aarjaneiro/planetmath huggingface dataset:
hyperreal

hyperreal

An

ultrafilter

ℱ

on a set

I

is called

nonprincipal

if no finite subsets of

I

are in

ℱ

.

Fix once and for all a

nonprincipal ultrafilter

ℱ

on the set

ℕ

of

natural numbers

. Let

∼

be the

equivalence relation

on the set

ℝ

ℕ

of

sequences

of

real numbers

given by

{

a

n

}

∼

{

b

n

}

⇔

{

n

∈

ℕ

∣

a

n

=

b

n

}

∈

ℱ

Let

ℝ

*

be the set of

equivalence classes

of

ℝ

ℕ

under the equivalence relation

∼

. The set

ℝ

*

is called the set of

hyperreals

. It is a field under coordinatewise

addition

and

multiplication

:

{

a

n

}

+

{

b

n

}

=

{

a

n

+

b

n

}

{

a

n

}

⋅

{

b

n

}

=

{

a

n

⋅

b

n

}

The field

ℝ

*

is an

ordered field

under the

ordering relation

{

a

n

}

≤

{

b

n

}

⇔

{

n

∈

ℕ

∣

a

n

≤

b

n

}

∈

ℱ

The real numbers embed into

ℝ

*

by the map sending the real number

x

∈

ℝ

to the equivalence class of the

constant

sequence given by

x

n

:=

x

for all

n

. In what follows, we adopt the convention of treating

ℝ

as a subset of

ℝ

*

under this

embedding

.

A hyperreal

x

∈

*

ℝ

is:

•

limited

if

a

<

x

<

b

for some real numbers

a

,

b

∈

ℝ

•

positive

unlimited

if

x

>

a

for all real numbers

a

∈

ℝ

•

negative unlimited

if

x

<

a

for all real numbers

a

∈

ℝ

•

unlimited

if it is either positive unlimited or negative unlimited

•

positive

infinitesimal

if

0

<

x

<

a

for all positive real numbers

a

∈

ℝ

+

•

negative infinitesimal

if

a

<

x

<

0

for all negative real numbers

a

∈

ℝ

-

•

infinitesimal

if it is either positive infinitesimal or negative infinitesimal

For any subset

A

of

ℝ

, the set

A

*

is defined to be the subset of

ℝ

*

consisting of equivalence classes of sequences

{

a

n

}

such that

{

n

∈

ℕ

∣

a

n

∈

A

}

∈

ℱ

.

The sets

ℕ

*

,

ℤ

*

, and

ℚ

*

are called

hypernaturals

,

hyperintegers

, and

hyperrationals

, respectively. An element of

ℕ

*

is also sometimes called

hyperfinite

.

Title

hyperreal

Canonical

name

Hyperreal

Date of creation

2013-03-22 12:35:45

Last modified on

2013-03-22 12:35:45

Owner

djao (24)

Last modified by

djao (24)

Numerical id

4

Author

djao (24)

Entry type

Definition

Classification

msc 26E35

Synonym

nonstandard real

Synonym

non-standard real

Related topic

Infinitesimal2

Defines

nonprincipal ultrafilter

Defines

infinitesimal

Defines

hypernatural

Defines

hyperinteger

Defines

hyperrational

Defines

hyperfinite

Generated on Fri Feb 9 20:03:20 2018 by

LaTeXML
properties of hyperreals under field operations

properties of hyperreals under field operations

Let

ℝ

b

*

denote the set of finite (or limited)

hyperreal numbers

and

ℝ

0

*

the set of

infinitesimal

hyperreal numbers.

-

We have that

1.

ℝ

b

*

and

ℝ

0

*

are subrings of

ℝ

*

.

2.

ℝ

0

*

is an ideal of

ℝ

b

*

.

3.

the sum of an

infinite

hyperreal with a finite hyperreal is infinite.

4.

the

inverse

of a non-zero infinitesimal hyperreal is infinite.

5.

the inverse of an infinite hyperreal is infinitesimal.

