aarjaneiro/planetmath huggingface dataset:
Lewy extension theorem

Lewy extension theorem

Let

M

⊂

𝐂

n

be a smooth

real hypersurface

.
Let

ρ

be a defining

function

for

M

near

p

.

That is, for some

neighbourhood

of

p

,

the

submanifold

M

is defined by

ρ

=

0

.

For a neighbourhood

U

⊂

ℂ

n

,

define the set

U

+

to be the set

U

∩

{

ρ

>

0

}

.

We will say that

M

has at least one negative eigenvalue if the

Levi form

defined by

ρ

has at least one negative
eigenvalue. That is, if

∑

j

,

k

=

1

n

∂

2

⁡

ρ

⁢

(

p

)

∂

⁡

z

j

⁢

∂

⁡

z

¯

k

⁢

w

j

⁢

w

¯

k

<

0

⁢

for some

⁢

w

∈

ℂ

n

⁢

such that

⁢

∑

j

=

1

n

w

j

⁢

∂

⁡

ρ

⁢

(

p

)

∂

⁡

z

j

=

0

.

Theorem

.

Let

f

be a smooth

CR function

on

M

.

Suppose that
near

p

∈

M

the Levi form of

M

has at least one positive eigenvalue at

p

.

Then there exists a
neighbourhood

U

of

p

,

such that for every smooth CR function

f

on

M

,

there exists a
function

F

holomorphic

in

U

+

and

C

1

up to

M

,

such that

F

|

U

∩

M

=

f

|

U

∩

M

.

By considering

-

ρ

instead of

ρ

as a defining function, we get the corresponding result for
at least one negative eigenvalue.
If the Levi form of

M

has both positive and negative eigenvalues at a point, then

f

extends to both sides
of

M

and is then a restriction of a

holomorphic function

.

A point is the fact that

U

is fixed and does not depend on

f

.

To see why this is necessary, imagine a

Levi flat

example. Let

M

be defined in

ℂ

2

in coordinates

(

z

,

w

)

by

Im

⁡

w

=

0

.

The domains

U

ϵ

:=

{

|

Im

w

|

<

ϵ

}

,

for

ϵ

>

0

,

are

pseudoconvex

and hence
there exist functions
holomorphic on

Ω

ϵ

(and hence CR on

M

) that do not extend past any point of the

boundary

. No neighbourhood of a point on

M

fits in all

U

ϵ

.

So at least one nonzero eigenvalue of the Levi form
is needed.

The statement of this theorem is not exactly the theorem that Lewy formulated

[

4

]

, but this is generally called the

Lewy extension

. There have been many results
in this direction since Lewy’s original paper, but this is the most result.

References

1

M. Salah Baouendi,
Peter Ebenfelt,
Linda Preiss Rothschild.
,
Princeton University Press,
Princeton, New Jersey, 1999.

2

Albert Boggess.
,
CRC, 1991.

3

Lars Hörmander.
,
North-Holland Publishing Company, New York, New York, 1973.

4

Hans Lewy.

Ann. of Math.

(2)

64

(1956), 514–522.

Title

Lewy extension theorem

Canonical name

LewyExtensionTheorem

Date of creation

2013-03-22 17:39:44

Last modified on

2013-03-22 17:39:44

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

4

Author

jirka (4157)

Entry type

Theorem

Classification

msc 32V25

Synonym

Lewy extension

Generated on Fri Feb 9 20:34:03 2018 by

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D’Angelo finite type

D’Angelo finite type

Let

M

⊂

ℂ

n

be a

smooth submanifold

of real codimension 1. Let

p

∈

M

and let

r

p

denote the generator of the

principal ideal

of germs at

p

of

smooth functions

vanishing on

M

.
Define the number

Δ

1

⁢

(

M

,

p

)

=

sup

z

⁡

v

⁢

(

z

*

⁢

r

p

)

v

⁢

(

z

)

,

where

z

ranges over all parametrized holomorphic curves

z

:

𝔻

→

ℂ

n

(where

𝔻

is the unit disc) such that

z

⁢

(

0

)

=

0

,

v

is the

order of vanishing

at the origin, and

z

*

⁢

r

p

is the composition of

r

p

and

z

.
The order of vanishing

v

⁢

(

z

)

is

k

if

k

is the smallest integer such that the

k

th

derivative

of

z

is nonzero at the origin and all derivatives of smaller order are zero at the origin. Infinity is allowed for

v

⁢

(

z

)

if all derivatives vanish.

