La notació de Landau es defineix de la següent manera:
Si f, g són funcions complexes definides en un entorn d'un punt
x
0
{\displaystyle x_{0}}
F
=
O
(
g
)
{\displaystyle F=O(g)\,\!}
x
→
x
0
{\displaystyle x\rightarrow x_{0}}
ϵ
>
0
{\displaystyle \epsilon >0}
|
f
(
x
)
|
≤
ϵ
|
g
(
x
)
|
{\displaystyle |f(x)|\leq \epsilon |g(x)|}
x
{\displaystyle x}
x
0
{\displaystyle x_{0}\,\!}
F
=
o
(
g
)
{\displaystyle F=o(g)\,\!}
x
→
x
0
{\displaystyle x\rightarrow x_{0}}
ϵ
>
0
{\displaystyle \epsilon >0}
|
f
(
x
)
|
<
ϵ
|
g
(
x
)
|
{\displaystyle |f(x)|<\epsilon |g(x)|\,\!}
x
{\displaystyle x}
x
0
{\displaystyle x_{0}\,\!}
Una versió una mica més restrictiva però més manejable que la definició anterior és la següent:
Siguin
f
(
x
)
{\displaystyle f(x)\,\!}
g
(
x
)
{\displaystyle g(x)\,\!}
x
>
x
0
.
{\displaystyle x>x_{0}.\,\!}
g
(
x
)
≠
0
∀
x
>
x
0
{\displaystyle g(x)\neq 0\ \forall x>x_{0}\,\!}
f
=
O
(
g
)
{\displaystyle f=O(g)\,\!}
f
=
o
(
g
)
{\displaystyle f=o(g)\,\!}
signifiquen respectivament que
f
(
x
)
/
g
(
x
)
→
0
{\displaystyle f(x)/g(x)\to 0\,\!}
x
→
−
∞
{\displaystyle x\to -\infty \,\!}
f
(
x
)
/
g
(
x
)
{\displaystyle f(x)/g(x)\,\!}
x
{\displaystyle x\,\!}
x
{\displaystyle x\,\!}
−
∞
{\displaystyle -\infty \,\!}
x
{\displaystyle x\,\!}
o
(
1
)
{\displaystyle o(1)\,\!}
O
(
1
)
{\displaystyle O(1)\,\!}
Dues funcions
f
(
x
)
{\displaystyle f(x)\,\!}
g
(
x
)
{\displaystyle g(x)\,\!}
x
0
{\displaystyle x_{0}\,\!}
asimptòticament iguals
f
(
x
)
/
g
(
x
)
→
1
{\displaystyle f(x)/g(x)\to 1\,\!}
x
→
x
0
{\displaystyle x\to x_{0}\,\!}
Si les fraccions
f
(
x
)
/
g
(
x
)
{\displaystyle f(x)/g(x)\,\!}
g
(
x
)
/
f
(
x
)
{\displaystyle g(x)/f(x)\,\!}
x
0
{\displaystyle x_{0}\,\!}
f
(
x
)
{\displaystyle f(x)\,\!}
g
(
x
)
{\displaystyle g(x)\,\!}
són del mateix ordre
x
→
x
0
{\displaystyle x\to x_{0}\,\!}
Context de les propietats
Siguin
a
,
b
∈
R
{\displaystyle a,b\in \mathbb {R} \,\!}
f
{\displaystyle f\,\!}
a
≤
x
<
b
{\displaystyle a\leq x<b\,\!}
(
a
,
b
′
)
{\displaystyle (a,b')\,\!}
b
′
<
b
{\displaystyle b'<b\,\!}
F
(
x
)
=
∫
a
x
f
(
t
)
d
t
x
∈
(
a
,
b
′
)
{\displaystyle F(x)=\int _{a}^{x}f(t)\mathrm {d} t\quad x\in (a,b')\,\!}