The above properties can be described more informally like:

1.

finite

+

finite

=

finite

2.

infinitesimal

+

infinitesimal

=

infinitesimal

3.

infinite

+

finite

=

infinite

4.

finite

×

finite

=

finite

5.

infinitesimal

×

finite

=

infinitesimal

6.

infinitesimal

-

1

=

infinite

7.

infinite

-

1

=

infinitesimal

Title

properties of hyperreals under field operations

Canonical name

PropertiesOfHyperrealsUnderFieldOperations

Date of creation

2013-03-22 17:26:19

Last modified on

2013-03-22 17:26:19

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

4

Author

asteroid (17536)

Entry type

Result

Classification

msc 26E35

Generated on Fri Feb 9 20:03:22 2018 by

LaTeXML
structure of finite hyperreal numbers

structure of finite hyperreal numbers

Theorem

-

Every (or )

hyperreal number

x

∈

ℝ

*

admits a unique of the form

x

=

a

+

ϵ

where

a

∈

ℝ

and

ϵ

is

infinitesimal

.

Remark :

This theorem just says that every finite hyperreal number has a

real part

and an infinitesimal part (just like real and

imaginary parts

in

complex numbers

).

Title

structure of finite hyperreal numbers

Canonical name

StructureOfFiniteHyperrealNumbers

Date of creation

2013-03-22 17:26:02

Last modified on

2013-03-22 17:26:02

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

4

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 26E35

Generated on Fri Feb 9 20:03:24 2018 by

LaTeXML
arithmetic-geometric mean as a product

arithmetic-geometric mean as a product

Recall that, given two

real numbers

0

<

x

≤

y

, their

arithmetic-geometric
mean

may be defined as

M

⁢

(

x

,

y

)

=

lim

n

→

∞

⁡

g

n

, where

g

0

=

x

a

0

=

y

g

n

+

1

=

a

n

⁢

g

n

a

n

+

1

=

a

n

+

g

n

2

.

In this entry, we will re-express this quantity as an infinite product.
We begin by rewriting the

recursion

for

g

n

:

g

n

+

1

=

a

n

⁢

g

n

=

a

n

g

n

⋅

g

n

2

=

g

n

⁢

a

n

g

n

From this, it follows that

g

n

=

g

0

⁢

∏

m

=

0

n

-

1

h

m

where

h

n

=

a

n

/

g

n

.

As it stands, this is not so interesting because no way has been given
to determine the factors

h

n

other than first computing

a

n

and

g

n

. We shall now correct this defect by deriving a recursion which
may be used to compute the

h

n

’s directly:

h

n

+

1

=

a

n

+

1

g

n

+

1

=

a

n

+

g

n

2

⁢

a

n

⁢

g

n

=

1

2

⁢

(

a

n

g

n

+

g

n

a

n

)

=

1

2

⁢

(

h

n

+

1

h

n

)

=

h

n

2

+

1

2

⁢

h

n

Taking the limit

n

→

∞

, we then have the

formula

M

⁢

(

x

,

y

)

=

x

⁢

∏

m

=

0

∞

h

n

where

h

0

=

y

x

and

h

n

+

1

=

h

n

2

+

1

2

⁢

h

n

.

Title

arithmetic-geometric mean as a

product

Canonical name

ArithmeticgeometricMeanAsAProduct

Date of creation

2013-03-22 17:09:59

Last modified on

2013-03-22 17:09:59

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Derivation

Classification

msc 26E60

Classification

msc 33E05

Generated on Fri Feb 9 20:03:29 2018 by

LaTeXML
contrageometric proportion

contrageometric proportion

Just as one converts the proportion equation

m

-

x

y

-

m

=

x

y

defining the

harmonic mean

of

x

and

y

into the proportion equation

m

-

x

y

-

m

=

y

x

defining their

contraharmonic
mean

(

http://

planetmath

.org/ContraharmonicProportion

), one also may convert the proportion
equation

m

-

x

y

-

m

=

m

y

defining the

geometric mean

into a new equation

m

-

x

y

-

m

=

y

m

(1)

defining the

contrageometric mean

m

of

x

and

y

.  Thus, the
three positive numbers

x

,

m

,

y

satisfying (1) are in

contrageometric proportion

.  One

integer

example is

1

,

4

,

6

.

