aarjaneiro/planetmath huggingface dataset:
uniform convergence of integral

uniform convergence of integral

Let the function

f

⁢

(

x

,

t

)

be

continuous

in the domain

a

≦

x

<

b

,

c

≦

t

≦

d

,

where

b

is a

real number

or

∞

, and let the

improper integral

F

⁢

(

t

)

:=

∫

a

b

f

⁢

(

x

,

t

)

⁢

𝑑

x

=

lim

u

→

b

-

⁡

∫

a

u

f

⁢

(

x

,

t

)

⁢

𝑑

x

(1)

be

convergent

(

http://planetmath.org/ImproperIntegral

) in every point

t

of the

interval

[

c

,

d

]

.  We say that the on the interval

[

c

,

d

]

,  if for each positive number

ε

there is a value

x

ε

∈

[

a

,

b

]

such that

|

∫

x

b

f

⁢

(

x

,

t

)

⁢

𝑑

x

|

<

ε

∀

t

∈

[

c

,

d

]

when

x

ε

≦

x

<

b

.

Title

uniform convergence of integral

Canonical name

UniformConvergenceOfIntegral

Date of creation

2013-03-22 14:40:30

Last modified on

2013-03-22 14:40:30

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Definition

Classification

msc 26A42

Related topic

SumFunctionOfSeries

Related topic

ConvergenceOfIntegrals

Defines

integral converging uniformly

Defines

uniformly converging integral

Generated on Fri Feb 9 19:54:31 2018 by

LaTeXML
using convolution to find Laplace transform

using convolution to find Laplace transform

We start from the (see the table of

Laplace transforms

)

e

α

⁢

t

↶

1

s

-

α

,

1

t

↶

π

s

(

s

>

α

)

(1)

where the curved from the Laplace-transformed

functions

to the original functions.  Setting

α

=

a

2

and dividing by

π

in (1), the

convolution property of Laplace transform

yields

1

(

s

-

a

2

)

⁢

s

↷

e

a

2

⁢

t

*

1

π

⁢

t

=

∫

0

t

e

a

2

⁢

(

t

-

u

)

⁢

1

π

⁢

u

⁢

𝑑

u

.

The

substitution

(

http://planetmath.org/ChangeOfVariableInDefiniteIntegral

)

a

2

⁢

u

=

x

2

then gives

1

(

s

-

a

2

)

⁢

s

↷

e

a

2

⁢

t

p

⁢

i

⁢

∫

0

a

⁢

t

e

-

x

2

⋅

a

x

⋅

2

⁢

x

a

2

⁢

𝑑

x

=

e

a

2

⁢

t

a

⋅

2

π

⁢

∫

0

a

⁢

t

e

-

x

2

⁢

𝑑

x

=

e

a

2

⁢

t

a

⁢

erf

⁢

a

⁢

t

.

Thus we may write the formula

ℒ

{

e

a

2

⁢

t

erf

a

t

}

=

a

(

s

-

a

2

)

⁢

s

(

s

>

a

2

)

.

(2)

Moreover, we obtain

1

(

s

+

a

)

⁢

s

=

s

-

a

(

s

-

a

2

)

⁢

s

=

1

s

-

a

2

-

a

(

s

-

a

2

)

⁢

s

↷

e

a

2

⁢

t

-

e

a

2

⁢

t

⁢

erf

⁢

a

⁢

t

=

e

a

2

⁢

t

⁢

(

1

-

erf

⁢

a

⁢

t

)

,

whence we have the other formula

ℒ

⁢

{

e

a

2

⁢

t

⁢

erfc

⁢

a

⁢

t

}

=

1

(

a

+

s

)

⁢

s

.

(3)

0.1

An improper integral

One can utilise the formula (3) for evaluating the improper integral

∫

0

∞

e

-

x

2

a

2

+

x

2

⁢

𝑑

x

.

