## 3 The Analytic Technique of Carleman Regularization
### The Carleman Regularization Technique  
To realize the Carleman regularization technique, at first, the first-kind singular integral equations' system (14) can be rewritten in the form of the simplest log-singular integral equation:  
\[\int_{-1}^{1}j_{n}(t)\ln|\tau-t|dt=f_{n}(\tau,\mu,\stackrel{{ \rightarrow}}{{q}}),\quad|\tau|<1. \tag{16}\]  
Where the right-hand part functions are determined by the formulas:  
\[f_{n}(\tau,\mu,\stackrel{{\rightarrow}}{{q}})=\frac{\pi}{2i}M_{n} (\mu,\stackrel{{\rightarrow}}{{q}})\cdot\exp(iq_{1}a_{n}\tau)- \sum_{m=1}^{N}\int_{-1}^{1}j_{m}(t)R_{nm}(\tau,t)dt,\quad n=1,\ldots,N. \tag{17}\]Secondly, the Carleman inversion formula for the simplest log-singular integral equation has to be used [1]. Due to it we have the expression:  
\[j_{n}(x)=\frac{1}{\pi^{2}\sqrt{1-x^{2}}}\left[p.v.\int\limits_{-1}^{1}\frac{ \sqrt{1-\tau^{2}}}{\tau-x}f^{\prime}_{n}(\tau,\mu,\overset{\rightarrow}{ \vec{q}})d\tau\right.-\frac{1}{\ln 2}\int\limits_{-1}^{1}f_{n}(\tau,\mu,\overset{ \rightarrow}{\vec{q}})\frac{d\tau}{\sqrt{1-\tau^{2}}}\right]. \tag{18}\]  
Derivatives under the first integral are, with respect to the variable of integration and \"\(p.v.\)\", before that integral means principal value of improper Cauchy integral. Then, we substitute expressions of the right-hand part functions, presented by the Formula (17) and change order of integration. Introducing new regular kernel functions and new right-hand part functions, we find a second-kind integral equations set of the form:  
\[j_{n}(x)+\sum_{m=1}^{N}\int_{-1}^{1}j_{m}(t)\Re_{nm}(x,t)dt=g_{n}(x),\quad|x| \leq 1,\quad n=1,\ldots,N. \tag{19}\]  
It will be noted that these regular kernel functions and right-hand part functions have expressions similar to the case of flat gratings with finite number of impedance strips [6]. Having that second-kind integral equations system with smoother kernel functions, it is easy to make correct and effective calculation algorithms. To show productivity of that full-wave mathematical model, let us focus our attention on an asymptotical case. To realize that asymptotic mathematical model, two principal assumptions with respect to normalized input parameters have to be true. The first assumption deals with the grating's strips: they have to be electrically narrow (\(\alpha_{n}=ka_{n}<<1\)). The second principal assumption deals with distances between the grating's strips: they have to be sufficient greatly spaced. In the case that the plane \(E\)-polarized wave scattering from sparsely filled coplanar electrically conducting flat strips system, the second-kind integral equations system (19) has an explicit asymptotic solution. Let us prove that statement in the main approximation. The main terms of regular kernel functions, presented by the expressions (15), do not dependent on variables \(t\) and \(\tau\) and we have simple expressions: \(R_{nm0}=\ln\frac{\gamma ka_{n}}{2i}\), \(R_{nm0}=\pi\cdot H_{0}^{(1)}(\rho_{nm})/2i,\quad m\
eq n\), here \(\rho_{nm}=k\sqrt{(x_{n0}-x_{nm0})^{2}+(z_{n0}-z_{m0})^{2}}\). The last normalized input parameters deal with distances between intervals' centers (see Figure 1). So, due to the second principal assumption for these normalized input parameters the relationships: \(\rho_{nm}=\text{O}(1)\) have to be true. The main terms of right-hand part functions of the second-kind integral equations system (19) are simple as well. In the main approximation, they are presented by the expressions: \(M_{n0}(\mu,\overset{\rightarrow}{\vec{q}})=2\pi\cdot\exp(ik(q_{1}x_{n0}+q_{2 }z_{n0}))\cdot(q_{2}\mu-1)^{-1}\). Thus, the Formulas (17)-(19) give us explicit asymptotic solutions of the following form: \(j_{n0}(x)=\frac{j_{n0}}{\pi\sqrt{1-x^{2}}}\). The constants in the numerator \(j_{n0}\) are solutions of rather simple linear algebraic equations system:  
\[ \begin{split} j_{n0}\cdot\ln\frac{\gamma ka_{n}}{4i}+\frac{ \gamma}{2i}\sum\limits_{m=-1,m\
eq n}^{N}j_{m0}H_{0}^{(1)}(\rho_{nm})=\frac{2 \gamma}{q_{2}i-1}\cdot\exp(ik(q_{1}x_{n0}+q_{2}z_{n0})),\\ n=1,\ldots,N.\end{split}