We
say

M

is of

(or finite 1-type)
at

p

∈

M

in the sense of D’Angelo if

Δ

1

⁢

(

M

,

p

)

<

∞

.

If

M

is

real analytic

, then

M

is

finite type

at

p

if and only if
there does not exist any germ of a

complex analytic subvariety

at

p

∈

M

, that is contained in

M

.
If

M

is only smooth, then it is possible that

M

is not finite type, but does not contain a germ of a holomorphic curve. However, if

M

is not finite type, then there exists a holomorphic curve which “touches”

M

to infinite order.

The

Diederich-Fornaess theorem

can be then restated to say that every

compact

real analytic
subvariety

of

ℂ

n

is of D’Angelo finite type at every point.

References

1

M. Salah Baouendi,
Peter Ebenfelt,
Linda Preiss Rothschild.
,
Princeton University Press,
Princeton, New Jersey, 1999.

2

D’Angelo, John P.
,
CRC Press, 1993.

Title

D’Angelo finite type

Canonical name

DAngeloFiniteType

Date of creation

2013-03-22 17:39:57

Last modified on

2013-03-22 17:39:57

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 32V35

Generated on Fri Feb 9 20:34:05 2018 by

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Diederich-Fornaess theorem

Diederich-Fornaess theorem

Theorem

(Diederich-Fornaes).

Let

X

⊂

C

n

be a compact

real analytic subvariety

. Then

X

contains
no germ of a nontrivial

complex analytic subvariety

.

In particular, all compact real analytic subvarieties (or

submanifolds

) are D’Angelo

finite type

at every point.

References

1

M. Salah Baouendi,
Peter Ebenfelt,
Linda Preiss Rothschild.
,
Princeton University Press,
Princeton, New Jersey, 1999.

2

D’Angelo, John P.
,
CRC Press, 1993.

3

Klas Diederich,
John E. Fornaess.

Ann. Math. (2)

107

(1978), no. 2, 371–384.

Title

Diederich-Fornaess theorem

Canonical name

DiederichFornaessTheorem

Date of creation

2013-03-22 17:40:00

Last modified on

2013-03-22 17:40:00

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

4

Author

jirka (4157)

Entry type

Theorem

Classification

msc 32V40

Classification

msc 32C07

Generated on Fri Feb 9 20:34:06 2018 by

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hypersurface

hypersurface

Definition

.

Let

M

be a subset of

ℝ

n

such that for every point

p

∈

M

there exists a

neighbourhood

U

p

of

p

in

ℝ

n

and a

continuously differentiable function

ρ

:

U

→

ℝ

with

grad

⁡

ρ

≠

0

on

U

,
such that

M

∩

U

=

{

x

∈

U

∣

ρ

⁢

(

x

)

=

0

}

.

Then

M

is called a

hypersurface

.

If

ρ

is in fact smooth then

M

is a

smooth hypersurface

and
similarly if

ρ

is

real analytic

then

M

is a

real analytic
hypersurface

. If
we identify

ℝ

2

⁢

n

with

ℂ

n

and we have a
hypersurface there it is called a

real hypersurface

in

ℂ

n

.

ρ

is usually called the

local defining function

.
Hypersurface is really special name for a

submanifold

of codimension 1. In fact if

M

is just a

topological manifold

of codimension 1, then it is often also called a hypersurface.

