aarjaneiro/planetmath huggingface dataset:
derivation of zeroth weighted power mean

derivation of zeroth weighted power mean

Let

x

1

,

x

2

,

…

,

x

n

be positive real numbers, and let

w

1

,

w

2

,

…

,

w

n

be positive real numbers such that

w

1

+

w

2

+

⋯

+

w

n

=

1

. For

r

≠

0

, the

r

-th

weighted power mean

of

x

1

,

x

2

,

…

,

x

n

is

M

w

r

⁢

(

x

1

,

x

2

,

…

,

x

n

)

=

(

w

1

⁢

x

1

r

+

w

2

⁢

x

2

r

+

⋯

+

w

n

⁢

x

n

r

)

1

/

r

.

Using the

Taylor series

expansion

e

t

=

1

+

t

+

𝒪

⁢

(

t

2

)

, where

𝒪

⁢

(

t

2

)

is

Landau notation

for terms of order

t

2

and higher, we can
write

x

i

r

as

x

i

r

=

e

r

⁢

log

⁡

x

i

=

1

+

r

⁢

log

⁡

x

i

+

𝒪

⁢

(

r

2

)

.

By substituting this into the

definition

of

M

w

r

, we get

M

w

r

⁢

(

x

1

,

x

2

,

…

,

x

n

)

=

[

w

1

⁢

(

1

+

r

⁢

log

⁡

x

1

)

+

⋯

+

w

n

⁢

(

1

+

r

⁢

log

⁡

x

n

)

+

𝒪

⁢

(

r

2

)

]

1

/

r

=

[

1

+

r

⁢

(

w

1

⁢

log

⁡

x

1

+

⋯

+

w

n

⁢

log

⁡

x

n

)

+

𝒪

⁢

(

r

2

)

]

1

/

r

=

[

1

+

r

⁢

log

⁡

(

x

1

w

1

⁢

x

2

w

2

⁢

⋯

⁢

x

n

w

n

)

+

𝒪

⁢

(

r

2

)

]

1

/

r

=

exp

⁡

{

1

r

⁢

log

⁡

[

1

+

r

⁢

log

⁡

(

x

1

w

1

⁢

x

2

w

2

⁢

⋯

⁢

x

n

w

n

)

+

𝒪

⁢

(

r

2

)

]

}

.

Again using a Taylor series, this time

log

⁡

(

1

+

t

)

=

t

+

𝒪

⁢

(

t

2

)

, we get

M

w

r

⁢

(

x

1

,

x

2

,

…

,

x

n

)

=

exp

⁡

{

1

r

⁢

[

r

⁢

log

⁡

(

x

1

w

1

⁢

x

2

w

2

⁢

⋯

⁢

x

n

w

n

)

+

𝒪

⁢

(

r

2

)

]

}

=

exp

⁡

[

log

⁡

(

x

1

w

1

⁢

x

2

w

2

⁢

⋯

⁢

x

n

w

n

)

+

𝒪

⁢

(

r

)

]

.

Taking the limit

r

→

0

, we find

M

w

0

⁢

(

x

1

,

x

2

,

…

,

x

n

)

=

exp

⁡

[

log

⁡

(

x

1

w

1

⁢

x

2

w

2

⁢

⋯

⁢

x

n

w

n

)

]

=

x

1

w

1

⁢

x

2

w

2

⁢

⋯

⁢

x

n

w

n

.

In particular, if we choose all the weights to be

1

n

,

M

0

⁢

(

x

1

,

x

2

,

…

,

x

n

)

=

x

1

⁢

x

2

⁢

⋯

⁢

x

n

n

,

the

geometric mean

of

x

1

,

x

2

,

…

,

x

n

.

Title

derivation of zeroth weighted power mean

Canonical

name

DerivationOfZerothWeightedPowerMean

Date of creation

2013-03-22 13:10:29

Last modified on

2013-03-22 13:10:29

Owner

pbruin (1001)

Last modified by

pbruin (1001)

Numerical id

6

Author

pbruin (1001)

Entry type

Derivation

Classification

msc 26B99

Related topic

PowerMean

Related topic

GeometricMean

Related topic

GeneralMeansInequality

Related topic

DerivationOfHarmonicMeanAsTheLimitOfThePowerMean

Generated on Fri Feb 9 19:59:57 2018 by

LaTeXML
harmonic mean in trapezoid

harmonic mean in trapezoid

Theorem.