Solving

m

from (1) one gets the expression

m

=

x

-

y

+

(

x

-

y

)

2

+

4

⁢

y

2

2

=

:

f

(

x

,

y

)

.

(2)

Suppose now that

0

≤

x

≤

y

.  Using (2) we see that

m

≥

x

-

y

+

0

2

+

4

⁢

y

2

2

=

x

+

y

2

≥

x

,

y

2

-

m

2

=

-

(

y

-

x

)

2

+

(

y

-

x

)

⁢

(

x

-

y

)

2

+

4

⁢

y

2

2

=

(

y

-

x

)

⁢

[

(

y

-

x

)

2

+

4

⁢

y

2

-

(

y

-

x

)

]

2

≥

0

,

accordingly

x

≤

f

⁢

(

x

,

y

)

≤

y

.

(3)

Thus the contrageometric mean of

x

and

y

also is at least equal to
their

arithmetic mean

.  We can also compare

m

with their

quadratic mean

by watching the

difference

(

x

2

+

y

2

2

)

2

-

m

2

=

(

y

-

x

)

⁢

(

1

2

⁢

(

y

-

x

)

2

+

4

⁢

y

2

-

y

)

≥

0

.

So we have

x

+

y

2

≤

f

⁢

(

x

,

y

)

≤

x

2

+

y

2

2

.

(4)

Cf. this result with the

comparison of Pythagorean means

(

http://planetmath.org/ComparisonOfPythagoreanMeans

); there the brown curve is the graph of

f

⁢

(

x

,

1

)

.

It’s clear that the contrageometric mean (2) is not symmetric with
respect to the variables

x

and

y

, contrary to the other types of
means in general.  On the other hand, the contrageometric mean is, as other types of means, a first-degree

homogeneous function

its

arguments

:

f

⁢

(

t

⁢

x

,

t

⁢

y

)

=

t

⁢

f

⁢

(

x

,

y

)

.

(5)

References

1

Mabrouk K. Faradj

:

http://etd.lsu.edu/docs/available/etd-07082004-091436/unrestricted/Faradj_thesis.pdf

What mean do you mean? An exposition on means

.  Louisiana State University (2004).

2

Georghe Toader & Silvia Toader

:

http://rgmia.org/papers/monographs/Grec.pdf

Greek means and the

arithmetic-geometric mean

.  RGMIA (2010).

Title

contrageometric proportion

Canonical name

ContrageometricProportion

Date of creation

2013-04-19 7:14:46

Last modified on

2013-04-19 7:14:46

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

17

Author

pahio (2872)

Entry type

Definition

Classification

msc 26E60

Classification

msc 11-00

Classification

msc 01A20

Classification

msc 01A17

Related topic

ContraharmonicProportion

Defines

contrageometric mean

Generated on Fri Feb 9 20:03:31 2018 by

LaTeXML
link between infinite products and sums

link between infinite products and sums

Let

∏

k

=

1

∞

p

k

be an infinite product such that

p

k

>

0

for all

k

.
Then the infinite product

converges

if and only if the infinite sum

∑

k

=

1

∞

log

⁡

p

k

converges. Moreover

∏

k

=

1

∞

p

k

=

exp

⁢

∑

k

=

1

∞

log

⁡

p

k

.

Proof.

Simply notice that

∏

k

=

1

N

p

k

=

exp

⁢

∑

k

=

1

N

log

⁡

p

k

.

If the infinite sum converges then (by

continuity

of

exp

function)

lim

N

→

∞

⁡

∏

k

=

1

N

p

k

=

lim

N

→

∞

⁡

exp

⁢

∑

k

=

1

N

log

⁡

p

k

=

exp

⁢

∑

k

=

1

∞

log

⁡

p

k

and also the infinite product converges.