We have

e

-

t

⁢

x

2

↶

1

s

+

x

2

(see the table of Laplace transforms (

http://planetmath.org/TableOfLaplaceTransforms

)).  Dividing this by

a

2

+

x

2

and integrating from 0 to

∞

, we can continue as follows:

∫

0

∞

e

-

t

⁢

x

2

a

2

+

x

2

⁢

𝑑

x

↶

∫

0

∞

d

⁢

x

(

a

2

+

x

2

)

⁢

(

s

+

x

2

)

=

1

s

-

a

2

⁢

∫

0

∞

(

1

a

2

+

x

2

-

1

s

+

x

2

)

⁢

𝑑

x

=

1

s

-

a

2

⁢

/

x

=

0

∞

⁡

(

1

a

⁢

arctan

⁡

x

a

-

1

s

⁢

arctan

⁡

x

s

)

=

1

s

-

a

2

⋅

π

2

⁢

(

1

a

-

1

s

)

=

π

2

⁢

a

⋅

1

(

a

+

s

)

⁢

s

↷

π

2

⁢

a

⁢

e

a

2

⁢

t

⁢

erfc

⁢

a

⁢

t

Consequently,

∫

0

∞

e

-

t

⁢

x

2

a

2

+

x

2

⁢

𝑑

x

=

π

2

⁢

a

⁢

e

a

2

⁢

t

⁢

erfc

⁢

a

⁢

t

,

and especially

∫

0

∞

e

-

x

2

a

2

+

x

2

⁢

𝑑

x

=

π

2

⁢

a

⁢

e

a

2

⁢

erfc

⁢

a

.

Title

using

convolution

to find Laplace transform

Canonical name

UsingConvolutionToFindLaplaceTransform

Date of creation

2013-03-22 18:44:05

Last modified on

2013-03-22 18:44:05

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Example

Classification

msc 26A42

Classification

msc 44A10

Related topic

ErrorFunction

Related topic

SubstitutionNotation

Related topic

IntegrationOfLaplaceTransformWithRespectToParameter

Generated on Fri Feb 9 19:54:34 2018 by

LaTeXML
continuous derivative implies bounded variation

continuous derivative implies bounded variation

Theorem

.

If the

real function

f

has

continuous

derivative

on the

interval

[

a

,

b

]

,  then on this interval,

•

f

is of

bounded variation

,

•

f

can be expressed as

difference

of two

continuously differentiable

monotonic functions

.

Proof.

1

o

¯

. The

continuous function

|

f

′

|

has its greatest value

M

on the

closed interval

[

a

,

b

]

,  i.e.

|

f

′

⁢

(

x

)

|

≦

M

∀

x

∈

[

a

,

b

]

.

Let

D

be an arbitrary

partition

of

[

a

,

b

]

,  with the points

x

0

=

a

<

x

1

<

x

2

<

…

<

x

n

-

1

<

b

=

x

n

.

Consider

f

on a

subinterval

[

x

i

-

1

,

x

i

]

.  By the

mean-value theorem

, there exists on this subinterval a point

ξ

i

such that

f

⁢

(

x

i

)

-

f

⁢

(

x

i

-

1

)

=

f

′

⁢

(

ξ

i

)

⁢

(

x

i

-

x

i

-

1

)

.  Then we get

S

D

:=

∑

i

=

1

n

|

f

⁢

(

x

i

)

-

f

⁢

(

x

i

-

1

)

|

=

∑

i

=

1

n

|

f

′

⁢

(

ξ

i

)

|

⁢

(

x

i

-

x

i

-

1

)

≦

M

⁢

∑

i

=

1

n

(

x

i

-

x

i

-

1

)

=

M

⁢

(

b

-

a

)

.

Thus the

total variation

satisfies

sup

D

⁡

{

all

⁢

S

D

⁢

’s

}

≦

M

⁢

(

b

-

a

)

<

∞

,

whence

f

is of bounded variation on the interval

[

a

,

b

]

.

2

o

¯

. Define the

functions

G

and

H

by setting

G

:=

|

f

′

|

+

f

′

2

,

H

:=

|

f

′

|

-

f

′

2

.

We see that these are non-negative and that

f

′

=

G

-

H

.  Define then the functions

g

and

h

on

[

a

,

b

]

by

g

⁢

(

x

)

:=

f

⁢

(

a

)

+

∫

a

x

G

⁢

(

t

)

⁢

𝑑

t

,

h

⁢

(

x

)

:=

∫

a

x

H

⁢

(

t

)

⁢

𝑑

t

.