A real (

http://planetmath.org/RealAnalyticSubvariety

) or

complex analytic subvariety

of codimension 1 (the

zero set

of a real or complex analytic function) is called a

singular hypersurface

. That is the definition is the same as above, but
we do not require

grad

⁡

ρ

≠

0

. Note that some authors leave out the word

singular

and then use

non-singular hypersurface

for a hypersurface which is also a

manifold

. Some authors use the word

hypervariety

to describe a singular hypersurface.

An example of a hypersurface is the hypersphere (of radius 1 for simplicity) which has the defining equation

x

1

2

+

x

2

2

+

…

+

x

n

2

=

1

.

Another example of a hypersurface would be the

boundary

of a domain in

ℂ

n

with smooth boundary.

An example of a singular hypersurface in

ℝ

2

is for example the zero set
of

ρ

⁢

(

x

1

,

x

2

)

=

x

1

⁢

x

2

which is really just the two axis. Note that this
hypersurface fails to be a manifold at the origin.

References

1

M. Salah Baouendi,
Peter Ebenfelt,
Linda Preiss Rothschild.
,
Princeton University Press,
Princeton, New Jersey, 1999.

Title

hypersurface

Canonical name

Hypersurface

Date of creation

2013-03-22 14:32:56

Last modified on

2013-03-22 14:32:56

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

8

Author

jirka (4157)

Entry type

Definition

Classification

msc 32V40

Classification

msc 14J70

Related topic

Submanifold

Defines

smooth hypersurface

Defines

real analytic hypersurface

Defines

real hypersurface

Defines

local defining function

Defines

singular hypersurface

Defines

non-singular hypersurface

Defines

hypervariety

Generated on Fri Feb 9 20:34:08 2018 by

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Lewy hypersurface

Lewy hypersurface

The

real hypersurface

in

(

z

1

,

…

,

z

n

)

∈

ℂ

n

given by

Im

⁡

z

n

=

∑

j

=

1

n

-

1

|

z

j

|

2

is called the

Lewy hypersurface

. Note that this is a real hypersurface of real dimension

2

⁢

n

-

1

. This is an example of a non-trivial real hypersurface in complex space. For example it is not

biholomorphically equivalent

to the

hyperplane

defined by

Im

⁡

z

n

=

0

, but it is locally (not globally) biholomorphically equivalent to a

unit sphere

.

References

1

M. Salah Baouendi,
Peter Ebenfelt,
Linda Preiss Rothschild.
,
Princeton University Press,
Princeton, New Jersey, 1999.

Title

Lewy hypersurface

Canonical name

LewyHypersurface

Date of creation

2013-03-22 14:49:01

Last modified on

2013-03-22 14:49:01

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

4

Author

jirka (4157)

Entry type

Example

Classification

msc 32V99

Generated on Fri Feb 9 20:34:12 2018 by

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index of special functions

index of special functions

The term is not a completely precise mathematical term. It usually refers to a

function

of one or more real or complex variables which is either of use in some application or interesting in its own right, and hence has been studied enough to warrant giving it a name.

Special functions

are usually named after the mathematician who first introduced them or contributed much to their theory although, as in the rest of mathematics, such attributions are not always accurate, and they should be taken with a grain of salt.

0.1

http://planetmath.org/node/6420

Elementary Functions

•

exponential

•

logarithm

•

http://planetmath.org/node/4676

trigonometric functions

•

http://planetmath.org/node/6169

cyclometric functions

•

hyperbolic functions

•

area functions

•

Gudermannian function

•

http://planetmath.org/node/5744

sinc

function

0.2

Antiderivatives of elementary functions

•

error function

•

logarithmic integral

•

exponential integral

•

cosine integral

•

sine integral

•

hyperbolic sine integral

•

Fresnel integrals

•

elliptic integrals

0.3

Gamma and related functions

•

gamma function

•

beta function

•

polygamma functions

•

Barnes function

0.4

Functions defined as solutions of linear differential equations

•

Bessel functions

•

hypergeometric function

•

confluent hypergeometric function

•

generalized hypergeometric functions

•

Hermite polynomials

•

Legendre polynomials

•

associated

Laguerre polynomials

•

spherical harmonics

•

surface harmonics

•

Lamé function

•

Heun function

0.5

Functions defined as solutions of non-linear equations

•

Painlevé transcendents

•

Emden function

•

Lambert W function

•

Airy functions

0.6

Abelian functions

•

Jacobi theta functions

•

Weierstrass sigma function

•

Jacobi Zeta function

•

Weierstrass zeta function

0.7

Zeta Functions

•

general

Zeta functions

(in the sense of Jorgensen and Lang)