Title

link between infinite products and sums

Canonical name

LinkBetweenInfiniteProductsAndSums

Date of creation

2013-03-22 13:41:38

Last modified on

2013-03-22 13:41:38

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

7

Author

paolini (1187)

Entry type

Theorem

Classification

msc 26E99

Classification

msc 40A20

Generated on Fri Feb 9 20:03:36 2018 by

LaTeXML
order of vanishing

order of vanishing

Definition

.

Let

x

0

be a zero (

http://planetmath.org/ZeroOfAFunction

) of the real function

Δ

.  The

order of vanishing

of

Δ

at

x

0

is

n

, if

lim

x

→

x

0

⁡

Δ

⁢

(

x

)

x

n

has a non-zero finite value.

Usually,

x

0

of the definition is 0.

Example.

If the curves

y

=

f

⁢

(

x

)

and

y

=

g

⁢

(

x

)

have in the point

(

x

0

,

y

0

)

the order of contact

n

, then the

difference

Δ

⁢

(

h

)

:=

g

⁢

(

x

0

+

h

)

-

f

⁢

(

x

0

+

h

)

of the ordinates has

n

+

1

-order of vanishing.

Title

order of vanishing

Canonical

name

OrderOfVanishing

Date of creation

2013-03-22 17:57:15

Last modified on

2013-03-22 17:57:15

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Definition

Classification

msc 26E99

Synonym

vanishing order

Related topic

Multiplicity

Related topic

OsculatingCurve

Generated on Fri Feb 9 20:03:37 2018 by

LaTeXML
proof of convergence criterion for infinite product

proof of convergence criterion for infinite product

Consider the partial

product

P

n

=

∏

i

=

1

n

p

i

.

By definition we say that the infinite product

∏

n

=

1

∞

p

n

is

convergent

iff

P

n

is convergent.

Suppose every

p

n

>

0

ln

is a

continuous

bijection

from

ℝ

+

to

ℝ

, therefore

lim

n

→

∞

⁡

a

n

=

a

⇔

lim

n

→

∞

⁡

ln

⁡

(

a

n

)

=

ln

⁡

(

a

)

, provided

a

n

>

0

and

a

>

0

.

so saying

P

n

→

P

>

0

is

equivalent

to saying that

ln

⁡

(

P

n

)

converges

.

Since

ln

⁡

(

P

n

)

=

ln

⁡

(

∏

i

=

1

n

p

i

)

=

∑

i

=

1

n

ln

⁡

(

p

i

)

, the infinite product converges to a positive value iff the series

∑

n

=

1

∞

ln

⁡

(

p

n

)

is convergent.

In particular, if the infinite product converges to a positive value, then

ln

⁡

(

p

n

)

→

0

⟹

p

n

→

1

.

P

n

→

0

, is equivalent to saying

∑

n

=

1

∞

ln

⁡

(

p

n

)

=

-

∞

For the second part of the

theorem

:

∏

n

=

1

∞

(

1

+

p

n

)

converges absolutely to a positive value iff

∑

n

=

1

∞

p

n

converges absolutely.

as we have seen,

1

+

p

n

→

1

⟹

p

n

→

0

consider:

lim

x

→

0

⁡

ln

⁡

(

1

+

x

)

x

=

1

(this is easy to prove since by Taylor’s

expansion

ln

⁡

(

1

+

x

)

=

x

+

O

⁢

(

x

2

)

)

Since

p

n

→

0

we can say that

lim

n

→

∞

⁡

ln

⁡

(

1

+

p

n

)

p

n

=

1

and by the

limit comparison test

, either both

∑

n

=

1

∞

ln

⁡

(

1

+

p

n

)

and

∑

i

=

1

n

p

i

converge or diverge.

Title

proof of convergence criterion for infinite product

Canonical name

ProofOfConvergenceCriterionForInfiniteProduct

Date of creation

2013-03-22 15:35:36

Last modified on

2013-03-22 15:35:36

Owner

cvalente (11260)

Last modified by

cvalente (11260)

Numerical id

10

Author

cvalente (11260)

Entry type

Proof

Classification

msc 26E99

Generated on Fri Feb 9 20:03:39 2018 by

LaTeXML
zero of a function

zero of a function

\PMlinkescapephrase

closed set

Suppose

X

is a set and

f

a

complex

(

http://planetmath.org/Complex

)-valued

function

f

:

X

→

ℂ

.  Then a

zero

of

f

is an

element

x

∈

X

such that

f

⁢

(

x

)

=

0

.  It is also said that

f

vanishes

at

x

.