Because

G

and

H

are non-negative, the functions

g

and

h

are

monotonically nondecreasing

.  We have also

(

g

-

h

)

⁢

(

x

)

=

f

⁢

(

a

)

+

∫

a

x

(

G

⁢

(

t

)

-

H

⁢

(

t

)

)

⁢

𝑑

t

=

f

⁢

(

a

)

+

∫

a

x

f

′

⁢

(

t

)

⁢

𝑑

t

=

f

⁢

(

x

)

,

whence

f

=

g

-

h

.  Since

G

and

H

are by their

definitions

continuous, the monotonic functions

g

and

h

have continuous derivatives

g

′

=

G

,

h

′

=

H

.  So

g

and

h

fulfil the requirements of the theorem.

Remark.

It may be proved that each

function of bounded variation

is difference of two

bounded

monotonically increasing

functions.

Title

continuous derivative implies bounded variation

Canonical

name

ContinuousDerivativeImpliesBoundedVariation

Date of creation

2013-03-22 17:56:32

Last modified on

2013-03-22 17:56:32

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26A45

Related topic

ProductAndQuotientOfFunctionsSum

Generated on Fri Feb 9 19:54:37 2018 by

LaTeXML
function of not bounded variation

function of not bounded variation

Example.

We show that the function

f

:

x

↦

{

x

⁢

cos

⁡

π

x

when

⁢

x

≠

0

,

0

when

⁢

x

=

0

,

which is

continuous

in the whole

ℝ

, is not of

bounded variation

on any

interval

containing the zero.

Let us take e.g. the interval

[

0

,

a

]

.  Chose a positive integer

m

such that

1

m

<

a

and the

partition

of the interval with the points

1

m

,

1

m

+

1

,

1

m

+

2

,

…

,

1

n

into the

subintervals

[

0

,

1

n

]

,

[

1

n

,

1

n

-

1

]

,

…

,

[

1

m

+

1

,

1

m

]

,

[

1

m

,

a

]

.  For each positive integer

ν

we have (see this (

http://planetmath.org/CosineAtMultiplesOfStraightAngle

))

f

⁢

(

1

ν

)

=

1

ν

⁢

cos

⁡

ν

⁢

π

=

(

-

1

)

ν

ν

.

Thus we see that the

total variation

of

f

in all partitions of

[

0

,

a

]

is at least

1

n

+

(

1

n

+

1

n

-

1

)

+

…

+

(

1

m

+

1

+

1

m

)

=

1

m

+

2

⁢

∑

ν

=

m

+

1

n

1

ν

.

Since the harmonic series

diverges

, the above sum increases to

∞

as

n

→

∞

.  Accordingly, the total variation must be

infinite

, and the function

f

is not of bounded variation on

[

0

,

a

]

.

It is not difficult to justify that

f

is of bounded variation on any finite interval that does not contain 0.

References

1

E. Lindelöf

:

Differentiali- ja integralilasku
ja sen sovellutukset III. Toinen osa.

Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).

Title

function of not bounded variation

Canonical name

FunctionOfNotBoundedVariation

Date of creation

2013-03-22 17:56:29

Last modified on

2013-03-22 17:56:29

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Example

Classification

msc 26A45

Synonym

example of unbounded variation

Synonym

function of unbounded variation

Generated on Fri Feb 9 19:54:46 2018 by

LaTeXML
total variation

total variation

Let

γ

:

[

a

,

b

]

→

X

be a function mapping an

interval

[

a

,

b

]

to a

metric space

(

X

,

d

)

. We say that

γ

is of

bounded variation

if there is a constant

M

such that, for each

partition

P

=

{

a

=

t

0

<

t

1

<

⋯

<

t

n

=

b

}

of

[

a

,

b

]

,

v

⁢

(

γ

,

P

)

=

∑

k

=

1

n

d

⁢

(

γ

⁢

(

t

k

)

,

γ

⁢

(

t

k

-

1

)

)

≤

M

.