•

Hecke zeta function

•

Hurwitz zeta function

•

L

-functions

•

Riemann zeta function

•

Zeta functions of

surfaces

•

Zeta functions of graphs

•

Zeta functions of operators

Title

index of special functions

Canonical name

IndexOfSpecialFunctions

Date of creation

2013-03-22 14:40:06

Last modified on

2013-03-22 14:40:06

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

35

Author

rspuzio (6075)

Entry type

Topic

Classification

msc 33-00

Related topic

ComplexFunction

Related topic

ExponentialIntegral

Related topic

SpecialCasesOfHypergeometricFunction

Related topic

PropertiesOfOrthogonalPolynomials

Generated on Fri Feb 9 20:34:31 2018 by

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addition and subtraction formulas for tangent

addition and subtraction formulas for tangent

The

addition formula for tangent

will be achieved via brute from the

addition formulas for sine and cosine

.

tan

⁡

(

α

+

β

)

=

sin

⁡

(

α

+

β

)

cos

⁡

(

α

+

β

)

=

sin

⁡

α

⁢

cos

⁡

β

+

cos

⁡

α

⁢

sin

⁡

β

cos

⁡

α

⁢

cos

⁡

β

-

sin

⁡

α

⁢

sin

⁡

β

=

sin

⁡

α

cos

⁡

α

⋅

cos

⁡

β

cos

⁡

β

+

cos

⁡

α

cos

⁡

α

⋅

sin

⁡

β

cos

⁡

β

cos

⁡

α

cos

⁡

α

⋅

cos

⁡

β

cos

⁡

β

-

sin

⁡

α

cos

⁡

α

⋅

sin

⁡

β

cos

⁡

β

=

tan

⁡

α

⋅

1

+

1

⋅

tan

⁡

β

1

⋅

1

-

tan

⁡

α

⁢

tan

⁡

β

=

tan

⁡

α

+

tan

⁡

β

1

-

tan

⁡

α

⁢

tan

⁡

β

Note that

tan

is an

odd function

, i.e. (

http://planetmath.org/Ie

)

tan

⁡

(

-

x

)

=

-

tan

⁡

x

. This fact enables us to obtain the subtraction formula for tangent.

tan

⁡

(

α

-

β

)

=

tan

⁡

(

α

+

(

-

β

)

)

=

tan

⁡

α

+

tan

⁡

(

-

β

)

1

-

tan

⁡

α

⁢

tan

⁡

(

-

β

)

=

tan

⁡

α

-

tan

⁡

β

1

+

tan

⁡

α

⁢

tan

⁡

β

Title

addition and subtraction formulas for tangent

Canonical name

AdditionAndSubtractionFormulasForTangent

Date of creation

2013-03-22 16:59:04

Last modified on

2013-03-22 16:59:04

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

8

Author

Wkbj79 (1863)

Entry type

Derivation

Classification

msc 33B10

Classification

msc 26A09

Synonym

addition and subtraction formulae for tangent

Synonym

addition formula for tangent

Synonym

subtraction formula for tangent

Related topic

AdditionFormula

Related topic

DefinitionsInTrigonometry

Related topic

AngleBetweenTwoLines

Related topic

AdditionFormulas

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

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application of logarithm series

application of logarithm series

The integrand of the improper integral

I

:=

∫

0

1

ln

⁡

(

1

+

x

)

x

⁢

𝑑

x

(1)

is not defined at the

lower limit

0.  However, from the

Taylor series

expansion

ln

(

1

+

x

)

=

x

-

x

2

2

+

x

3

3

-

x

4

4

+

-

…

(

-

1

<

x

≦

1

)

of the

natural logarithm

we obtain the expansion of the integrand

ln

⁡

(

1

+

x

)

x

=

1

-

x

2

+

x

2

3

-

x

3

4

+

-

…

(

-

1

<

x

<

0

,

0

<

x

≦

1

)

whence

lim

x

→

0

⁡

ln

⁡

(

1

+

x

)

x

=

1

.