The

zero set

of

f

is the set

Z

⁢

(

f

)

:=

{

x

∈

X

∣

f

⁢

(

x

)

=

0

}

.

Remark.

When

X

is a “simple” space, such as

ℝ

or

ℂ

a zero is also called a

root

.  However, in pure mathematics and especially if

Z

⁢

(

f

)

is

infinite

, it seems to be customary to talk of zeroes and the zero set instead of roots.

Examples

•

For any

z

∈

ℂ

, define

z

^

:

X

→

ℂ

by

z

^

⁢

(

x

)

=

z

. Then

Z

⁢

(

0

^

)

=

X

and

Z

⁢

(

z

^

)

=

∅

if

z

≠

0

.

•

Suppose

p

is a

polynomial

(

http://planetmath.org/Polynomial

)

p

:

ℂ

→

ℂ

of degree

n

≥

1

.  Then

p

has at most

n

zeroes. That is,

|

Z

⁢

(

p

)

|

≤

n

.

•

If

f

and

g

are functions

f

:

X

→

ℂ

and

g

:

X

→

ℂ

, then

Z

⁢

(

f

⁢

g

)

=

Z

⁢

(

f

)

∪

Z

⁢

(

g

)

,

Z

⁢

(

f

⁢

g

)

⊇

Z

⁢

(

f

)

,

where

f

⁢

g

is the function

x

↦

f

⁢

(

x

)

⁢

g

⁢

(

x

)

.

•

For any

f

:

X

→

ℝ

, then

Z

⁢

(

f

)

=

Z

⁢

(

|

f

|

)

=

Z

⁢

(

f

n

)

,

where

f

n

is the defined

f

n

⁢

(

x

)

=

(

f

⁢

(

x

)

)

n

.

•

If

f

and

g

are both real-valued functions, then

Z

⁢

(

f

)

∩

Z

⁢

(

g

)

=

Z

⁢

(

f

2

+

g

2

)

=

Z

⁢

(

|

f

|

+

|

g

|

)

.

•

If

X

is a

topological space

and

f

:

X

→

ℂ

is a function, then the

support

(

http://planetmath.org/SupportOfFunction

) of

f

is given by:

supp

⁡

f

=

Z

⁢

(

f

)

∁

¯

Further, if

f

is

continuous

, then

Z

⁢

(

f

)

is closed (

http://planetmath.org/ClosedSet

) in

X

(assuming that

ℂ

is given the

usual topology

of the

complex plane

where

{

0

}

is a closed set).

Title

zero of a function

Canonical name

ZeroOfAFunction

Date of creation

2013-03-22 14:00:58

Last modified on

2013-03-22 14:00:58

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

30

Author

mathcam (2727)

Entry type

Definition

Classification

msc 26E99

Synonym

zero

Synonym

vanish

Synonym

vanishes

Related topic

SupportOfFunction

Defines

zero set

Generated on Fri Feb 9 20:03:42 2018 by

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extended real numbers

extended real numbers

The

extended real numbers

are the

real numbers

together with

+

∞

(or simply

∞

) and

-

∞

.
This set is usually denoted by

ℝ

¯

or

[

-

∞

,

∞

]

,
and the

elements

+

∞

and

-

∞

are called

plus

and

minus infinity

, respectively.  (N.B.,  “

ℝ

¯

” may sometimes the

algebraic closure

of

ℝ

; see the

special notations in algebra

.)

The real numbers are in certain

contexts

called

finite

as contrast to

∞

.

