# New Possibilities for Constructing Heuristic Solutions to Problems of Electromagnetic Diffraction
## 3 Formulas Used in the MFC
### Modifying Functions for a Perfectly Conducting Plane Angular Sector
Additionally, (in accordance with the principles of the MFC), additional multipliers (modifying functions) obtained as a result of the adjustment were applied, as follows:  
\[ \begin{array}{l}D^{\prime\prime}\,_{\theta\theta}=D_{\theta\theta}2 cr\gamma ^{0.7},\,\,D^{\prime\prime}\,_{\theta\varphi}=D_{\varphi\theta}0.5 cr\gamma^ {-0.7},\\ D^{\prime\prime}\,_{\varphi\theta}=D_{\varphi\theta}2 cr\gamma^{0.7},\,\,D^{ \prime\prime}\,_{\varphi\varphi}=D_{\varphi\theta}0.5 cr\gamma^{-0.7}.\end{array}  \tag{22}\]  
Figure 5: Scheme of using the imaginary edge. (**a**) a scheme for solving 2D problem for diffraction of a plane wave at an infinite edge, (**b**) an infinite edge, source position and viewpoint that are outside the diffraction cone, relative to the source, (**c**) the wave vector of a wave incident on an edge can be divided into two components, (**d**) a lilac dotted line shows an imaginary edge. Formula (22) contain the following functions:  
\[cp\gamma=\frac{f(\varphi\gamma,\varphi\gamma_{0})}{f^{0}(\varphi\gamma,\varphi \gamma_{0})}=\frac{\sin\frac{\varphi\gamma}{2}}{\cos\frac{\varphi\gamma_{0}}{2} }=cp\gamma^{-1},\quad cr\gamma=\frac{g(\varphi\gamma,\varphi\gamma_{0})}{g^{0} (\varphi\gamma,\varphi\gamma_{0})}=\frac{\cos\frac{\varphi\gamma_{0}}{2}}{ \sin\frac{\varphi\gamma}{2}}. \tag{23}\]  
The index \"\(\gamma\)\" means that the diffraction coefficients were taken on an imaginary edge located at an angle \(\gamma/2\) to the coordinate axis (Figure 4). The use of modifying functions clarifies the heuristic solution of the MFC in comparison with the MEW. The calculation results are shown in Figure 6.  
In Figure 6, the worst coincidence with the strict solution [17] (solid gray line) gives the PO approximation, since the \(D\) coefficients for the cross polarizations \(D\varphi\vartheta\) and \(D\varphi\varphi\) vanish. The best match with the strict solution is given by the modified formula in (22). The curves for the \(D\) coefficients calculated using the indicated heuristic approaches are shown by red dots and blue dashes next to the strict solution curves. Diffraction Coefficients GDC (Generalized Diffraction Coefficients) and PODC (Physical Optics Diffraction Coefficients) for a Semitransparent Half-Plane  
The boundary conditions for the impedance half-plane were considered in [38; 39; 40; 41; 42; 43]. In papers [4; 44], using the MFC, a heuristic solution to the problem of diffraction of an electromagnetic TH--a polarized wave on a half-plane with impedance boundary conditions, such as a thin layer--was obtained. In [45], a heuristic solution of the same problem was obtained for the case of TE polarization. Figure 6: _Cont._  
Primary heuristic formulas are constructed on the basis of diffraction coefficients (16), (17), (19) and (20), adding to these expressions the reflection, \(R\), and transmission, \(T\), coefficients for an unbounded plane surface. The subscripts \"TH\" and \"TE\" indicate the type of polarization when the corresponding vectors are perpendicular to the edge. As the degree of translucency of the half-plane changes from opaque to completely transparent, the moduli of the reflection coefficients \(R\) change from 1 to 0.