Equivalent

Equivalence for the continuations

Definition

Are

u_n

and

v_n

two continuations with values in same a vector Space normalized

E

or, if one wants, to be less general, in a body

\ mathbb {K}

(

\ mathbb {R}

\ mathbb {C}

), these is besides the case, least general, which is most current and most useful.

It is said that $u_n$ and $v_n$ are equivalent if and only if, $u_n-v_n$ is Négligeable in front of $v_n$ , or, which returns to same, $v_n-u_n$ Négligeable in front of $u_n$ .

An equivalent definition can be: there exists a definite continuation $\ varepsilon_n$ with values in $\ mathbb {K}$ starting from a certain row, which tends towards zero and checks $u_n= (1+ \ varepsilon_n) v_n$ .

One notes $u_n \ sim v_n$ then.

Simpler particular case

In the particular case where the continuation

u_n

is not cancelled starting from a certain row, the continuations

u_n

and

v_n

are equivalent if, and only if,

\ frac {v_n} {u_n} \ underset {N \ rightarrow + \ infty} {\ longrightarrow} =1

Another formulation

One can formulate the things differently: two continuations

u_n

and

v_n

are equivalent if, and only if, there is

u_n=v_n+o (v_n)

(by using the notation small " o").

Properties

the relation " to be equivalent à" is a Relation of equivalence.
If $u_n$ converges towards $l \ neq 0$ , then, it is equivalent after constant equal to $l$ .

Operations on the equivalents

Examples

an equivalent of the Somme partial $H_n$ of order N of the harmonic Série is $H_n \ sim \ ln (N)$ .
a famous equivalent is the Formule of Stirling: $n \! \ sim \ sqrt {2 \ pi N} \, \ left ({N \ over E} \ right) ^n$ .
Is $\ pi$ the continuation whose nth term is equal to the number of prime numbers lower or equal to $n$ . The Théorème of the prime numbers, which is a difficult result, affirms that $\ pi (N) \ sim \ frac {N} {\ ln N}$ .

Equivalence for the functions

Elementary definition

Are

I

part of

\ mathbb {R}

a

a point of the adherence of

I

f

and

g

of the applications of

I

towards

\ mathbb {R}

g

nonnull with the Voisinage of

a

It is said that $f$ is equivalent with $g$ to the Voisinage of $a$ if and only if

\ lim_ {X \ to has} \ frac {F (X)}{G (X)} = 1

One then writes $f \ sim g$ which is read “ $f$ is equivalent to $g$ ”. If there is an ambiguity on the point has that it is considered, one uses the more precise notation: $f\sim_a g$

More erudite definition

Either

X

a topological Space, or

A

a Under-part of

X

. Either

a \ in \ overline {has}

an adherent element of

X

with

A

. This space is the starting space of the functions which one will study. To some extent, it is the space of the parameters.

That is to say $\ mathbb {K} = \ mathbb {R}$ or $\ mathbb {C}$ , provided with its usual absolute value. That is to say $E$ a $\ mathbb {K}$ - vector Space normalized, called to be the space of the values of our functions.

Are $f$ and $g$ two functions of $A$ in $E$ .

It is said that $f$ and $g$ are equivalent in $a$ and one notes $f \ sim_a g$ if and only if, there exists a Voisinage $V$ of $a$ in $A \ cup \ {has \}$ and a function $\ varepsilon$ definite on $V$ such as:

$\ lim_a \ varepsilon = 0$
$\ forall X \ in V \ setminus \ {has \}, F (X) = (1+ \ varepsilon (X))G (X)$

Remarks

If one wants to be even more complete, one can place within the framework of the valués Corps or the topological vector spaces above a Corps valué.
the preceding definition is more natural and is expressed better within the framework of spaces of germs of functions.

Properties

the relation $\ sim$ is a Relation of equivalence.

$f \ sim_a G \ Rightarrow \ exists V \ in \ mathcal {V} (A) | \ forall X \ in V \ \ frac {F (X)}{G (X)} > 0$ , where $\ mathcal {V} (a)$ represents the whole of the Voisinage S of has .

In particular, F and G have even sign around A. locally.

If $f \ sim_a g$ and that $\ lim_ {has} F = L, L \ in \ overline {\ mathbb R}$ , then $\ lim_ {has} g=l$ : two equivalent functions has some have the same limit there.

Operations on the equivalents

See too

Negligible

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