For the purpose of illustration, a complete solution in terms of elementary functions for a rigid circularly shaped inclusion is given and discussed from the point of view of material failure.
In this subsection, we will find some upper bounds for (lambda_{p}) in terms of elementary functions to obtain some computable upper bounds for (W_{v} ( x ) ) by using relation (3.7), that is, lambda_{p}=frac{f_{1} ( x_{0} ) }{f_{2} ( x_{0} ) }>min biggl{ frac{2v+2-p}{v+2},1 biggr}, and an analogous technique used in the proof of Subcase 2.4 of Theorem 1.1.
It seems that explicit computation of (alpha(x)) and so that of (X_{2,1/2}(a,b)), for all (a,b>0), in terms of elementary functions is impossible.
Exact expressions for various physical quantities are obtained in terms of elementary functions under a flat stamp.
Between them they developed most of the standard material found in calculus courses: the rules for differentiation, the integration of rational functions, the theory of elementary functions, applications to mechanics, and the geometry of curves.
As an illustration, a complete solution for a rigid circular inclusion is obtained in terms of elementary functions and analyzed.
Second, the extended Boussinesq and Cerruti solutions for the magnetoelectroelastic half-space are obtained in terms of elementary functions by utilizing this general solution.
It is well known that the function Φ cannot be expressed as the composition of elementary functions, therefore, it is valuable to find sharp bounds for the Mills ratio by certain simple and elementary functions.
As applications, we get two new bounds for the complete elliptic integral of the second kind in terms of elementary functions.
Definition 13 We say that the integral is elementary if it can be expressed in terms of elementary functions.