If a line

parallel

to the bases of a

trapezoid

passes through

the intersecting point of the

diagonals

, then the portion of the line inside the trapezoid is the

harmonic mean

of the bases.

Proof.

Let

A

⁢

B

and

D

⁢

C

be the bases of a trapezoid

A

⁢

B

⁢

C

⁢

D

and

E

the intersecting point of the diagonals of

A

⁢

B

⁢

C

⁢

D

. Denote the cutting point of

A

⁢

D

and the line through

E

and parallel to the bases by

P

, and the cutting point of

B

⁢

C

and the same line by

Q

.  Then we have

Δ

⁢

C

⁢

D

⁢

E

∼

Δ

⁢

A

⁢

B

⁢

E

with

line ratio

k

h

=

C

⁢

D

A

⁢

B

, where

h

and

k

are the heights of the

triangles

A

⁢

B

⁢

E

and

C

⁢

D

⁢

E

, respectively, when

h

+

k

equals the height of the trapezoid.  We have also

Δ

⁢

P

⁢

E

⁢

D

∼

Δ

⁢

A

⁢

B

⁢

D

with line ratio

P

⁢

E

:

A

⁢

B

=

k

h

+

k

=

k

h

1

+

k

h

=

C

⁢

D

A

⁢

B

1

+

C

⁢

D

A

⁢

B

.

Thus we can express the length of

P

⁢

E

as

P

⁢

E

=

A

⁢

B

⋅

C

⁢

D

A

⁢

B

1

+

C

⁢

D

A

⁢

B

=

C

⁢

D

1

+

C

⁢

D

A

⁢

B

=

A

⁢

B

⋅

C

⁢

D

A

⁢

B

+

C

⁢

D

.

Similarly we may determine

E

⁢

Q

and that

E

⁢

Q

=

P

⁢

E

.  Consequently,

P

⁢

Q

=

P

⁢

E

+

E

⁢

Q

=

2

⋅

A

⁢

B

⋅

C

⁢

D

A

⁢

B

+

C

⁢

D

,

which is the harmonic mean of the bases

A

⁢

B

and

C

⁢

D

.

Title

harmonic mean in trapezoid

Canonical name

HarmonicMeanInTrapezoid

Date of creation

2013-03-22 17:49:22

Last modified on

2013-03-22 17:49:22

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

14

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26B99

Classification

msc 51M04

Classification

msc 51M15

Related topic

HarmonicMean

Related topic

SimilarityOfTriangles

Related topic

CorrespondingAnglesInTransversalCutting

Related topic

SimilarityInGeometry

Related topic

MedianOfTrapezoid

Related topic

ConstructionOfContraharmonicMeanOfTwoSegments

Related topic

IntegerHarmonicMeans

Generated on Fri Feb 9 20:00:00 2018 by

LaTeXML
Heronian mean is between geometric and arithmetic mean

Heronian mean is between geometric and arithmetic mean

Theorem

.

For non-negative numbers

x

and

y

, the

inequalities

x

⁢

y

≦

x

+

x

⁢

y

+

y

3

≦

x

+

y

2

are in , i.e. the

Heronian mean

is always at least equal to the

geometric mean

and at most equal to the

arithmetic mean

.  The equality signs are true if and only if

x

=

y

.

Proof.

1

∘

.

x

⁢

y

≦

x

+

x

⁢

y

+

y

3

⇔

3

x

⁢

y

≦

x

+

x

⁢

y

+

y

⇔

2

x

⁢

y

≦

x

+

y

⇔

4

x

y

≦

x

2

+

2

x

y

+

y

2

⇔

0

≦

x

2

-

2

x

y

+

y

2

⇔

0

≦

(

x

-

y

)

2

2

∘

.

x

+

x

⁢

y

+

y

3

≦

x

+

y

2

⇔

2

x

+

2

x

⁢

y

+

2

y

≦

3

x

+

3

y

⇔

2

x

⁢

y

≦

x

+

y

⇔

4

x

y

≦

x

2

+

2

x

y

+

y

2

⇔

0

≦

(

x

-

y

)

2

All inequalities of both chains are

equivalent

(

http://

planetmath

.org/Equivalent3

) since

x

and

y

are non-negative.  As for the equalities, the chains are valid with the mere equality signs.

