21) in (60) and (61) and rearranging the results, we are led to  
\[\frac{U_{(+)}(\alpha)}{b}=\frac{M_{+}(\alpha)}{b^{1/2}}\bigg{[}-\frac{A_{u}}{b( \alpha-k\cos\theta_{0})}\ -\sum_{n=1}^{\infty}\frac{\delta_{2n-1}a_{n}p_{n}u_{n}^{+}}{b(\alpha+i\gamma_{2 n-1})}\bigg{]}, \tag{62}\]  
\[\frac{V_{(+)}(\alpha)}{b}=\frac{N_{+}(\alpha)}{b^{1/2}}\bigg{[}-\frac{A_{v}}{b (\alpha-k\cos\theta_{0})}\ -\sum_{n=1}^{\infty}\frac{\
u_{2n-2}\delta_{2n-2}b_{n}q_{n}v_{n}^{+}}{b(\alpha +i\gamma_{2n-2})}\bigg{]}, \tag{63}\]  
where  
\[A_{u}=-\bigg{(}\frac{2b}{\pi}\bigg{)}^{1/2}\frac{k\sin\theta_{0}\cos(kb\sin \theta_{0})}{M_{+}(k\cos\theta_{0})},\ A_{v}=\bigg{(}\frac{2\mathrm{b}}{\pi} \bigg{)}^{1/2}\frac{ik\sin\theta_{0}\sin(kb\sin\theta_{0})}{N_{+}(k\cos\theta _{0})}, \tag{64}\]  
\[a_{n}=(bi\gamma_{2n-1})^{-1},\ n\geq 0,\ b_{n}=(bi\gamma_{2n-2})^{-1},\ n\geq 1, \tag{65}\]  
\[p_{n}=b^{1/2}M_{+}(i\gamma_{2n-1}),\ n\geq 0,\ q_{n}=b^{1/2}N_{+}(i\gamma_{2n-2}), \ n\geq 1, \tag{66}\]  
\[u_{n}^{+}=U_{(+)}(i\gamma_{2n-1}),\ v_{n}^{+}=V_{(+)}(i\gamma_{2n-2}),\ n\geq 0. \tag{67}\]  
Equations (62) and (63) provide the exact solutions to the Wiener-Hopf Equations (38) and (39), respectively. However, they remain formal as they entail infinite series with unknown coefficients \(u_{n}^{+}\) and \(v_{n}^{+}\) for \(n=1,\ 2,\ 3,\ldots\) are involved. Using (58) and (67), we find that  
\[\begin{array}{l}u_{n}^{+}\sim-2^{1/2}K_{u}(b\gamma_{2n-1})^{-1/2},\ v_{n}^{+} \sim-2^{1/2}K_{v}(b\gamma  
where \((\rho_{1,2},\theta_{1,2})\) are the cylindrical coordinates given by  
\[\left. \begin{array}{l}x-b=\rho_{1}\sin\theta_{1},\;z=\rho_{1}\cos\theta_{1}\; \;\text{for}\;0<\theta_{1}<\pi,\\ x+b=\rho_{2}\sin\theta_{2},\;z=\rho_{2}\cos\theta_{2}\;\;\text{for}\;-\pi< \theta_{2}<0,\end{array} \right\} \tag{81}\]  
and \(F(\cdot)\) is the Fresnel integral defined by  
\[F(\omega)=\frac{e^{-i\pi/4}}{\pi^{1/2}}\int_{\omega}^{\infty}e^{it^{2}}dt. \tag{82}\]  
Equation (80) gives an asymptotic expression of the scattered field as \(k\rho_{1,2}\rightarrow\infty\), which is uniformly valid in observation angles \(\theta_{1,2}\). Introducing the cylindrical coordinate \((\rho,\theta)\) as \(x=\rho\sin\theta,\;z=\rho\cos\theta\) for \(-\pi<\theta<\pi\), It is evident that the following approximate relationship holds in the far field. \[\cos\theta_{1}\approx\cos\theta\approx\cos\theta_{2}, \tag{83}\]  
\[\rho_{1}\approx\rho-b\sin\theta,\;\text{for}\;0<\theta<\pi, \tag{84}\]  
\[\rho_{2}\approx\rho+b\sin\theta,\;\text{for}\;-\pi<\theta<0. \tag{85}\]  
By substituting an asymptotic expansion of the Fresnel integral in (80) with large \(|k|\rho_{1,2}\) and using (83)-(85), we can obtain an alternative form for the scattered far field  
\[\phi(\rho,\theta)\sim\phi^{s}(\rho,\theta)+\phi^{d}(\rho,\theta),\;\theta_{1, 2}\
cong\pi\mp\theta_{0}. \tag{86}\]  
where \(\phi^{s}(\rho,\theta)\) and \(\phi^{d}(\rho,\theta)\) are the geometrical optics field and the diffraction field, respectively, and are defined by  
\[\phi^{s}(\rho,\theta)= \begin{cases}-e^{-ik\rho\cos(\theta-\theta_{0})},&\text {for}\;-\pi<\theta_{2}<\pi+\theta_{0},\\ 0,&\text{for}\;-\pi+\theta_{0}<\theta_{2}<0,\;0<\theta_{1}<\pi-\theta_{0},\\ -e^{-2 ikb\sin\theta_{0}}e^{-ik\rho\cos(\theta+\theta_{0})},&\text{for}\;\pi- \theta_{0}<\theta_{1}<\pi.\end{cases}  \tag{87}\]  
\[\phi^{d}(\rho,\theta)=\Psi_{(+)}(\pm b,-k\cos\theta)k\sin\theta e^{\mp ikb\sin \theta}\frac{e^{i(k\rho-3\pi/4)}}{(k\rho)^{1/2}},\;\text{for}\;\theta\gtrgtr 0. \tag{88}\]