Preamble

This section presents a mathematical analysis of electromagnetic fields in the presence of specific boundary conditions, building upon the foundational work of T. Senior \cite{1} and Yu. V. Vandakurov \cite{2}. The methods developed by these authors have been further refined in subsequent studies \cite{3, 4, 5, 6}, as well as in the works of G. D. Malyuzhinets \cite{7, 8, 9}.

We consider a source located at the point $r_0 = (\rho_0, \phi_0, z_0)$ in a cylindrical coordinate system. The primary field is generated by a dipole with moment $P$, which can be expressed as:
$$P = P (\cos \theta_0 e_1 + \sin \theta_0 e_2)$$
where $e_1$ and $e_2$ are unit vectors. Using the representation (1.12) for the function $G_\nu(k, R)$, we have:
$$G_{1/2}(k, R) = \sqrt{\frac{\pi}{2kR}} H_{1/2}^{(1)}(kR) = \frac{e^{ikR}}{R}$$ (1.13)
The potential $\Pi(\rho, \phi, z)$ can be expressed as:
$$\Pi(\rho, \phi, z) = G_{1/2}(k, R(\alpha)) e_\alpha + \dots$$ (1.14)
where $R(\alpha)$ is the distance function. The components of the electric and magnetic fields $E$ and $H$ are derived in Section 2 and Section 3, following the methodology established in \cite{14}.

2. Potential Representation

The potential $\Pi(\rho, \phi, z)$ is defined as:
$$\Pi(\rho, \phi, z) = \frac{iP}{8\pi} \cos \frac{\pi}{n} \int_C \left[ e(\phi/2 - \phi_0), e_j \right] H_0^{(1)}(kR) \, d\alpha$$
$$\Pi(\rho, \phi, z) = \frac{iP}{8\pi} \sin \frac{\pi}{n} \int_C \left[ e(\phi/2 - \phi_0) H_0^{(1)}(kR) + \dots \right]$$ (2.1)
For the case of a wedge, the potential $\Pi(\rho, \phi, z)$ is given by:
$$\Pi(\rho, \phi, z) = \frac{iP}{8\pi} \sin \frac{\pi}{n} \left[ e(\phi/2 - \phi_0) H_0^{(1)}(kR(\phi - \phi_0)) + \dots \right]$$ (2.2)
Using the results from \cite{8, 9}, we can express the integral in terms of the function:
$$H_0^{(1)}(z) = \frac{2}{i\pi} \int_0^\infty e^{iz \cosh t} \, dt$$ (2.3)
The function $\Pi(\rho, \delta)$ is then represented as:
$$\Pi(\rho, \delta) = \sum_{k} \frac{1}{2k+1} G_{1/2}(k, R(\phi + \delta)) \frac{\cos \frac{\pi}{n}}{\cosh t + \cos(\phi + \delta)}$$
$$\int_0^\infty \frac{G_{1/2}(k, R(\pi + it)) (\cosh t - 1) \cosh \frac{t}{n} \, dt}{\cosh t + \cos(\phi + \delta)}$$ (2.4)
According to (1.14), the potential $\Pi(\rho, \delta)$ can be decomposed as:
$$\Pi(\rho, \delta) = \Gamma(\rho, \delta) + \Delta(\rho, \delta)$$ (2.5)
where the geometric part $\Gamma(\rho, \delta)$ is:
$$\Gamma(\rho, \delta) = \sum G_{1/2}(k, R(\alpha)) e_{\alpha+\phi} \cot \frac{\alpha + \phi + \delta}{2n}$$ (2.6)
and the diffraction part $\Pi_d(\rho, \delta)$ is:
$$\Pi_d(\rho, \delta) = -\frac{1}{4\pi n} \int_{-\pi+i\infty}^{\pi+i\infty} G_{1/2}(k, R(\alpha)) e_{\alpha+\phi+\delta} \cot \frac{\alpha + \phi + \delta}{2n} \, d\alpha$$ (2.7)
As shown by A. A. Tuzhilin in (2.6), the poles of the cotangent occur at $\alpha = -\phi - \delta + 4\pi n$. For the range $-\pi < \text{Re } \alpha < \pi$, this condition is satisfied only for specific values of $n$.

