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It should be pointed out that the longitudinal component of the field is small for both the TE and TM modes so that the field is predominantly transverse and the modes are nearly TEM.
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When cos(πH) =0, higher order terms in the expansion of Si(α) and Ci(α) must be taken into consideration. However, these terms lead to an expression for Bjk′ that tends to zero as (1/j4) and (1/k4) with increasing j and k. Similar considerations apply to the expression for Bjk″ when sin(πH) =0.
As in Sec. 2, we say that Eqs. (41) and (43) exist if the sum Σ |βm,jk | over all j and k is finite. Now, we recognize immediately that[equation](42*)is a majorant of βm,jk. Since the terms Bmjk are obtained by solving the homogeneous integral equation γ¯mRm(ρ) = πHʃ10Rm(ρ1)Jm(πHρρ1)ρ1dρ1 (36*)which is symmetrical in Hermitian sense, the algebraic equation [Bm,jk-γ¯mδj,k] =0 exists and so do Eqs. (43) and (41).
An expansion of Eq. (52) gives [equation]
An expansion of Eq. (69) gives [equation]