External Version of Laplace Method

The aim of this paper is to use some concepts of nonstandard analysis introduced by Robinson A. and axiomatized by Nelson E. to give an external method of Laplace. More precisely under certain conditions we get the following result. 1 Γ 1 1 Where , , , are standard constants and ≃ 0 , .


Introduction
Throughout this paper the following definitions and notations will be used.A real number is called unlimited if | | for all real 0, [7].A real number is called infinitesimal if is unlimited, [7].
Two real number and are said to be infinitely near, denoted by ≃ , if is infinitesimal [7].A real number is called appreciable if it is neither unlimited nor infinitesimal, and the set of all positive appreciable real number is denoted by [2].The collections of limited, unlimited real numbers and infinitesimals are said to be external sets ( [1] , [2] , [6]).

Tahir H. Ismail
The external set of infinitesimal real numbers only is called the monad of 0 , denoted by 0 .In general the set of real numbers which are infinitely near to a standard real number is called the monad of , denoted by , [3], [5].The external set of limited real numbers is called the galaxy of 0, denoted by gal(0).In general, if is a standard real number, then the set of real numbers such that is limited is called the galaxy of , denoted by gal( ) [4], [9].If 0, the external set of real numbers such that ≃ 0 is called the , and it is strictly included 0 [8], [9].If ≃ 0, the external set of real numbers such that ≃ 0 for all standard ∈ is called -micromonad, denoted by , [2], [6].If is a limited point of ( standard), then it is infinitly near to a unique standard point of , this unique point is called the standard part of , or the shadow of , denoted by or 0 [2].If is a subset of ( standard), then shadow of , denoted by ° , is defined as the unique standard subset of obtained by taking the standardization of the collection of shadows of the limited points of , ( [4], [6]).
Any set or formula in internal set theory, denoted by IST, is called internal in case it does not involve the predicate "standard", otherwise it is called external ( [4], [5]).
A function : ⟶ is called an internal function if is an internal set ( [2]).
A function : , ⟶ , is called piecewise continuous on , if (i) When ever ≃ and ≃ implies exists (ii) is continuous at all , but a finite number , of points in , .
(iii) Left and right limits exist at all points of , .[9] A standard function : ⟶ is called s-continuous at the standard point in , if for all ( ≃ ⟶ ≃ , ([2], [8]).If ≃ 0 and is a real number, we define the as follows: If ≃ 0 and is a real number, we define the as follows: ∈ : [2].

The Main Result Theorem (2.1):
Let be an increasing standard function defined on 0, ∞ such that , , , ≃ 0, ≃ 0 and that , 0 for certain 0. Let be an internal piecewise continuous function defined on 0, ∞ such that , 0 , 1 are standard, ≃ 0 for ≃ 0 and for

External version of Laplace method
1225 each 0 there exist standard constants , such that | | Let be unlimited positive number, then Suppose that for every standard positive integer .

,
We observe that the passage from appreciable within infinitesimals for integrand workout when leaving the galaxy.We obtain an integrand, which is an , near an integrand, whose integral is continuous.Putting , we get Consequently

Remark (2.2):
The assertion of the above theorem remains true if the functions ∅ and satisfy the conditions mentioned on a standard interval 0, , 0, and if we integrate from 0 to .
We will assume that the function and satisfy always the always the conditions of theorem 2.1.Consider the following examples as illustration of the theorem 2.1 .
The relation remains valid if we integrate over a standard interval , such that 0 and 0. (3) From the result of example (2) we deduce the stirling formula.
Because if ∈ is unlimited , we get

!
We will bring back the last integral to are of the given in example (2).To identify the maximum value of the integral of at , put , then ≃ 0 (4) We return to integrals of type given in example (1) but with nonstandard ∈ , and determine the principle part of the integrals of type where is limited and the function is internal, contiunous and appreciable on 0 , and such that there exist standard constants , such that for every 0. We notice that the maximum factor to fast variation is reached for , we return to an integral of the type given in example (2) by putting .Then Since is limited ≃ 0 and is s-continuous function of for every limited 1, then from the formula of example ( 2) and the striting formula we deduce that

Remark (2.3):
We notice that as in the case of example (1) where the factor has its maximum value contributes to a sensible manner the value of the integral .
(5) Let be uninternal function such that (i) is s -continuous on the 0 .(ii) ≃ 0 on 0 .(iii) there exists standard constants and such that for every 0. Let ≃ 0 and ∈ be unlimited such that is limited .Then

Remark (2.4):
The proof of the above formula consists of (a) where is unlimited , and return to the previous example by putting , (b) in the case where is limit-