## 3 Fast Evaluation of the Matrix Coefficients
Let us assume the real axis of the complex plane \(u\) as the inverse Fourier transform integration path \(C\). As a first task, observing that the CHF1 can be written as [45]  
\[{}_{1}F_{1}(a;b;u)=\frac{\Gamma(b)e^{\pm j\pi a}}{\Gamma(b-a)}U(a;b;u)+\frac{ \Gamma(b)e^{u}e^{\mp j\pi(b-a)}}{\Gamma(a)}U(b-a;b;-u) \tag{10}\]where \(U(\cdot\ ;\ \cdot\ ;\ \cdot)\) denotes the CHF2 [45], the upper sign has to be chosen for \(-\pi/2\leq\arg(u)<3\pi/2\) and the lower sign for \(-3\pi/2<\arg(u)\leq\pi/2\), the following alternative expression for the expansion functions in the spectral domain defined in Equation (6b) can be introduced:  
\[\widetilde{\phi}_{n}^{(\alpha,\beta)}(u)=\widetilde{\psi}_{n}^{(\alpha,\beta)} (u)+(-1)^{n}\widetilde{\psi}_{n}^{(\beta,\alpha)}(-u) \tag{11}\]  
where  
\[\widetilde{\psi}_{n}^{(\alpha,\beta)}(u)=\frac{\overline{\xi}_{n}^{(\alpha, \beta)}\Gamma(2n+\alpha+\beta+1)}{\Gamma(n+\alpha+1)}e^{\pm j\pi(n+\beta+1)}( 2 ju)^{n}e^{-jz}U(n+\beta+1;2n+\alpha+\beta+2;2 ju) \tag{12}\]  
the upper sign has to be chosen for \(-\pi/2<\arg(u)\leq\pi/2\) and the lower sign for \(\pi/2<\arg(u)\leq 3\pi/2\). The properties of the function in Equation (12) can be immediately deduced from the following relation between the CHF1 and the CHF2 [45]:  
\[U(a,b,u)=\frac{\pi}{\sin(\pi b)}\bigg{[}\frac{{}_{1}F_{1}(a,b,u)}{\Gamma(1+a- b)\Gamma(b)}-u^{1-b}\frac{{}_{1}F_{1}(1+a-b,2-b,u)}{\Gamma(a)\Gamma(2-b)} \bigg{]} \tag{13}\]  
As such, \(\widetilde{\psi}_{n}^{(\alpha,\beta)}(u)\) is a many-valued function if \(\alpha+\beta\) is a non-integer number with a branch-cut for \(u_{R}=0\) and \(u_{I}>0\), and is singular for \(u=0\) except when \(\alpha=\beta=-1/2\) and \(n=0\) simultaneously. Let us introduce the auxiliary function:  
\[ \begin{array}{l}\overline{F}_{n,m,w,s,t}^{(i,j)}(u)=(-1)^{ns+mt}f_{i}\big{(} (-1)^{w}u_{\tau}\overline{x}_{j},\overline{z}_{j}\big{)}\cdot\\ \cdot\widetilde{\psi}_{n}^{(t_{i})}\widetilde{\iota}_{t_{i}}^{(i-1)}\big{(}(- 1)^{w+s}a_{i}u\big{)}\widetilde{\psi}_{m}^{(t_{i})}\widetilde{\iota}_{j}^{(t_ {i})}\big{(}-(-1)^{t}a_{j}\big{[}T_{i,j}\big{(}(-1)^{w}u_{\tau}\overline{x}_{ j},\overline{z}_{j}\big{)}\big{]}^{*}\big{)}\end{array}  \tag{14}\]  
where \(w,s,t\in\{0,1\}\), \(t_{1}^{(0)}=\alpha_{l}\) and \(t_{1}^{(1)}=\beta_{l}\), with \(l\in\{i,j\}\). Such a function is generally a many-valued function with two extra branch-cuts associated with the involved CHF2. The first one is along the imaginary axis of the complex plane \(u\) (\(u_{R}=0\)) with a branch point at \(u=0\), while the second one lies on the hyperbole described by the equation \(s_{i,j}^{2}u_{R}^{2}-c_{i,j}^{2}u_{I}^{2}+s_{i,j}^{2}c_{i,j}^{2}\big{(}v_{R}^ {2}-k_{2}^{2}\big{)}=0\), where \(s_{i,j}\) and \(c_{i,j}\) are defined in the Appendix A, with a branch point at \(u=-j\text{sgn}(z_{i}(x,z))s_{i,j}\sqrt{v_{R}^{2}-k_{2}^{2}}\). By setting  
\[F_{n,m}^{(i,j)}(u)=f_{i}\big{(}u,\overline{x}_{j},\overline{z}_{j}\big{)} \widetilde{\phi}_{n}^{(\alpha_{i},\beta_{i})}(a_{i}u)\widetilde{\phi}_{m}^{( \alpha_{i},\beta_{j})}\left(-a_{j}\big{[}T_{i,j}\big{(}u,\overline{x}_{j}, \overline{z}_{j}\big{)}\big{]}^{*}\right) \tag{15}\]  
the following new representation for the integral in Equation (7) can be immediately stated  
\[\int\limits_{-\infty}^{+\infty}F_{n,m}^{(i,j)}(u)du=\int\limits_{-r}^{r}F_{n,m}^{(i,j)}(u)du+\sum\limits_{w,s,t=0}^{1}\int\limits_{r}^{+\infty}\overline{F }_{n,m,w,s,t}^{(i,j)}(u)du \tag{16}\]  
where \(r>0\) in order to avoid the singularity of \(\overline{F}_{n,m,w,s,t}^{(i,j)}(u)\) for \(u=0\).