0.0.1

Order on

ℝ

¯

The order (

http://planetmath.org/TotalOrder

) relation on

ℝ

extends to

ℝ

¯

by
defining that for any

x

∈

ℝ

, we have

-

∞

<

x

,

x

<

∞

,

and that

-

∞

<

∞

.  For

a

∈

ℝ

, let us also define

intervals

(

a

,

∞

]

=

{

x

∈

ℝ

:

x

>

a

}

,

[

-

∞

,

a

)

=

{

x

∈

ℝ

:

x

<

a

}

.

0.0.2

Addition

For any real number

x

, we define

x

+

(

±

∞

)

=

(

±

∞

)

+

x

=

±

∞

,

and for

+

∞

and

-

∞

, we define

(

±

∞

)

+

(

±

∞

)

=

±

∞

.

It should be pointed out that sums like

(

+

∞

)

+

(

-

∞

)

are left undefined.  Thus

ℝ

¯

is not an

ordered ring

although

ℝ

is.

0.0.3

Multiplication

If

x

is a positive real number, then

x

⋅

(

±

∞

)

=

(

±

∞

)

⋅

x

=

±

∞

.

Similarly, if

x

is a negative real number, then

x

⋅

(

±

∞

)

=

(

±

∞

)

⋅

x

=

∓

∞

.

Furthermore, for

∞

and

-

∞

, we define

(

+

∞

)

⋅

(

+

∞

)

=

(

-

∞

)

⋅

(

-

∞

)

=

+

∞

,

(

+

∞

)

⋅

(

-

∞

)

=

(

-

∞

)

⋅

(

+

∞

)

=

-

∞

.

In many areas of mathematics,

products

like

0

⋅

∞

are left undefined.  However, a special case is
measure theory, where it is convenient to define

0

⋅

(

±

∞

)

=

(

±

∞

)

⋅

0

=

0

.

0.0.4

Absolute value

For

∞

and

-

∞

, the

absolute value

is defined as

|

±

∞

|

=

+

∞

.

0.0.5

Topology

The

topology

of

R

¯

is given by the usual base of

ℝ

together with with intervals of type

[

-

∞

,

a

)

,

(

a

,

∞

]

.
This makes

ℝ

¯

into a

compact

topological space

.

ℝ

¯

can also be seen to be

homeomorphic

to the interval

[

-

1

,

1

]

, via
the map

x

↦

(

2

/

π

)

⁢

arctan

⁡

x

.
Consequently, every

continuous function

f

:

ℝ

¯

→

ℝ

¯

has
a

minimum

and

maximum

.

0.0.6

Examples

1.

By taking

x

=

-

1

in the , we obtain
the

relations

(

-

1

)

⋅

(

±

∞

)

=

∓

∞

.

Title

extended real numbers

Canonical name

ExtendedRealNumbers

Date of creation

2013-03-22 13:44:44

Last modified on

2013-03-22 13:44:44

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

21

Author

matte (1858)

Entry type

Definition

Classification

msc 28-00

Classification

msc 12D99

Related topic

ImproperLimits

Related topic

IntermediateValueTheoremForExtendedRealNumbers

Related topic

ExampleOfNonCompleteLatticeHomomorphism

Defines

plus infinity

Defines

minus infinity

Defines

ℝ

¯

Defines

infinite

Defines

infinity

Defines

finite

Generated on Fri Feb 9 20:04:33 2018 by

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partition

partition

Let

a

,

b

∈

ℝ

with

a

<

b

. A

partition

of an

interval

[

a

,

b

]

is a set of nonempty

subintervals

{

[

a

,

x

1

)

,

[

x

1

,

x

2

)

,

…

,

[

x

n

-

1

,

b

]

}

for some positive integer

n

. That is,

a

<

x

1

<

x

2

<

…

<

x

n

-

1

<

b

. Note that

n

is the number of subintervals in the partition.

Subinterval partitions are useful for defining

Riemann integrals

.

Note that subinterval partition is a specific case of a partition (

http://planetmath.org/Partition

) of a set since the subintervals are defined so that they are

pairwise disjoint

.