Title

Heronian mean is between geometric and arithmetic mean

Canonical

name

HeronianMeanIsBetweenGeometricAndArithmeticMean

Date of creation

2013-03-22 17:49:14

Last modified on

2013-03-22 17:49:14

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26B99

Classification

msc 26D07

Classification

msc 01A20

Classification

msc 00A05

Synonym

Heronian mean inequalities

Related topic

ArithmeticGeometricMeansInequality

Related topic

ComparisonOfPythagoreanMeans

Related topic

SquareOfSum

Related topic

Equivalent3

Related topic

HeronsPrinciple

Generated on Fri Feb 9 20:00:02 2018 by

LaTeXML
weighted power mean

weighted power mean

If

w

1

,

w

2

,

…

,

w

n

are positive real numbers such that

w

1

+

w

2

+

⋯

+

w

n

=

1

, we define the

r

-th

weighted power mean

of the

x

i

as:

M

w

r

⁢

(

x

1

,

x

2

,

…

,

x

n

)

=

(

w

1

⁢

x

1

r

+

w

2

⁢

x

2

r

+

⋯

+

w

n

⁢

x

n

r

)

1

/

r

.

When all the

w

i

=

1

n

we get the standard

power mean

.
The weighted power mean is a

continuous function

of

r

, and taking limit when

r

→

0

gives us

M

w

0

=

x

1

w

1

⁢

x

2

w

2

⁢

⋯

⁢

w

n

w

n

.

We can weighted use power means to generalize the

power means inequality

:
If

w

is a set of weights, and if

r

<

s

then

M

w

r

≤

M

w

s

.

Title

weighted power mean

Canonical

name

WeightedPowerMean

Date of creation

2013-03-22 11:47:20

Last modified on

2013-03-22 11:47:20

Owner

drini (3)

Last modified by

drini (3)

Numerical id

12

Author

drini (3)

Entry type

Definition

Classification

msc 26B99

Classification

msc 00-01

Classification

msc 26-00

Related topic

ArithmeticGeometricMeansInequality

Related topic

ArithmeticMean

Related topic

GeometricMean

Related topic

HarmonicMean

Related topic

PowerMean

Related topic

ProofOfArithmeticGeometricHarmonicMeansInequality

Related topic

RootMeanSquare3

Related topic

ProofOfGeneralMeansInequality

Related topic

DerivationOfHarmonicMeanAsTheLimitOfThePowerMean

Generated on Fri Feb 9 20:00:04 2018 by

LaTeXML
continuity of natural power

continuity of natural power

Theorem

.

Let

n

be arbitrary

positive integer

.  The

power function

x

↦

x

n

from

ℝ

to

ℝ

(or

ℂ

to

ℂ

) is

continuous at

each point

x

0

.

Proof.

Let

ε

be any positive number.  Denote

x

0

+

h

=

x

and

x

n

-

x

0

n

=

Δ

.  Then identically

Δ

=

(

x

-

x

0

)

⁢

(

x

n

-

1

+

x

n

-

2

⁢

x

0

+

…

+

x

0

n

-

1

)

.

Taking the

absolute value

and using the

triangle inequality

give

|

Δ

|

=

|

h

|

⋅

|

x

n

-

1

+

x

n

-

2

⁢

x

0

+

…

+

x

0

n

-

1

|

≦

|

h

|

⋅

(

|

x

n

-

1

|

+

|

x

n

-

2

⁢

x

0

|

+

…

+

|

x

0

n

-

1

|

)

.

But since

|

x

|

=

|

x

0

+

h

|

≦

|

x

0

|

+

|

h

|

and also

|

x

0

|

≦

|

x

0

|

+

|

h

|

,  so each summand in the parentheses is at most equal to

(

|

x

0

|

+

|

h

|

)

n

-

1

,  and since there are

n

summands, the sum is at most equal to

n

⁢

(

|

x

0

|

+

|

h

|

)

n

-

1

.  Thus we get

|

Δ

|

≦

n

⁢

|

h

|

⁢

(

|

x

0

|

+

|

h

|

)

n

-

1

.

