properties of O and o

properties of

O

and

o

The following properties of

Landau notation

hold:

1.

o

⁢

(

f

)

and

O

⁢

(

f

)

are vector spaces, i.e. if

g

,

h

∈

o

⁢

(

f

)

(resp. in

O

⁢

(

f

)

) then

λ

⁢

g

+

μ

⁢

h

∈

o

⁢

(

f

)

(resp. in

O

⁢

(

f

)

) whenever

λ

,

μ

∈

ℝ

;
In particular

o

⁢

(

f

)

+

o

⁢

(

f

)

=

o

⁢

(

f

)

and

λ

⁢

o

⁢

(

f

)

=

o

⁢

(

f

)

;

2.

if

λ

≠

0

then

λ

⁢

o

⁢

(

f

)

=

o

⁢

(

f

)

and

λ

⁢

O

⁢

(

f

)

=

O

⁢

(

f

)

;

3.

f

⁢

o

⁢

(

g

)

=

o

⁢

(

f

⁢

g

)

,

f

⁢

O

⁢

(

g

)

=

O

⁢

(

f

⁢

g

)

;

4.

o

⁢

(

g

)

α

=

o

⁢

(

g

α

)

,

O

⁢

(

g

)

α

=

O

⁢

(

g

α

)

;

5.

o

⁢

(

f

)

⊆

O

⁢

(

f

)

; on the other hand if

f

∈

o

⁢

(

g

)

then

O

⁢

(

f

)

⊆

o

⁢

(

g

)

;

6.

o

⁢

(

f

)

⊆

o

⁢

(

g

)

if

f

∈

O

⁢

(

g

)

; analogously

O

⁢

(

f

)

⊆

O

⁢

(

g

)

if

f

∈

O

⁢

(

g

)

;

7.

o

⁢

(

o

⁢

(

f

)

)

=

o

⁢

(

f

)

,

O

⁢

(

O

⁢

(

f

)

)

=

O

⁢

(

f

)

,

o

⁢

(

O

⁢

(

f

)

)

=

o

⁢

(

f

)

,

O

⁢

(

o

⁢

(

f

)

)

=

o

⁢

(

f

)

.

Here are some examples.
First of all we consider Taylor formula.
If

x

0

∈

(

a

,

b

)

⊂

ℝ

and

f

:

(

a

,

b

)

→

ℝ

has

n

derivatives

, then

f

⁢

(

x

)

∈

∑

k

=

0

n

f

(

k

)

⁢

(

x

0

)

k

!

⁢

(

x

-

x

0

)

k

+

o

⁢

(

(

x

-

x

0

)

n

)

.

As a consequence, if

f

has

n

+

1

derivatives, we can replace

o

⁢

(

(

x

-

x

0

)

n

)

with

O

⁢

(

(

x

-

x

0

)

n

+

1

)

in the previous formula.

For example:

e

x

∈

1

+

x

+

1

2

⁢

x

2

+

1

6

⁢

x

4

+

O

⁢

(

x

5

)

⊂

1

+

x

+

1

2

⁢

x

2

+

1

6

⁢

x

4

+

o

⁢

(

x

4

)

.

Using the properties stated above we can compose and

iterate

Taylor expansions

.
For example from the expansions

sin

⁡

x

∈

x

+

x

3

3

!

+

o

⁢

(

x

4

)

,

e

x

∈

1

+

x

+

x

2

2

+

O

⁢

(

x

3

)

,

cos

⁡

x

∈

1

-

x

2

2

+

x

4

4

!

+

o

⁢

(

x

5

)

⊆

1

-

x

2

2

+

O

⁢

(

x

4

)

,

log

⁡

(

1

+

x

)

∈

x

-

x

2

2

+

o

⁢

(

x

2

)

we get

(

x

⁢

sin

⁡

x

-

e

(

x

2

)

)

⁢

log

⁡

(

cos

⁡

x

)

∈

(

x

(

x

-

x

3

3

!

+

o

(

x

4

)

)

-

(

1

+

x

2

+

x

4

2

+

O

(

(

x

2

)

3

)

)

log

(

1

-

x

2

2

+

x

4

4

!

+

o

(

x

5

)

)

=

(

x

2

-

x

4

3

!

+

o

⁢

(

x

4

)

-

1

-

x

2

-

x

4

2

+

O

⁢

(

x

6

)

)

⁢

(

-

x

2

2

+

x

4

4

!

+

o

⁢

(

x

5

)

-

(

-

x

2

2

+

o

⁢

(

x

3

)

)

2

2

+

o

⁢

(

(

-

x

2

2

+

o

⁢

(

x

3

)

)

2

)

)

=

(

-

1

-

2

3

⁢

x

4

+

o

⁢

(

x

4

)

+

O

⁢

(

x

6

)

)

⁢

(

-

x

2

2

+

x

4

4

!

+

o

⁢

(

x

5

)

-

x

4

4

-

2

⁢

x

2

2

⁢

o

⁢

(

x

3

)

+

(

o

⁢

(

x

3

)

)

2

2

+

o

⁢

(

x

4

4

+

o

⁢

(

x

4

)

)

)

=

(

-

1

-

2

3

⁢

x

4

+

o

⁢

(

x

4

)

)

⁢

(

-

x

2

2

+

x

4

4

!

+

o

⁢

(

x

5

)

+

x

4

8

+

o

⁢

(

x

5

)

+

o

⁢

(

x

6

)

+

o

⁢

(

x

4

)

)

=

(

-

1

-

2

3

⁢

x

4

+

o

⁢

(

x

4

)

)

⁢

(

-

x

2

2

+

6

⁢

x

4

+

o

⁢

(

x

4

)

)

=

-

x

2

2

-

6

⁢

x

4

+

o

⁢

(

x

4

)

+

x

4

⁢

O

⁢

(

x

2

)

+

o

⁢

(

x

4

)

⁢

O

⁢

(

x

2

)

=

-

x

2

2

-

6

⁢

x

4

+

o

⁢

(

x

4

)

+

O

⁢

(

x

6

)

+

o

⁢

(

x

6

)

=

-

x

2

2

-

6

⁢

x

4

+

o

⁢

(

x

4

)

Title

properties of

O

and

o

Canonical name

PropertiesOfOAndO

Date of creation

2013-03-22 15:15:45

Last modified on

2013-03-22 15:15:45

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

7

Author

paolini (1187)

Entry type

Result

Classification

msc 26A12

Related topic

FormalDefinitionOfLandauNotation

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