Title

partition

Canonical name

Partition1

Date of creation

2013-03-22 15:57:50

Last modified on

2013-03-22 15:57:50

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

8

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 28-00

Classification

msc 26A42

Synonym

subinterval partition

Related topic

Subinterval

Generated on Fri Feb 9 20:04:35 2018 by

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quadrature

quadrature

Quadrature

is the computation of a univariate definite integral. It can refer to either numerical or analytic techniques; one must gather from context which is meant. The term refers to the geometric origin of integration in determining the area of a plane figure by approximating it with squares.

Cubature

refers to higher-dimensional definite integral computation. Likewise, this term refers to the geometric operation of approximating the volume of a solid by means of cubes (and has since been extended to higher

dimensions

).

The terms “quadrature” and “cubature” are typically used in numerical analysis
to denote the

approximation

of a definite integral, typically by a suitable
weighted sum. Perhaps the simplest possibility is approximation by a sum of
values at equidistant points, i.e. approximate

∫

0

1

f

⁢

(

x

)

⁢

𝑑

x

by

∑

k

=

0

n

f

⁢

(

k

/

n

)

/

n

. More complicated approximations involve

variable

weights and evaluation of the function at points which may not be spaced
equidistantly. Some such numerical quadrature methods are Simpson’s rule, the

trapezoidal rule

, and

Gaussian quadrature

.

Title

quadrature

Canonical

name

Quadrature

Date of creation

2013-03-22 12:07:35

Last modified on

2013-03-22 12:07:35

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

13

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 28-00

Classification

msc 65D32

Classification

msc 41A55

Classification

msc 26A42

Defines

cubature

Generated on Fri Feb 9 20:04:37 2018 by

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Riemann integral

Riemann integral

Let

I

=

[

a

,

b

]

be an

interval

of

ℝ

and let

f

:

I

→

ℝ

be a

bounded function

. For any

finite set

of points

{

x

0

,

x

1

,

x

2

,

…

,

x

n

}

such that

a

=

x

0

<

x

1

<

x

2

⁢

⋯

<

x

n

=

b

, there is a corresponding

partition

P

=

{

[

x

0

,

x

1

)

,

[

x

1

,

x

2

)

,

…

,

[

x

n

-

1

,

x

n

]

}

of

I

.

Let

C

⁢

(

ϵ

)

be the set of all partitions of

I

with

max

⁡

(

x

i

+

1

-

x

i

)

<

ϵ

. Then let

S

*

⁢

(

ϵ

)

be the

infimum

of the set of

upper Riemann sums

with each partition in

C

⁢

(

ϵ

)

, and let

S

*

⁢

(

ϵ

)

be the

supremum

of the set of lower Riemann sums with each partition in

C

⁢

(

ϵ

)

. If

ϵ

1

<

ϵ

2

, then

C

⁢

(

ϵ

1

)

⊂

C

⁢

(

ϵ

2

)

, so

S

*

⁢

(

ϵ

)

is

decreasing

(

http://planetmath.org/IncreasingdecreasingmonotoneFunction

) and

S

*

⁢

(

ϵ

)

is increasing (

http://planetmath.org/IncreasingdecreasingmonotoneFunction

). Moreover,

|

S

*

⁢

(

ϵ

)

|

and

|

S

*

⁢

(

ϵ

)

|

are

bounded

by

(

b

-

a

)

⁢

sup

x

⁡

|

f

⁢

(

x

)

|

. Therefore, the limits

S

*

=

lim

ϵ

→

0

⁡

S

*

⁢

(

ϵ

)

and

S

*

=

lim

ϵ

→

0

⁡

S

*

⁢

(

ϵ

)

exist and are finite. If

S

*

=

S

*

, then

f

is Riemann-integrable over

I

, and the

Riemann integral

of

f

over

I

is defined by

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

=

S

*

=

S

*

.

Title

Riemann integral

Canonical name

RiemannIntegral

Date of creation

2013-03-22 11:49:24

Last modified on

2013-03-22 11:49:24

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

14

Author

bbukh (348)

Entry type

Definition

Classification

msc 28-00

Classification

msc 26A42

Related topic

RiemannSum

Related topic

Integral2

Defines

Riemann integrable

Generated on Fri Feb 9 20:04:39 2018 by

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