We may choose

|

h

|

<

1

;  this implies

|

Δ

|

≦

n

⁢

|

h

|

⁢

(

|

x

0

|

+

1

)

n

-

1

.

The right hand side of this

inequality

is less than

ε

as soon as we still require

|

h

|

<

ε

n

⁢

(

|

x

0

|

+

1

)

n

-

1

.

This means that the power function

x

↦

x

n

is continuous at the point

x

0

.

Note.

Another way to prove the theorem is to use induction on

n

and the rule 2 in

limit rules of functions

.

Title

continuity of natural power

Canonical name

ContinuityOfNaturalPower

Date of creation

2013-03-22 15:39:25

Last modified on

2013-03-22 15:39:25

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26C05

Classification

msc 26A15

Related topic

Exponentiation2

Generated on Fri Feb 9 20:00:17 2018 by

LaTeXML
elimination of unknown

elimination of unknown

Consider the simultaneous polynomial equations

{

a

(

x

,

y

)

=

:

∑

i

=

0

m

a

i

(

y

)

x

i

=

0

,

b

(

x

,

y

)

=

:

∑

j

=

0

n

b

j

(

y

)

x

j

=

0

(1)

in two unknowns

x

and

y

, where e.g.

m

≥

n

.  It is possible to eliminate one of the unknowns from (1), i.e. derive an

equivalent

(

http://planetmath.org/Equivalent3

) pair of polynomial equations

{

f

⁢

(

y

)

=

0

,

g

⁢

(

x

,

y

)

=

0

.

First we form the

polynomial

c

(

x

,

y

)

=

:

b

n

(

y

)

a

(

x

,

y

)

-

a

m

(

y

)

x

m

-

n

b

(

x

,

y

)

,

(2)

the degree of which is less than

m

.  When

(

x

0

,

y

0

)

is a solution of (1), then it

satisfies

{

c

⁢

(

x

,

y

)

=

0

b

⁢

(

x

,

y

)

=

0

.

(3)

On the other hand, when

(

x

1

,

y

1

)

is a solution of (3), then (2) implies that it satisfies also (1), except possibly in the case

b

n

⁢

(

y

1

)

=

0

.

We can continue similarly until we arrive at a pair of equations

{

f

⁢

(

y

)

=

0

,

g

⁢

(

x

,

y

)

=

0

(4)

Substituting the roots of the former of the equations (4) into the latter one, which in practice is usually of first degree with respect to

x

, one can get the corresponding values of

x

.  Hence one obtains all solutions of the original system of equations (1).  Since the cases

b

n

⁢

(

y

1

)

=

0

may yield wrong solutions, one should check them by substituting into (1).

Note.

One can derive from the equations (1) an equation of lower degree also by eliminating from them the constant terms; the terms of resulting equation have as common factor

x

or its higher power, which is removed by dividing.

Title

elimination of unknown

Canonical name

EliminationOfUnknown

Date of creation

2013-03-22 19:20:27

Last modified on

2013-03-22 19:20:27

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Algorithm

Classification

msc 26C05

Classification

msc 13P10

Classification

msc 12D99

Synonym

elementary method of elimination

Related topic

FactorizationOfPrimitivePolynomial

Generated on Fri Feb 9 20:00:27 2018 by

LaTeXML
homogeneous equation

homogeneous equation

The

homogeneous equation

f

⁢

(

x

,

y

)

=

0

,

where the left hand is a

homogeneous polynomial

of degree

r

in

x

and

y

,  determines the ratio

x

/

y

between the

indeterminates

.  One can be persuaded of this by dividing both of the equation by

y

r

.  Then the left depends only on

x

/

y

(which may be denoted e.g. by

t

).

Examples

•

The equation

5

⁢

x

+

8

⁢

y

=

0

determines that

x

/

y

=

-

8

5

.

•

The equation

x

2

-

7

⁢

x

⁢

y

+

10

⁢

y

2

=

0

determines that

x

/

y

=

2

or

x

/

y

=

5

(these values are obtained by first dividing both of the equation by

y

2

and then solving the equation

(

x

/

y

)

2

-

7

⁢

(

x

/

y

)

+

10

=

0

).