(2)

This implies that the integrand of (1) is

bounded

on the

interval

[

0

,

1

]

and also

continuous

, if we think that (2) defines its value at

x

=

0

.  Accordingly, the integrand is

Riemann integrable

on the interval, and we can determine the improper integral by integrating termwise:

I

=

∫

0

1

(

1

-

x

2

+

x

2

3

-

x

3

4

+

-

…

)

d

x

=

/

0

1

(

x

-

x

2

2

2

+

x

3

3

2

-

x

4

4

2

+

-

…

)

=

1

-

1

2

2

+

1

3

2

-

1

4

2

+

-

…

By the entry on

Dirichlet eta function

at 2 (

http://planetmath.org/ValueOfDirichletEtaFunctionAtS2

), the sum of the obtained series is

η

⁢

(

2

)

=

π

2

12

.  Thus we have the result

∫

0

1

ln

⁡

(

1

+

x

)

x

⁢

𝑑

x

=

π

2

12

.

(3)

Similarly, using the series

ln

(

1

-

x

)

=

-

x

-

x

2

2

-

x

3

3

-

x

4

4

-

…

(

-

1

≦

x

<

1

)

and the result in the entry

Riemann zeta function

at 2 (

http://planetmath.org/ValueOfTheRiemannZetaFunctionAtS2

), one can calculate that

∫

0

1

ln

⁡

(

1

-

x

)

x

⁢

𝑑

x

=

-

π

2

6

.

(4)

Title

application of logarithm series

Canonical name

ApplicationOfLogarithmSeries

Date of creation

2013-03-22 18:56:09

Last modified on

2013-03-22 18:56:09

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Application

Classification

msc 33B10

Related topic

DilogarithmFunction

Related topic

ExamplesOnHowToFindTaylorSeriesFromOtherKnownSeries

Related topic

SubstitutionNotation

Generated on Fri Feb 9 20:34:36 2018 by

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duality of Gudermannian and its inverse function

duality of Gudermannian and its inverse function

There are a lot of formulae concerning the

Gudermannian function

and its

inverse function

containing a

hyperbolic function

or a

trigonometric function

or both, such that if we change

functions

of one kind to the corresponding functions of the other kind, then the new formula also is true.

Some exemples:

gd

⁢

x

=

∫

0

x

d

⁢

t

cosh

⁡

t

,

gd

-

1

⁢

x

=

∫

0

x

d

⁢

t

cos

⁡

t

(1)

d

d

⁢

x

⁢

gd

⁢

x

=

1

cosh

⁡

x

,

d

d

⁢

x

⁢

gd

-

1

⁢

x

=

1

cos

⁡

x

(2)

tan

⁡

(

gd

⁢

x

)

=

sinh

⁡

x

,

tanh

⁡

(

gd

-

1

⁢

x

)

=

sin

⁡

x

(3)

sin

⁡

(

gd

⁢

x

)

=

tanh

⁡

x

,

sinh

⁡

(

gd

-

1

⁢

x

)

=

tan

⁡

x

(4)

tan

⁡

gd

⁢

x

2

=

tanh

⁡

x

2

,

tanh

⁡

gd

-

1

⁢

x

2

=

tan

⁡

x

2

(5)

For proving (5) we can check that

d

d

⁢

x

⁢

[

2

⁢

arctan

⁡

(

tanh

⁡

x

2

)

]

=

1

cosh

⁡

x

,

and since both the expression in the brackets and the

http://planetmath.org/node/11997

Gudermannian

vanish in the origin, we have

gd

⁢

x

≡

2

⁢

arctan

⁡

(

tanh

⁡

x

2

)

.

