2.1. Statement of the Problem and Decoupling of the Functional Difference Equations
 A wedge is placed in a circular cylindrical coordinate system (r, ϕ, z) in such a way that its edge coincides with the z axis and its faces are the half-planes ϕ = ±Φ with π > Φ > 0 (Figure 1). On the wedge faces the following boundary conditions must be met by the tangential components of the electric (Ez, Er) and the magnetic (Hz, Hr) fields:
Z0 being the intrinsic impedance of the surrounding medium, a12± and a21± the normalized axially anisotropic surface impedances. The passivity requires that Re a12± ≥ 0 and Re a21± ≥ 0 hold good. Obviously, a scalar impedance wedge with a12± = a21± studied by Lyalinov and Zhu  represents a special case of this work.
 Let a plane electromagnetic wave fall on this wedge. The z components of the incident field are given by (a time-dependence exp(−iωt) has been assumed, but is dropped in the remaining part of this work)
Here k0 stands for the wave number in the surrounding medium, the angles ϑ0 and ϕ0 characterize the incident direction of the plane wave, U10/Z0 and U20 denote the amplitude of the magnetic and electric components along the edge, respectively.
 Equation (7) can be rewritten as
Auxiliary angles ϑ± and χ± have been introduced above according to
 Eliminating f2(−α) from (8) one obtains a wholly decoupled FD equation for f1(α),
After having obtained a solution for f1(α) in the strip Π(−2Φ, 2Φ), f2(α) in the same strip is given by, on use of (8),
The spectra (α) can be analytically extended to α outside of the basic strip Π(−2Φ, 2Φ) by virtue of (8).
2.2. Simplified Difference Equation
 It is worth noting that there exists only one nonconstant coefficient at the left-hand side of (16), namely,
and this coefficient can be written in an equivalent form,
with the auxiliary function Ψ0(α) given by
 In case of Φ > π/2, the two zeros of Ψ0(α) inside the basic strip can be eliminated,
This new function F0(α) is free of both zeros and poles in the basic strip.
2.3. Fredholm Integral Equation of the Second Kind
 Expressing both the spectrum 1(α) and the right-hand side of (23) in their even and odd parts, 1(α) = 1e(α) + 1o(α), H1(α) = Q1(α)1(−α) = H1e(α) + H1o(α), it is evident that (23) is identical to
To get the particular solution to the functional difference equation (25), which is equivalent to a system of equations,
we resort to the so-called S integral [Tuzhlin, 1973; Buldyrev and Lyalinov, 2001; Babich et al., 2004] [see also Lyalinov and Zhu, 2003a, 2006].
 To find the particular solution to (24) by using the same procedure, it is advantageous to introduce a new odd function 1o(α) via 1e(α) = cot(να)1o(α). The respective difference equation follows from (24),
and can be dealt with in the same way as (25). Hence the particular solution is given by
The sought-for equivalent integral representation of (23) in the basic strip Π(−2Φ, 2Φ) reads
The nonintegral terms at the right-hand side recover the geometrical-optics poles and the asymptotic behavior of 1(α) at infinity.
 The constants A1± are determined by the radiation condition according to which the second spectrum f2(α) given in the basic strip by (17) must be free of nonphysical poles. This leads to two additional conditions,
 As indicated clearly by (30), especially by the last term at its right-hand side (the particular solution), 1(α) in the basic strip depends upon its value along the imaginary axis of the complex plane. In line with (31), the same is true for the constants A1±. Therefore, for points on the imaginary axis of the complex α plane, (30), together with (31), amounts to a Fredholm integral equation of the second kind.
 The kernel of the integral equation (30), Q1(−t)/cos[ν(α + t)], is free from singularity and does not contain the wave number k0 which may be a large parameter, as can be verified by recalling the definition of Q1(t) and the properties of F0(t) and q1(t). Furthermore, this kernel decreases exponentially with ∣Im t∣, or more precisely,
as t ±i∞. It is precisely these properties that render the above described solution procedure a very efficient, that is, fast and accurate one; see section 3.
 From the asymptotic behavior of Q1(α) and 1(α) it can be inferred that the kernel of (30) is square integrable. Hence its solvability follows from the uniqueness of solution to the present problem, which has been shown elsewhere [cf. Lyalinov and Zhu, 2003b, 2006].
2.4. Uniform Asymptotic Solution
 Together with (31), (30) is solved by quadrature method, with the asymptotic behavior of 1 accounted for explicitly [see Lyalinov and Zhu, 2003b, 2006]. Evaluating the Sommerfeld integrals (6) by the saddle-point method, a uniform asymptotic solution for the problem under study has been derived,
where the superscripts “go”, “sw” and “d” stand for the geometrical-optics part, the surface waves and diffracted waves, respectively. The formulae for these wave ingredients are given explicitly by Lyalinov and Zhu , where now in the residues of the surface waves the following multiplication factors have to be used instead:
 On setting a12± = a21± = η±, the above formulae reduce precisely to the ones for diffraction of a plane electromagnetic wave by a wedge with scalar impedance faces [Lyalinov and Zhu, 2006].