•

The equation

2

⁢

x

3

-

x

2

⁢

y

-

6

⁢

x

⁢

y

2

+

3

⁢

y

3

=

0

determines that

x

/

y

=

1

2

or

x

/

y

=

±

3

(first divide the equation by

y

3

and then solve

2

⁢

(

x

/

y

)

3

-

(

x

/

y

)

2

-

6

⁢

(

x

/

y

)

+

3

=

0

).

Title

homogeneous equation

Canonical

name

HomogeneousEquation

Date of creation

2013-03-22 15:14:41

Last modified on

2013-03-22 15:14:41

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Topic

Classification

msc 26C05

Classification

msc 26B35

Classification

msc 00A99

Related topic

Variation

Related topic

HomogeneousPolynomial

Related topic

Equation

Related topic

RegularDecagonInscribedInCircle

Generated on Fri Feb 9 20:00:29 2018 by

LaTeXML
Descartes’ rule of signs

Descartes’ rule of signs

Descartes’s rule of signs is a method for determining the number of positive or negative roots of a

polynomial

.

Let

p

⁢

(

x

)

=

∑

i

=

0

m

a

i

⁢

x

i

be a polynomial with real

coefficients

such that

a

m

≠

0

.

Define

v

to be the number of

variations

in sign

of the sequence of coefficients

a

m

,

…

,

a

0

. By ’variations in sign’ we mean the number of values of

n

such that the sign of

a

n

differs from the sign of

a

n

-

1

, as

n

ranges from

m

down to 1.

For example, consider

p

⁢

(

x

)

=

x

2

-

4

⁢

x

+

4

. The coefficients are

1

,

-

4

,

4

, so there are 2 variations in sign (since the sign of

1

differs from that of

-

4

, which in turn differs from that of

4

.)

Then the number of positive

real roots

of

p

⁢

(

x

)

is

v

-

2

⁢

N

for some integer

N

satisfying

0

≤

N

≤

v

2

.

The number

N

represents the number of

irreducible

factors of degree

2

in the

factorization

of

p

⁢

(

x

)

. Thus

N

=

0

if is known that

p

⁢

(

x

)

splits over the

real numbers

.

The number of negative roots of

p

⁢

(

x

)

may be obtained by the same method by applying the rule of signs to

p

⁢

(

-

x

)

.

History

This result is believed to have been first described by Réné Descartes in his 1637 work

La Géométrie

. In 1828, Carl Friedrich Gauss improved the rule by proving that when there are fewer roots of polynomials than there are variations of sign, the parity of the

difference

between the two is even.

Example

Let

p

⁢

(

x

)

=

x

3

+

3

⁢

x

2

-

x

-

3

. Looking at the list of coefficients, we have

1

,

3

,

-

1

,

-

3

, so there is only one variation in sign (from

3

to

-

1

).

Thus

v

=

1

. Since

0

≤

N

≤

1

2

then we must have

N

=

0

. Thus

v

-

2

⁢

N

=

1

and so there is exactly one positive real root of

p

⁢

(

x

)

.

To find the negative roots, we examine

p

⁢

(

-

x

)

=

-

x

3

+

3

⁢

x

2

+

x

-

3

. The coefficient list is

-

1

,

3

,

1

,

-

3

, so here are are two variations in sign (from

-

1

to

3

and

1

to

-

3

. Thus

v

=

2

and so

0

≤

N

≤

2

2

=

1

.

Thus we have two possible

solutions

,

N

=

0

and

N

=

1

, and two possible values of

v

-

2

⁢

N

. Therefore there are either two negative real roots or none at all.

We note that

p

⁢

(

-

1

)

=

(

-

1

)

3

+

3

⋅

(

-

1

)

2

-

(

-

1

)

-

3

=

0

, hence there is at least one negative root. Therefore there must be exactly two.

Title

Descartes’ rule of signs

Canonical name

DescartesRuleOfSigns

Date of creation

2013-03-22 14:28:31

Last modified on

2013-03-22 14:28:31

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

14

Author

PrimeFan (13766)

Entry type

Theorem

Classification

msc 26C10

Classification

msc 26A06

Generated on Fri Feb 9 20:00:31 2018 by

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existence of nth root

existence of

n

th root

Theorem

.

