S-continued fractions for complex transport coefficients of two-phase composites

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Authors

  • Stanisław Tokarzewski Institute of Fundamental Technological Research Polish Academy of Sciences, Poland
  • Józef Joachim Telega Institute of Fundamental Technological Research Polish Academy of Sciences, Poland

Abstract

We examine S-continued fraction bounds on the effective dielectric constant ee of a two-phase composite for the case where the dielectric coefficients e1 and e2 are complex. The starting point for our study is a power series expansion of e(z) at Z = 0, Z = e2/e 1-1. The S-continued fractions to the expansions of e(z) have an interesting mathematical structure. Its convergents represent the best bounds derived earlier by Milton [24-25], and independently by Bergman [5] . Specific examples of calculation of complex S-continued fraction bounds on ee are provided.

References

[1] G.A. Baker. Essentials of Padé Approximants. Academic Press, London, 1975.
[2] G.A. Baker, P. Graves-Morris. Padé Approximants, Part I: Basic Theory. Encyclopedia of Mathematics and its Applications, 13, Addison- Wesley Publ. Co., London, 1981.
[3] G.A. Baker, P. Graves-Morris. Padé Approximants, Part II: Extensions and Applications. Encyclopedia of Math· ematics and its Applications, 14, Addison-Wesley Publ. Co., London, 1981.
[4] D.J. Bergman. The dielectric constant of a composite material - A problem in classical physics, Phys. Rep., 43: 377- 407, 1978.
[5] D.J . Bergman. Rigorous bounds for the complex dielectric constant of a two-component composite. Ann. Phys., 138: 78-114, 1982.