This equation implies (5).

The duality (

http://planetmath.org/DualityInMathematics

) of the formula pairs may be explained by the equality

gd

⁢

i

⁢

x

=

i

⁢

gd

-

1

⁢

x

.

(6)

Title

duality of Gudermannian and its inverse function

Canonical name

DualityOfGudermannianAndItsInverseFunction

Date of creation

2013-03-22 19:06:41

Last modified on

2013-03-22 19:06:41

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Topic

Classification

msc 33B10

Classification

msc 26E05

Classification

msc 26A09

Classification

msc 26A48

Related topic

InverseGudermannianFunction

Related topic

IdealInvertingInPruferRing

Generated on Fri Feb 9 20:34:39 2018 by

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inverse Gudermannian function

inverse Gudermannian function

Since the real

Gudermannian function

gd is

strictly increasing

and forms a

bijection

from

ℝ

onto the

open interval

(

-

π

2

,

π

2

)

,  it has an

inverse function

gd

-

1

:

(

-

π

2

,

π

2

)

→

ℝ

.

The

function

gd

-

1

is denoted also

arcgd

.

If

x

=

gd

⁢

y

, which may be explicitly written e.g.

x

=

arcsin

⁡

(

tanh

⁡

y

)

,

one can solve this for

y

, getting first

tanh

⁡

y

=

sin

⁡

x

and then

y

=

artanh

⁢

(

sin

⁡

x

)

(see the

area functions

).  Hence the

inverse Gudermannian

is expressed as

gd

-

1

⁢

(

x

)

=

arcgd

⁢

x

=

artanh

⁢

(

sin

⁡

x

)

(1)

It has other

equivalent

(

http://planetmath.org/Equivalent3

)

expressions

, such as

gd

-

1

⁢

(

x

)

=

arsinh

⁢

(

tan

⁡

x

)

=

1

2

⁢

ln

⁡

1

+

sin

⁡

x

1

-

sin

⁡

x

=

∫

0

x

d

⁢

t

cos

⁡

t

.

(2)

Thus its

derivative

is

d

d

⁢

x

⁢

gd

-

1

⁢

(

x

)

=

1

cos

⁡

x

.

(3)

Cf. the formulae (1)–(3) with the corresponding ones of gd.

Title

inverse Gudermannian function

Canonical

name

InverseGudermannianFunction

Date of creation

2013-03-22 19:06:28

Last modified on

2013-03-22 19:06:28

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Definition

Classification

msc 33B10

Classification

msc 26E05

Classification

msc 26A48

Classification

msc 26A09

Synonym

inverse Gudermannian

Related topic

HyperbolicFunctions

Related topic

AreaFunctions

Related topic

MercatorProjection

Related topic

EulerNumbers2

Related topic

DualityOfGudermannianAndItsInverseFunction

Generated on Fri Feb 9 20:34:50 2018 by

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limits of natural logarithm

limits of natural logarithm

The parent entry (

http://planetmath.org/NaturalLogarithm

) defines the

natural logarithm

as

ln

x

=

∫

1

x

1

t

d

t

(

x

>

0

)

(1)

and derives the

ln

⁡

x

⁢

y

=

ln

⁡

x

+

ln

⁡

y

which implies easily by

induction

that

ln

⁡

a

n

=

n

⁢

ln

⁡

a

.

(2)

Basing on (1), we prove here the

Theorem

.

The

function

x

↦

ln

⁡

x

is

strictly increasing

and

continuous

on

ℝ

+

.  It has the limits

lim

x

→

+

∞

⁡

ln

⁡

x

=

+

∞

and

lim

x

→

0

+

⁡

ln

⁡

x

=

-

∞

.

(3)

Proof.

By the above definition,

ln

⁡

x

is

differentiable

:

d

d

⁢

x

⁢

ln

⁡

x

=

1

x

>

0

Accordingly,

ln

⁡

x

is also continuous and strictly increasing.