If

a

∈

R

with

a

>

0

and

n

is a positive integer, then there exists a unique positive real number

u

such that

u

n

=

a

.

Proof.

The statement is clearly true for

n

=

1

(let

u

=

a

). Thus, it will be assumed that

n

>

1

.

Define

p

:

ℝ

→

ℝ

by

p

⁢

(

x

)

=

x

n

-

a

. Note that a positive real root of

p

⁢

(

x

)

corresponds to a positive real number

u

such that

u

n

=

a

.

If

a

=

1

, then

p

⁢

(

1

)

=

1

n

-

1

=

0

, in which case the

existence

of

u

has been established.

Note that

p

⁢

(

x

)

is a polynomial function and thus is

continuous

. If

a

<

1

, then

p

⁢

(

1

)

=

1

n

-

a

>

1

-

1

=

0

. If

a

>

1

, then

p

⁢

(

a

)

=

a

n

-

a

=

a

⁢

(

a

n

-

1

-

1

)

>

0

. Note also that

p

⁢

(

0

)

=

0

n

-

a

=

-

a

<

0

. Thus, if

a

≠

1

, then the

intermediate value theorem

can be applied to yield the existence of

u

.

For uniqueness, note that the

function

p

⁢

(

x

)

is

strictly increasing

on the

interval

(

0

,

∞

)

. It follows that

u

as described in the statement of the

theorem

exists uniquely.
∎

Title

existence of

n

th root

Canonical name

ExistenceOfNthRoot

Date of creation

2013-03-22 15:52:15

Last modified on

2013-03-22 15:52:15

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

21

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 26C10

Classification

msc 26A06

Classification

msc 12D99

Related topic

ExistenceOfNthRoot

Generated on Fri Feb 9 20:00:34 2018 by

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existence of nth root

existence of nth root

If

n

is a positive integer and

a

>

0

then the

polynomial

x

n

-

a

∈

ℝ

⁢

[

x

]

has one sign change so by Descartes’s Rule of Signs has
a unique positive

real root

.

Title

existence of nth root

Canonical name

ExistenceOfNthRoot1

Date of creation

2013-03-22 16:08:38

Last modified on

2013-03-22 16:08:38

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

10

Author

Mathprof (13753)

Entry type

Proof

Classification

msc 26C10

Classification

msc 12D99

Classification

msc 26A06

Related topic

ExistenceOfRoot

Related topic

ExistenceOfSquareRootsOfNonNegativeRealNumbers

Generated on Fri Feb 9 20:00:36 2018 by

LaTeXML
partial fractions of expressions

partial fractions of expressions

Let

R

⁢

(

z

)

=

P

⁢

(

z

)

Q

⁢

(

z

)

be a

fractional expression

, i.e., a

quotient

of the

polynomials

P

⁢

(

z

)

and

Q

⁢

(

z

)

such that

P

⁢

(

z

)

is not

divisible

by

Q

⁢

(

z

)

.  Let’s restrict to the case that the coefficients are real or

complex numbers

.

If the distinct

complex

zeros of the

denominator

are

b

1

,

b

2

,

…

,

b

t

with the

multiplicities

τ

1

,

τ

2

,

…

,

τ

t

(

t

≥

1

), and the numerator has not common zeros, then

R

⁢

(

z

)

can be decomposed uniquely as the sum

R

⁢

(

z

)

=

H

⁢

(

z

)

+

∑

j

=

1

t

(

A

j

⁢

1

z

-

b

j

+

A

j

⁢

2

(

z

-

b

j

)

2

+

…

+

A

j

⁢

τ

j

(

z

-

b

j

)

τ

j

)

,

where

H

⁢

(

z

)

is a polynomial and the

A

j

⁢

k

’s are certain complex numbers.

