Modern Problems in Applied Analysis pp 15-34 | Cite as

# Exact and “Exact” Formulae in the Theory of Composites

## 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.

## Keywords

Composite material Effective property Closed form solution Weierstrass function Eisenstein summation Asymptotic solution## Mathematics Subject Classification (2010)

74A40 35C05 35C20 33E05 30E25 74Q15## Notes

### 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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