The

total variation

V

γ

of

γ

is defined by

V

γ

=

sup

⁡

{

v

⁢

(

γ

,

P

)

:

P

is a partition of

[

a

,

b

]

}

.

It can be shown that, if

X

is either

ℝ

or

ℂ

, every

continuously differentiable

(or

piecewise

continuously differentiable) function

γ

:

[

a

,

b

]

→

X

is of bounded variation (

http://planetmath.org/ContinuousDerivativeImpliesBoundedVariation

), and

V

γ

=

∫

a

b

|

γ

′

⁢

(

t

)

|

⁢

𝑑

t

.

Also, if

γ

is of bounded variation and

f

:

[

a

,

b

]

→

X

is

continuous

, then the

Riemann-Stieltjes integral

∫

a

b

f

⁢

𝑑

γ

is finite.

If

γ

is also continuous, it is said to be a

rectifiable path

, and

V

⁢

(

γ

)

is the length of its trace.

If

X

=

ℝ

, it can be shown that

γ

is of bounded variation if and only if it is the difference of two

monotonic functions

.

Title

total variation

Canonical name

TotalVariation

Date of creation

2013-03-22 13:26:09

Last modified on

2013-03-22 13:26:09

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

8

Author

Koro (127)

Entry type

Definition

Classification

msc 26A45

Classification

msc 26B30

Related topic

BVFunction

Related topic

IntegralRepresentationOfLengthOfSmoothCurve

Related topic

OscillationOfAFunction

Defines

bounded variation

Defines

rectifiable path

Generated on Fri Feb 9 19:54:48 2018 by

LaTeXML
absolutely continuous on [0,1] versus absolutely continuous on [ε,1] for every ε>0

absolutely continuous on

[

0

,

1

]

versus absolutely continuous on

[

ε

,

1

]

for every

ε

>

0

Lemma

.

Define

f

:

R

→

R

by

f

⁢

(

x

)

=

{

0

if

⁢

x

=

0

x

⁢

sin

⁡

(

1

x

)

if

⁢

x

≠

0

.

Then

f

is

absolutely continuous

(

http://

planetmath

.org/AbsolutelyContinuousFunction2

) on

[

ε

,

1

]

for every

ε

>

0

but is not absolutely continuous on

[

0

,

1

]

.

Proof.

Note that

f

is

continuous

on

[

0

,

1

]

and

differentiable

on

(

0

,

1

]

with

f

′

⁢

(

x

)

=

sin

⁡

(

1

x

)

-

1

x

⁢

cos

⁡

(

1

x

)

.

Let

ε

>

0

. Then for all

x

∈

[

ε

,

1

]

:

|

f

′

⁢

(

x

)

|

=

|

sin

⁡

(

1

x

)

-

1

x

⁢

cos

⁡

(

1

x

)

|

≤

|

sin

⁡

(

1

x

)

|

+

|

1

x

|

⋅

|

cos

⁡

(

1

x

)

|

≤

1

+

1

ε

⋅

1

=

1

+

1

ε

Since

f

is continuous on

[

ε

,

1

]

and differentiable on

(

ε

,

1

)

, the mean value theorem (

http://planetmath.org/MeanValueTheorem

) can be applied to

f

. Thus, for every

x

1

,

x

2

∈

(

ε

,

1

)

with

x

1

≠

x

2

,

|

f

⁢

(

x

2

)

-

f

⁢

(

x

1

)

x

2

-

x

1

|

≤

1

+

1

ε

. This yields

|

f

⁢

(

x

2

)

-

f

⁢

(

x

1

)

|

≤

(

1

+

1

ε

)

⁢

|

x

2

-

x

1

|

, which also holds when

x

1

=

x

2

. Thus,

f

is

Lipschitz

on

(

ε

,

1

)

. It follows that

f

is absolutely continuous on

[

ε

,

1

]

.

