Definition

Are $I$ part of $\backslash \; R$, has a point of the adherence of $I$, $f$ and $g$ of the applications of $I$ towards $\backslash \; R$

When has is real, it is said that $f$ is negligible in front of $g$ in the vicinity of has if and only if:

$\backslash \; forall\; \backslash \; varepsilon\; >0,\; \backslash ,\; \backslash \; exists\; \backslash \; eta>0,\; \backslash ,\; X\; \backslash \; in]\; has\; \backslash \; eta,\; a+\; \backslash \; eta\; I\; \backslash \; Rightarrow\; \backslash ,\; F\; (X)\; \backslash \; the\; \backslash \; varepsilon\; \backslash ,\; G\; (X)$

If has is equal to $+\; \backslash \; infty$, it is said that $f$ is negligible in front of $g$ in the vicinity of has if and only if:

$\backslash \; forall\; \backslash \; varepsilon\; >0,\; \backslash ,\; \backslash \; exists\; A>0,\; \backslash ,\; X\; \backslash \; in\; I\; \backslash \; Rightarrow\; \backslash ,\; F\; (X)\; \backslash \; the\; \backslash \; varepsilon\; \backslash ,\; G\; (X)$

An equivalent and more useful definition to in practice show the equivalence of two functions in a point has is, if $g$ is (except perhaps in has) nonnull on a Voisinage of has : $f$ is negligible in front of $g$ in the vicinity of has if and only if:

$\backslash \; lim\_\; \{X\; \backslash \; rightarrow\; has,\; X\; \backslash \; not=a\}\; \{F\; (X)\; \backslash \; over\; G\; (X)\}\; =\; 0$

One writes $f\; =\_\; \{then\; has\}\; O\; (G)$ which is read “ F is a small O of G in the vicinity of has ”.

Properties

• If $f\_1=\_a\; O\; (G)\; \backslash ,$ and $f\_2=\_a\; O\; (G)\; \backslash ,$ then for all realities $\backslash \; alpha$ and $\backslash \; beta$, $\backslash \; alpha\; \backslash ,\; f\_1\; +\; \backslash \; beta\; \backslash ,\; f\_2=\_a\; O\; (G)\; \backslash ,$
• If $f\_1=\_a\; O\; (g\_1)\; \backslash ,$ and $f\_2=\_a\; O\; (g\_2)\; \backslash ,$ then $f\_1.\; f\_2=\_a\; O\; (g\_1.g\_2)\; \backslash ,$
• If $f=\_a\; O\; (G)\; \backslash ,$ and $h\; \backslash ,$ limited to the Voisinage of has then $h.f=\_a\; O\; (G)\; \backslash ,$
• If $f\_1\; =\_a\; O\; (G)\; \backslash ,$ and $f\_2\; =\_a\; O\; (f\_1)\; \backslash ,$ then $f\_2\; =\_a\; O\; (G)\; \backslash ,$ ( transitivity )
• $f\; \backslash \; sim\_a\; G\; \backslash \; Leftrightarrow\; F\; -\; G\; =\_a\; O\; (G)\; \backslash \; Leftrightarrow\; F\; =\_a\; G\; +\; O\; (G)$

Scale of comparison

A scale of comparison $E\_a$ is a family of functions defined in the Voisinage of has (except perhaps in has) such as:
$\backslash \; forall\; (F,\; G)\; \backslash \; in\; \{E\_a\}\; ^2,\; \backslash ,\; F\; \backslash \; G\; \backslash \; \backslash \; Rightarrow\; (f=\_a\; O\; (G)\; \backslash \; \backslash \; mathrm\; \{or\}\; \backslash \; g=\_a\; O\; (F))\backslash ,$

Principal part of a function compared to a scale

Definition

Are a function $f\; \backslash ,$ defined in a vicinity V of has (except perhaps in has), not cancelling itself on $V-\; \backslash \; \{has\; \backslash \}\; \backslash ,$, $E\_a\; \backslash ,$ a scale of comparison in has . It is said that $f\; \backslash ,$ admits the function $g\; \backslash \; in\; E\_a\; \backslash ,$ like principal part compared to the scale $E\_a\; \backslash ,$ if and only if there exists a reality has not no one such as $f\; \backslash \; sim\_a\; A.g$ (or $f\; =\_a\; A.g\; +\; O\; (G)$).

