# New Possibilities for Constructing Heuristic Solutions to Problems of Electromagnetic Diffraction
## 3 Formulas Used in the MFC  
### Partitioning the Formula for a 2D Edge  
The book [3] contains a general formula for the field \(V(\varphi)\) at the observation point (\(r\),\(\varphi\)), excited by a source located at the point (\(r_{0}\),\(\varphi_{0}\)) and scattered by a perfectly conducting wedge with an external aperture angle \(\pi\)n at normal incidence on the TH- or TE-polarized electromagnetic wave:  
\[ \begin{array}{l}V(\varphi)=v(\varphi-\varphi_{0})-v(\varphi+\varphi_{0})\ ( \mathrm{TH}),\\ V(\varphi)=v(\varphi-\varphi_{0})+v(\varphi+\varphi_{0})\ (\mathrm{TE}),\end{array}  \tag{1}\]  
where:  
\[v(\psi)\cong\frac{P(w_{sm})}{\sqrt{2\pi ik\frac{r_{0}}{r_{+}r_{0}}}}\frac{ \frac{i}{\pi}\sin\frac{\pi}{n}}{\cos\frac{\pi}{n}-\cos\frac{\psi}{n}}\frac{2i \sqrt{S(w_{sm})-S(\psi)}}{\exp[iS(w_{sm})-iS(\psi)]}\int\limits_{\infty\sqrt{ S(w_{sm})-S(\psi)}}^{\sqrt{S(w_{sm})-S(\psi)}}\exp\Bigl{(}iq^{2}\Bigr{)}dq \tag{2}\]  
Here, \(P(w_{sm})\) is the source field at the saddle point \(w_{sm}\) (from the geometric point of view, the saddle point corresponds to the situation when the edge is on the straight line connecting the observation point with the source, with \(\varphi\pm\varphi_{0}=\pi\)), and the eikonal \(S(\psi)\) and the eikonal at the saddle point \(S(w_{sm})\) are equal, as follows:  
\[S(w_{sm})=k(r+r_{0}),\ S(\psi)=k\rho=k\sqrt{(r+r_{0})^{2}-2 rr_{0}[1+\cos\psi]}, \ \psi=\varphi\mp\varphi_{0}. \tag{3}\]  
The area external to the wedge occupies the space of angles \(0<\varphi<\pi n\). The geometry of the problem is shown in Figure 1.  
In Figure 1, \(P\) and \(Q\) are the source and observation point, respectively, and other designations are as indicated in Formulas (1)-(3). Let us consider the integral representation for the scattered field in the case of two saddle points [3] and its particular case for diffraction by a wedge--Expression (2). Each of the two terms on the right-hand side of Expression (2) consists of four factors. The first factor is as follows:  
\[\frac{\sqrt{i}P(w_{sm})}{\sqrt{2\pi k\frac{r_{0}}{r_{+}r_{0}}}} \tag{4}\]  
Figure 1: Diagram for the incidence of a point source wave on a wedge. \(P\) and Q: source and observation point, \(\varphi\): scattering angle, \(r_{0}\): source distance, \(\varphi_{0}\): angle of incidence, \(\rho\): distance from source to observation point. which does not depend on the angular variable and is the product of the field value at the saddle point (i.e., on the shadow boundary) by the factor that determines the dependence of the solution on the distance to the source and the observation point. The second factor, as follows:  
\[\frac{\frac{1}{n}\sin\frac{\pi}{n}}{\cos\frac{\pi}{n}-\cos\frac{\Psi}{n}} \tag{5}\]  
represents half of the diffraction coefficient. The total diffraction coefficient for a certain type of polarization is obtained by adding or subtracting the values of this factor at the observation point, as follows:  
\[\frac{\sin\frac{\pi}{n}}{n}\left(\frac{1}{\cos\frac{\pi}{n}-\cos\frac{\theta- \varphi_{0}}{n}}\pm\frac{1}{\cos\frac{\pi}{n}-\cos\frac{\varphi+\varphi_{0}}{ n}}\right). \tag{6}\]  
Product of the third and fourth factors, as follows:  
\[\frac{2i\sqrt{S(w_{\rm sm})-S(\psi)}}{\exp[iS(w_{\rm sm})-iS(\psi)]}\int\limits _{\infty_{\sqrt{S(w_{\rm sm})-S(\psi)}}}^{\sqrt{S(w_{\rm sm})-S(\psi)}}\exp \Bigl{(}iq^{2}\Bigr{)}dq \tag{7}\]  
is the quotient of dividing the Fresnel integral by its asymptotics and characterizes the dependence of the field on the angular distance to the \"light-shadow\" boundary (or, that which is the same, the \"shadow\" boundary). Far from the shadow boundary, this product is equal to 1. At the shadow boundary, this factor is zero and compensates for the singularity of half of the diffraction coefficient (5). This compensation results in a field equal to half of the field of geometrical optics at the shadow boundary of a semi-infinite scatterer.