Let

M

be an arbitrary positive number.  We have

ln

⁡

2

=

∫

1

2

d

⁢

t

t

>

0

.  There exists a

positive integer

n

such that

n

⁢

ln

⁡

2

>

M

(see Archimedean property).  By (2) we thus get

ln

⁡

2

n

>

M

, and since

ln

⁡

x

is strictly increasing, we see that

ln

⁡

x

>

M

∀

x

>

2

n

.

Hence the first limit assertion is true.
Now

-

M

<

0

.  If

x

>

2

n

,  then

ln

⁡

x

>

M

and

0

<

1

x

<

2

-

n

,

ln

⁡

1

x

=

∫

1

1

x

d

⁢

t

t

=

∫

x

1

d

⁢

u

u

=

-

ln

⁡

x

<

-

M

(substitution (

http://planetmath.org/SubstitutionForIntegration

)

x

⁢

t

:=

u

).  From this we can infer the second limit assertion.

Title

limits of natural logarithm

Canonical name

LimitsOfNaturalLogarithm

Date of creation

2014-12-12 10:15:50

Last modified on

2014-12-12 10:15:50

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Theorem

Classification

msc 33B10

Related topic

ImproperLimits

Related topic

GrowthOfExponentialFunction

Related topic

FundamentalTheoremOfCalculusClassicalVersion

Related topic

DifferentiableFunctionsAreContinuous

Generated on Fri Feb 9 20:34:52 2018 by

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logarithm series

logarithm series

The

derivative

of

ln

⁡

(

1

+

x

)

is

1

1

+

x

, which can be represented as the sum of

geometric series

:

1

1

+

x

=

1

-

x

+

x

2

-

x

3

+

-

…

for

-

1

<

x

<

1

.

Integrating both from 0 to

x

gives

ln

(

1

+

x

)

=

x

-

x

2

2

+

x

3

3

-

x

4

4

+

-

…

for

-

1

<

x

<

1

.

(1)

which is valid on the whole

open interval

of

convergence

-

1

<

x

<

1

of this

power series

and in
for

x

=

1

, as one may prove.

Replacing

x

with

-

x

in (1) yields the series

ln

⁡

(

1

-

x

)

=

-

x

-

x

2

2

-

x

3

3

-

x

4

4

-

…

for

-

1

<

x

<

1

.

(2)

Subtracting (2) from (1) gives

ln

⁡

1

+

x

1

-

x

=

2

⁢

(

x

+

x

3

3

+

x

5

5

+

x

7

7

+

…

)

(3)

which also is true for

-

1

<

x

<

1

.  Here the

inner function

of the

logarithm

attains all positive real values when

0

<

x

<

1

(its graph (

http://planetmath.org/Graph2

) is a hyperbola (

http://planetmath.org/Hyperbola2

) with asymptotes (

http://planetmath.org/AsymptotesOfGraphOfRationalFunction

)

x

=

1

and

y

=

-

1

).  Thus, in principle, the series (3) can be used for calculating any values of

natural logarithm

(

http://planetmath.org/NaturalLogarithm2

).  For this purpose, one could denote

1

+

x

1

-

x

:=

t

,

which implies

x

=

t

-

1

t

+

1

,

and accordingly

ln

⁡

t

=

2

⁢

[

t

-

1

t

+

1

+

1

3

⁢

(

t

-

1

t

+

1

)

3

+

1

5

⁢

(

t

-

1

t

+

1

)

5

+

…

]

.

(4)

For example,

ln

⁡

3

=

2

⁢

(

1

2

+

1

3

⋅

2

3

+

1

5

⋅

2

5

+

…

)

.

The convergence of (4) is the slower the

greater

is

t

.

Title

logarithm series

Canonical name

LogarithmSeries

Date of creation

2013-03-22 18:56:13

Last modified on

2013-03-22 18:56:13

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Topic

Classification

msc 33B10

Related topic

TaylorSeriesOfArcusSine

Related topic

TaylorSeriesOfArcusTangent

Generated on Fri Feb 9 20:34:55 2018 by

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