Properties

• Unicité in the event of existence
• Is $f\_1\; \backslash ,$ and $f\_2\; \backslash ,$ admitting $g\_1\; \backslash \; respectively,$ and $g\_2\; \backslash ,$ like principal part compared to the scale of comparison $E\_a\; \backslash ,$. The principal part of $f\_1.f\_2\; \backslash ,$ compared to the scale of comparison $E\_a\; \backslash ,$ is the function $g\_1.g\_2\; \backslash ,$
• Are $f\_1\; \backslash ,$ and $f\_2\; \backslash ,$ admitting $g\_1\; \backslash \; respectively,$ and $g\_2\; \backslash ,$ like principal part compared to the scale of comparison $E\_a\; \backslash ,$.

If $g\_1\; =\_a\; O\; (g\_2)\; \backslash ,$ then $g\_2\; \backslash ,$ is the principal part of $f\_1\; +\; f\_2\; \backslash ,$ compared to the scale of comparison $E\_a\; \backslash ,$.

If $g\_2\; =\_a\; O\; (g\_1)\; \backslash ,$ then $g\_1\; \backslash ,$ is the principal part of $f\_1\; +\; f\_2\; \backslash ,$ compared to the scale of comparison $E\_a\; \backslash ,$.

If $g\_1\; =\; g\_2\; \backslash ,$ and that $A\_1\; +\; A\_2\; \backslash \; 0\; \backslash ,$ then $g\_1\; \backslash ,$ is the principal part of $f\_1\; +\; f\_2\; \backslash ,$ compared to the scale of comparison $E\_a\; \backslash ,$.

Comparison for the continuations

Definition

A continuation $(u\_n)\; \backslash ,$ of real numbers is known as negligible in front of a real continuation $(v\_n)\; \backslash ,$ when:

$\backslash \; forall\; \backslash \; varepsilon\; >0,\; \backslash ,\; \backslash \; exists\; NR\; \backslash \; in\; \backslash \; NR,\; \backslash ,\; N\; \backslash \; Ge\; NR\; \backslash ,\; \backslash \; Rightarrow\; \backslash ,\; u\_n\; \backslash \; the\; \backslash \; varepsilon\; \backslash ,\; v\_n$

An equivalent definition: a continuation $(u\_n)\; \backslash ,$ of real numbers is known as negligible in front of a real continuation $(v\_n)\; \backslash ,$ when there exists a continuation $(\backslash \; varepsilon\_n)\; \_\; \{N\; \backslash \; geq\; N\_0\}\; \backslash ,$ of null limit such as:

$\backslash \; forall\; N\; \backslash \; geq\; N\_0,\; \backslash \; qquad\; u\_n=\; \backslash \; varepsilon\_nv\_n\; \backslash ,$

One notes: $u\_n=o\; (v\_n)\; \backslash ,$

Property

A more useful definition to show the equivalence of two continuations in practice is:

$u\_n=o\; (v\_n)\; \backslash \; Leftrightarrow\; \backslash \; lim\_\; \{N\; \backslash \; to\; +\; \backslash \; infty\}\; \backslash \; frac\; \{u\_n\}\; \{v\_n\}\; =0\; \backslash \; Leftrightarrow\; \backslash \; forall\; \backslash \; varepsilon>0,\; \backslash \; exists\; NR\; \backslash \; in\; \backslash \; mathbb\; \{NR\},\; \backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; \{NR\},\; N\; \backslash \; geq\; NR\; \backslash \; Rightarrow\; |u\_n|\; \backslash \; Leq\; \backslash \; varepsilon\; |v\_n|$

See too

• Equivalent

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