4. Analysis of $G_{1/2}(k, R)$

The function $G_{1/2}(k, R(\phi + \delta))$ is periodic. Since $R(-\phi - \delta + 4\pi n) = R(\phi + \delta)$, we have:
$$\Gamma(\rho, \delta) = \sum G_{1/2}(k, R(\phi + \delta)) e_{\alpha}$$ (2.8)
The diffraction term $\Delta(\rho, \delta)$ is evaluated along the contours $\alpha = \pm \pi + it$. Using the symmetry $R(\pm \pi \pm it) = R(\pi + it)$, we obtain:
$$\Delta(\rho, \delta) = -\frac{1}{4\pi n} \int_0^\infty G_{1/2}(k, R(\pi + it)) \left[ \dots \right] \, dt$$ (2.9)
where the bracketed term involves the cotangent functions evaluated at the shifted arguments. This leads to the expression:
$$\frac{\sin \frac{\pi}{n} \cosh \frac{t}{n}}{\cosh t + \cos(\phi + \delta)}$$
which matches the form in (2.4). For $m=0$ and $\nu > 0$, the integral (2.10) can be evaluated. As $a \to \infty$, the limit of the function $f(a)$ is zero. For $m > 0$ and $\cos a = -1$, we use the identity:
$$\int_0^\infty (\cosh t - 1)^m \cosh \frac{t}{n} \, dt = (1 + \cos a) \frac{2^{3/2m}}{\Gamma(m + 1/2)} \dots$$
The final evaluation of the integral $J(m)$ yields:
$$J(m) = \frac{(-1)^{m-1} b^{m-1} \Gamma(l + 1/2) \Gamma(m + 1/2)}{\Gamma(1/2) J(0)}$$ (2.11)
For $m=0$, the expression simplifies. The function $f(a)$ is then:
$$f(a) = \frac{(-1)^{m-1} 2^{m-1}}{\sqrt{\pi}} \int \dots + \dots$$ (2.10)
This allows for the determination of the field behavior near the edge.

For $\nu > 0$, the function $G_\nu(k, R(\pi + it))$ can be expanded. For $m=0$:
$$\int_0^\infty G_\nu(k, R(\pi + it)) (\cosh t - 1)^m \cosh \frac{t}{n} \, dt$$ (2.12)
where $G_\nu$ is defined by (1.13) and $R_0 = R(\pi)$. As shown by A. A. Tuzhilin, (2.12) holds for $\text{Im } k \ge 0$. For $k \neq 0$, the integral converges. If $\text{Im } k > 0$ and $m=0$, we use the substitution $k = i\kappa$, leading to:
$$G_\nu(i\kappa, R(\pi + it)) = \exp\left[-i\frac{\pi}{2}(\nu+1)\right] K_\nu(\kappa R)$$
Using the properties of the modified Bessel function $K_\nu(z)$ from \cite{15}, valid for $|\arg z| < \pi/4$:
$$K_\nu(z) = \sqrt{\frac{\pi}{2z}} e^{-z} \dots$$
The integral representation for $G_\nu$ becomes:
$$\int_0^\infty \exp\left[ -\kappa \sqrt{R^2(\pi) + t^2} \right] (\cosh t - 1)^m \cosh \frac{t}{n} \, dt$$
Substituting the asymptotic forms, we find the behavior for $\cos \frac{a}{2} > 0$ and $\cos \frac{a}{2} < 0$. The potential near the edge is dominated by terms of the form:
$$\frac{P \rho_0 \cos(a/2)}{\kappa R(a)^{v+3/2}} K_{v+1/2}(\kappa R(a))$$
By setting $\kappa = -ik$, we recover the Hankel function representation $H_\nu^{(1)}(kR)$ as seen in (2.3) and (2.12).

For the specific case $v=1/2$ and $\mu=0, 1$, the potential $\Pi$ simplifies to:
$$\Pi = \frac{e^{ikR(\phi+\delta)}}{R(\phi+\delta)} + \dots$$ (2.13)
This result is consistent with the primary field (1.12) and the boundary conditions (2.1), (2.2). Section 3 details the electric and magnetic field components $E$ and $H$. We define the position vectors:
$$r = \rho(\cos \phi i + \sin \phi j) + z k$$ (3.1)
$$r_0(\alpha) = \rho_0(\cos \alpha i + \sin \alpha j) + z_0 k$$ (3.2)
where $i, j, k$ are the Cartesian unit vectors. The distance function is $R(\alpha - \phi) = |r - r_0(\alpha)|$ as per (1.8) and (3.3).