# Porblem Book in the Theory of Functions by Knopp K.

By Knopp K.

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Is the Bessel function of the ﬁrst kind of order 0. Formulate the scattered ﬁeld in z > 0, Πzs , and the transmitted ﬁeld in z < 0, Πzt . Answer The divergence of the Hertzian vector Π Πz^z Πz^z is @Πz @z (2:203) Hence, 1 @ 2 Πz ^ ϕ ρ @ϕ@z @ 2 Πz ^z @z2 where a spherical coordinate system is used. 18 Current source above the interface. 9 Integral Equation N5 for an Arbitrarily Shaped Wire Antenna Based on these results, the electromagnetic ﬁelds E Π k 2 Π and H homogeneous medium characterized by ε, μ, and k ω εμ are written as E @ 2 Πz ^ ρ @ρ@z 1 @ 2 Πz ^ ϕ ρ @ϕ@z @ 2 Πz ^z; @z2 k 2 Πz H 1 @Πz ^ ρ ρ @ϕ jωε jωε @Πz ^ ϕ @ρ 45 Π in a (2:206) Let the electromagnetic ﬁelds above interface i and the corresponding Hertzian vector be E 0 and H 0 and Πz0 , respectively, and let those below the interface be E 1 and H 1 and Πz1 , respectively.

IEEE Trans. , vol. 36, no. 12, pp. 1667–1674, 1988. 17. H. Nakano, K. Hirose, T. Suzuki, S. R. G. Alexopoulos, Numerical analyses of printed line antennas. IEE Proc. Microw. Antennas Propag. H, vol. 136, no. 2, pp. 98–104, 1989. 18. H. Nakano, A numerical approach to line antennas printed on dielectric materials. Comput. Phys. , vol. 68, pp. 441–450, 1991. 19. H. Nakano, T. Kawano, Y. Kozono, and J. Yamauchi, A fast MoM calculation technique using sinusoidal basis and testing functions for a wire on a dielectric substrate and its application to mean­ der loop and grid array antennas.

29) is written as Ez where β 1 jωε ∫ L L @ 2 V z; z @z2 β2 V z; z I z dz (2:30) ω με and I z are the current (line distribution) at z . 1 A straight-wire conductor. E iz 0 (2:33) 16 Chapter 2 Integral Equations and Method of Moments where E iz is the z-axis component of the incident electric ﬁeld or the applied electric ﬁeld. Substituting Eq. 30) into Eq. 33), we have L 1 Iz jωε ∫ L @ 2 V z; z @z2 β2 V z; z dz Eiz z 0 (2:34) Eq. 34) is called Pocklington’s integral equation [4–6]. Note that V z; z and E iz z in the integral equation are known functions, and I z is an unknown func­ tion.