A Sommerfeld integral for an impedance half-plane is one of the classical problems in electromagnetic theory. In this paper, the integral is evaluated into two series representations, which are expressed in terms of the exponential integral and the Lommel function, respectively. It is shown that the series expansions absolutely converge for any parameters, such as distance or surface impedance. Then based on the Lommel function expansion, an exact closed-form expression of the integral is formulated. This expression is written in terms of incomplete Weber integrals, which are directly related to incomplete cylindrical functions such as incomplete Lipschitz-Hankel integrals. Additionally, a complete asymptotic series of the integral is obtained based on the exponential integral expansion. With conventional methods such as steepest descent method, it is cumbersome to derive the divergent series. The validity of all the formulations derived in this paper is demonstrated through a comparison with a numerical integration of the integral for various situations.
Bibliographical noteFunding Information:
Manuscript received October 12, 2005; revised May 16, 2006. This work was supported by an INHA University Research Grant. I.-S. Koh is with the Graduate School of Information Technology and Telecommunication, Inha University, Incheon 402-751, Korea (e-mail: firstname.lastname@example.org). J.-G. Yook is with the Department of Electrical and Electronics Engineering, Yonsei University, Seoul 120-749, Korea. Color versions of Figs. 2–8 are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2006.880747
All Science Journal Classification (ASJC) codes
- Electrical and Electronic Engineering