Let us now take the special case that all coefficients of

P

⁢

(

z

)

and

Q

⁢

(

z

)

are real.  Then the (i.e. non-real) zeros of

Q

⁢

(

z

)

are pairwise

complex conjugates

, with same multiplicities, and the corresponding linear factors (

http://planetmath.org/

Product

) of

Q

⁢

(

z

)

may be pairwise multiplied to quadratic polynomials of the form

z

2

+

p

⁢

z

+

q

with real

p

’s and

q

’s and

p

2

<

4

⁢

q

.  Hence the above decomposition leads to the unique decomposition of the form

R

⁢

(

x

)

=

H

⁢

(

x

)

+

∑

i

=

1

m

(

A

i

⁢

1

x

-

b

i

+

A

i

⁢

2

(

x

-

b

i

)

2

+

…

+

A

i

⁢

μ

i

(

x

-

b

i

)

μ

i

)

+

∑

j

=

1

n

(

B

j

⁢

1

⁢

x

+

C

j

⁢

1

x

2

+

p

j

⁢

x

+

q

j

+

B

j

⁢

2

⁢

x

+

C

j

⁢

2

(

x

2

+

p

j

⁢

x

+

q

j

)

2

+

…

+

B

j

⁢

ν

j

⁢

x

+

C

j

⁢

ν

j

(

x

2

+

p

j

⁢

x

+

q

j

)

ν

j

)

,

where

m

is the number of the distinct real zeros and

2

⁢

n

the number of the distinct zeros of the denominator

Q

⁢

(

x

)

of the fractional expression

R

⁢

(

x

)

=

P

⁢

(

x

)

Q

⁢

(

x

)

.  The coefficients

A

i

⁢

k

,

B

j

⁢

k

and

C

j

⁢

k

are uniquely determined

real numbers

.

Cf. the

partial fractions

of

fractional numbers

.

Example.

-

x

5

+

6

⁢

x

4

-

7

⁢

x

3

+

15

⁢

x

2

-

4

⁢

x

+

3

(

x

-

1

)

3

⁢

(

x

2

+

1

)

2

=

-

1

x

-

1

+

3

(

x

-

1

)

3

+

x

x

2

+

1

+

2

⁢

x

-

1

(

x

2

+

1

)

2

Title

partial fractions of expressions

Canonical name

PartialFractionsOfExpressions

Date of creation

2013-03-22 14:20:27

Last modified on

2013-03-22 14:20:27

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

29

Author

pahio (2872)

Entry type

Definition

Classification

msc 26C15

Synonym

partial fractions

Related topic

ALectureOnThePartialFractionDecompositionMethod

Related topic

PartialFractionsForPolynomials

Related topic

ConjugatedRootsOfEquation2

Related topic

MixedFraction

Defines

fractional expression

Generated on Fri Feb 9 20:00:39 2018 by

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rational function

rational function

A real function

R

⁢

(

x

)

of a single

variable

x

is called

if it can be written as a

quotient

R

⁢

(

x

)

=

P

⁢

(

x

)

Q

⁢

(

x

)

,

where

P

⁢

(

x

)

and

Q

⁢

(

x

)

are

polynomials

in

x

with real

coefficients

. When one is only interested in

algebraic

properties of

R

⁢

(

x

)

or

P

⁢

(

x

)

and

Q

⁢

(

x

)

, it is convenient to forget that they define functions and simply treat them as algebraic expressions in

x

. In this case

R

⁢

(

x

)

is referred to as a

rational expression

.

In general, a

rational function

(expression)

R

⁢

(

x

1

,

…

,

x

n

)

has the form

R

⁢

(

x

1

,

…

,

x

n

)

=

P

⁢

(

x

1

,

…

,

x

n

)

Q

⁢

(

x

1

,

…

,

x

n

)

,

where

P

⁢

(

x

1

,

…

,

x

n

)

and

Q

⁢

(

x

1

,

…

,

x

n

)

are polynomials in the
variables

(

x

1

,

…

,

x

n

)

with coefficients in some field or
ring

S

.

In this sense,

R

⁢

(

x

1

,

…

,

x

n

)

can be regarded as an element of the

fraction
field

S

⁢

(

x

1

,

…

,

x

n

)

of the

polynomial ring

S

⁢

[

x

1

,

…

,

x

n

]

.

Title

rational function

Canonical name

RationalFunction

Date of creation

2013-03-22 13:38:54

Last modified on

2013-03-22 13:38:54

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 26C15

Synonym

rational expression

Related topic

PolynomialRing

Related topic

FractionField

Related topic

RealFunction

Related topic

PropertiesOfEntireFunctions

Related topic

IntegrationOfFractionPowerExpressions

Generated on Fri Feb 9 20:00:47 2018 by

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