Sigui
o
0
,
o
1
,
o
2
,
…
{\displaystyle o_{0},o_{1},o_{2},\ldots \,\!}
O
ν
=
o
0
+
o
1
+
⋯
+
O
ν
(
ν
=
0
,
1
,
…
)
{\displaystyle O_{\nu }=o_{0}+o_{1}+\cdots +O_{\nu }\ \ (\nu =0,1,\ldots )\,\!}
la mateixa notació serà utilitzada per altres lletres. Es tenen les següents propietats:
Suposeu que
f
(
x
)
{\displaystyle f(x)\,\!}
g
(
x
)
{\displaystyle g(x)\,\!}
a
≤
x
<
b
{\displaystyle a\leq x<b\,\!}
(
a
,
b
′
)
{\displaystyle (a,b')\,\!}
g
(
x
)
≥
0
{\displaystyle g(x)\geq 0\,\!}
G
(
x
)
→
+
∞
{\displaystyle G(x)\to +\infty \,\!}
x
→
b
{\displaystyle x\to b\,\!}
f
(
x
)
=
o
(
g
(
x
)
)
{\displaystyle f(x)=o(g(x))\,\!}
x
→
b
{\displaystyle x\to b\,\!}
F
(
x
)
=
o
(
G
(
x
)
)
{\displaystyle F(x)=o(G(x))\,\!}
Siguin
{
O
ν
}
,
{
v
ν
}
,
{\displaystyle \{O_{\nu }\},\ \{v_{\nu }\},\ \,\!}
{
O
ν
}
=
o
{
v
ν
}
,
{\displaystyle \{O_{\nu }\}=o\{v_{\nu }\},\ \,\!}
V
ν
→
+
∞
{\displaystyle V_{\nu }\to +\infty \,\!}
O
ν
=
o
(
V
ν
)
{\displaystyle O_{\nu }=o(V_{\nu })\,\!}
Suposeu que la sèrie
∑
v
ν
{\displaystyle \sum v_{\nu }\,\!}
v
{\displaystyle v\,\!}
O
ν
=
o
(
v
ν
)
{\displaystyle O_{\nu }=o(v_{\nu })\,\!}
O
ν
+
U
ν
+
1
+
⋯
=
o
(
v
ν
+
v
ν
+
1
+
⋯
)
{\displaystyle O_{\nu }+U_{\nu +1}+\cdots =o(v_{\nu }+v_{\nu +1}+\cdots )\,\!}
Sigui
f
(
x
)
{\displaystyle f(x)\,\!}
x
≥
0
{\displaystyle x\geq 0\,\!}
F
(
x
)
=
∫
0
x
f
d
t
,
F
n
=
f
(
0
)
+
f
(
1
)
+
⋯
+
f
(
n
)
.
{\displaystyle F(x)=\int _{0}^{x}f\mathrm {d} t,\quad F_{n}=f(0)+f(1)+\cdots +f(n).\,\!}
Llavors
(
i
)
{\displaystyle (i)}
f
(
x
)
{\displaystyle f(x)\,\!}
F
(
n
)
−
f
n
{\displaystyle F(n)-f_{n}}
(
i
i
)
{\displaystyle (ii)}
f
(
x
)
{\displaystyle f(x)\,\!}
F
(
n
)
≤
F
n
≤
F
(
n
)
−
f
(
n
)
n
→
+
∞
{\displaystyle F(n)\leq F_{n}\leq F(n)-f(n)_{n}\to +\infty }
Sigui
f
(
x
)
{\displaystyle f(x)\,\!}
x
≥
0
{\displaystyle x\geq 0\,\!}
(
i
)
{\displaystyle (i)}
f
(
x
)
{\displaystyle f(x)\,\!}
F
(
x
)
→
∞
{\displaystyle F(x)\to \infty \,\!}
(
i
i
)
{\displaystyle (ii)}
f
(
x
)
{\displaystyle f(x)\,\!}
f
(
x
)
=
o
(
F
(
x
)
)
{\displaystyle f(x)=o(F(x))\,\!}
F
n
{\displaystyle F_{n}\,\!}
F
(
n
)
{\displaystyle F(n)\,\!}
{\displaystyle \,\!}
