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Exact and “Exact” Formulae in the Theory of Composites

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Abstract

The effective properties of composites and review literature on the methods of Rayleigh, Natanzon–Filshtinsky, functional equations and asymptotic approaches are outlined. In connection with the above methods and new recent publications devoted to composites, we discuss the terms analytical formula, approximate solution, closed form solution, asymptotic formula, etc…frequently used in applied mathematics and engineering in various contexts. Though mathematicians give rigorous definitions of exact form solution the term “exact solution” continues to be used too loosely and its attributes are lost. In the present paper, we give examples of misleading usage of such a term.

Unfairly recognizes advertising, which advertises under the guise of one commodity another product.

—a shortened form of FTC Act, as amended, §15(a), 15 U.S.C.A. §55(a) (Supp. 1938)

The complexity of the model is a measure of misunderstanding the essence of the problem.

—A.Ya. Findlin [20]

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Notes

  1. 1.

    However, an integral is accepted. In the same time, Riemann’s integral is a limit of Riemann’s sums, i.e., it is a series.

  2. 2.

    Not “exact”.

  3. 3.

    The most unexpected paper concerning doubly periodic functions belongs to Wang & Wang [69] where in this one paper (i) rediscovered Weierstrass’s functions [69, (2.29) and (2.30)], (ii) rediscovered Eisenstein’s summation approach [69, (2.18)], and (iii) solved a boundary value problem easily solved by means of the standard conformal mapping.

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Acknowledgements

Authors thanks Dr Galina Starushenko for fruitful discussions and providing the working notes with the calculation data and figures presented in Supplement.

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Authors and Affiliations

  1. Institut für Allgemeine Mechanik, RWTH Aachen University, Templergraben 64, 52056, Aachen, Germany

    Igor Andrianov

  2. Institute of Computer Sciences, Pedagogical University, ul. Podchorazych 2, Krakow, 30-084, Poland

    Vladimir Mityushev

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  1. Igor Andrianov
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Correspondence to Igor Andrianov .

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Editors and Affiliations

  1. Department of Mathematics & Natural Sciences, Rzeszow University, Rzeszów, Poland

    Piotr Drygaś

  2. Department of Economics, Belarusian State University, Minsk, Belarus

    Sergei Rogosin

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Andrianov, I., Mityushev, V. (2018). Exact and “Exact” Formulae in the Theory of Composites. In: Drygaś, P., Rogosin, S. (eds) Modern Problems in Applied Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-72640-3_2

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