On Some Nonstandard Asymptotic Approximation of Integrals and Series

INTRODUCTION: Through this paper we consider properties of a quite general nature, which unify certain number of processes used to establish approximate expression of numbers and functions. Finally, we shall explain our nonstandard works and we treat the following problems. a) How to approximate certain functions to other functions? b) How to calculate the proper and improper integrals for approximate functions? c) What is the method to transform an unlimited integrand to another, everywhere limited? d) How to find an appreciable length for functions which are infinitesimal? e) How to calculate the sum of integral sequences under certain nonstandard conditions?


INTRODUCTION:
Through this paper we consider properties of a quite general nature, which unify certain number of processes used to establish approximate expression of numbers and functions.
Finally, we shall explain our nonstandard works and we treat the following problems. a) How to approximate certain functions to other functions? b) How to calculate the proper and improper integrals for approximate functions? c) What is the method to transform an unlimited integrand to another, everywhere limited? d) How to find an appreciable length for functions which are infinitesimal? e) How to calculate the sum of integral sequences under certain nonstandard conditions? Throughout this paper the following definitions and notations will be used: A real number x is called infinitesimal if and only if r x  for all rR.

If
x and y are real numbers, then x and y are said to be infinitely near, (denoted by The collection of limited, unlimited real numbers of infinitesimals are called external sets [3], [5]. The external set of infinitesimal real numbers is called the monad of 0 (denoted by m (0)). In general, the set of all real numbers, which are infinitely near to a standard real number a , is called the monad of a , (denoted by m(a)) [6], [7].
The set of all limited real numbers is called principal galaxy, (denoted by G).
For any real number a , the set of all real numbers x such that a x − limited is called the galaxy of a (denoted by G (a)) [2].

I. Asymptotic Approximation of Integrals
Theorem (I,1): If f and g are two measurable internal functions such that of limited length, then In the following theorem, we form a condition in order that the approximate equality of Theorem (I.1) holds for unbounded intervals.
Theorem(I,2): Let f and g be two measurable internal functions such that .
The above properties (Theorem (I,1) and (I,2)) permit, particularly to give an approximate value to the integral of a function which is near a function with known primitive. Notice that in the proof of Theorem (I.2), the major function h interferes only in the justification, and did not interfere in the approximation's quality.

Theorem (I, 3): Let f and g be two measurable internal functions such that
(the set of appreciable numbers). Let h be an integrable standard function such that Then p be standard, and  be unlimited positive as follows: The maximum of the integrand attains for 0 = v , putting this maximum as a factor, we obtain The new integrand is limited everywhere. By using the Euler formula

Comments on the strategy which comes out from example (I.4):
The above example is an illustration of the situation for which Theorem (I.2) is applied particularly, and of the procedure to follow. First of all we observe that Theorem (I.2) is mainly concerned with the functions f , which are: i) Limited every where, ii) noninfinitesimals on a subset of R , which contains at least one interval of appreciable length, and does not exceed the principal galaxy [2]. In fact, if   , we can not find a standard majoration of f , and for example if . Now such approximation of an infinitesimal number by another infinitesimal number is not meaningful In order to be able to treat the case of an integrand, which does not satisfy the conditions (i) and (ii), we examine the reason in the Example (I.4). The departure function is not necessarily limited everywhere, We obtain the integrand ) (v g which is limited everywhere by putting maximum of ) (v f in factor.

2)
We know that  

II. Asymptotic Approximation of Series
The following theorem links the series and integrals. Theorem (II.1): Let f be a positive, decreasing function such that