For each

elementary function, we searched for at least one solution based on a literature review, analysis of similar systems in the market or patented systems, and application of the brainstorming method as proposed by BACK et al.

In [1,5], the authors derived new uniform convergent expansions of the incomplete gamma function [gamma](a, z) and the Bessel functions [J.sub.v](z) and [Y.sub.v](z) in terms of

elementary functions of z that hold uniformly in unbounded regions of C containing the point z = 0.

The potential is in general given parametrically; however, in several cases the involved coordinate transformation allows inversion thus leading to particular potentials which are explicitly written in terms of

elementary functions. These reductions are achieved by particular specifications of a parameter standing for the third finite singularity of the general Heun equation.

Now we shall call the function defined in (1) [mathematical expression not reproducible] since it does not refer to anyone and it has unknown analytic representation as

elementary function using standard special functions and the RHS of (18) presents another representation of T(x) function using CDF of the normal distribution.

Therefore, one major consequence should be that the RTD function of this zone can theoretically be calculated by a proper but simple combination of the

elementary function of the EZ or KN elements at the same feed rate and speed.

We have shown, with the help of a conformally flat metric, that these integrals maybe evaluated in terms of

elementary functions. This example suggests that our approach may be extended to other physically relevant metrics.

Representing an image with a few

elementary functions is widely used in image processing and computer vision.

It is obvious that the term [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ds can not solved directly due to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ds can not be expressed by

elementary functions. But, applying Theorem 3.3 we can give an upper bound 10/3 [e.sup.2]+ [[square root of [e.sup.3]]/3] for (17).

In some way or another, each looks at

elementary functions from the perspective of experimental mathematics.

We see from Examples 1 to 4 that, in general, for most

elementary functions (such as polynomials, rational functions, radical functions, and trigonometric functions) we can use algebraic identities to figure out their respective derivatives (the largest constant) of respective functions; although sometimes we need to elaborate on finding proper inequalities.

It is easy to compute for usual

elementary functions. When no efficient algorithm exists for the computation of [f], it can be approximated by an inclusion function F satisfying the following definition.

where the integral for f(x) cannot be written in terms of

elementary functions. A use of the definite integral is to determine the area between a curve and the horizontal axis (see Figure 1).