On the other hand, it can be verified that

f

is not of

bounded variation

on

[

0

,

1

]

and thus cannot be absolutely continuous on

[

0

,

1

]

.
∎

Title

absolutely continuous on

[

0

,

1

]

versus absolutely continuous on

[

ε

,

1

]

for every

ε

>

0

Canonical name

AbsolutelyContinuousOn01VersusAbsolutelyContinuousOnvarepsilon1ForEveryvarepsilon0

Date of creation

2013-03-22 16:12:19

Last modified on

2013-03-22 16:12:19

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

11

Author

Wkbj79 (1863)

Entry type

Example

Classification

msc 26A46

Classification

msc 26B30

Generated on Fri Feb 9 19:54:51 2018 by

LaTeXML
almost convex

almost convex

A real function,

f

:

ℝ

→

ℝ

, is said to be

almost convex if

f

⁢

(

z

)

≤

max

⁡

{

f

⁢

(

x

)

,

f

⁢

(

y

)

}

,

∀

x

,

y

,

z

⁢

x

≤

z

≤

y

It is easy to see that a

convex function

is an almost convex function.

Title

almost convex

Canonical name

AlmostConvex

Date of creation

2013-03-22 15:20:55

Last modified on

2013-03-22 15:20:55

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

5

Author

georgiosl (7242)

Entry type

Definition

Classification

msc 26A51

Generated on Fri Feb 9 19:54:57 2018 by

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logarithmically convex function

logarithmically convex function

Definition

.

A

function

f

:

[

a

,

b

]

→

ℝ

such that

f

⁢

(

x

)

>

0

for all

x

is said
to be

logarithmically convex

if

log

⁡

f

⁢

(

x

)

is a

convex function

.

It is easy to see that a

logarithmically convex function

is a convex function, but the converse is not true. For example

f

⁢

(

x

)

=

x

2

is a convex function, but

log

⁡

f

⁢

(

x

)

=

log

⁡

x

2

=

2

⁢

log

⁡

x

is not a convex function and thus

f

⁢

(

x

)

=

x

2

is not logarithmically convex. On the other hand

e

x

2

is logarithmically convex since

log

⁡

e

x

2

=

x

2

is convex. A less trivial example of a logarithmically convex function is the

gamma function

, if restricted to the positive reals.

The definition is easily extended to functions

f

:

U

⊂

ℝ

→

ℝ

, for any connected set

U

(where still we have

f

>

0

), in the obvious way. Such a function is logarithmically convex if it is logarithmically convex on all

intervals

[

a

,

b

]

⊂

U

.

References

1

John B. Conway.
.
Springer-Verlag, New York, New York, 1978.

Title

logarithmically convex function

Canonical name

LogarithmicallyConvexFunction

Date of creation

2013-03-22 14:13:33

Last modified on

2013-03-22 14:13:33

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

7

Author

jirka (4157)

Entry type

Definition

Classification

msc 26A51

Synonym

logarithmically convex

Synonym

log-convex function

Synonym

log-convex

Synonym

log convex function

Synonym

log convex

Related topic

ConvexFunction

Related topic

BohrMollerupTheorem

Related topic

HadamardThreeCircleTheorem

Generated on Fri Feb 9 19:55:13 2018 by

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proof of {f⁢(t)-f⁢(s)}/{t-s}≤{f⁢(u)-f⁢(s)}/{u-s}≤{f⁢(u)-f⁢(t)}/{u-t} for convex f

proof of

f

⁢

(

t

)

-

f

⁢

(

s

)

t

-

s

≤

f

⁢

(

u

)

-

f

⁢

(

s

)

u

-

s

≤

f

⁢

(

u

)

-

f

⁢

(

t

)

u

-

t

for convex

f

We will prove

f

⁢

(

t

)

-

f

⁢

(

s

)

t

-

s

≤

f

⁢

(

u

)

-

f

⁢

(

s

)

u

-

s

.

(1)

The proof of the right-most

inequality

is

similar

.

Suppose (

1

) does not hold. Then for some

s

,

t

,

u

,

f

⁢

(

t

)

-

f

⁢

(

s

)

t

-

s

>

f

⁢

(

u

)

-

f

⁢

(

s

)

u

-

s

.

(2)

This inequality is just the statement of the slope of the

line
segment

A

⁢

B

¯

,

A

=

(

t

,

f

⁢

(

t

)

)

,

B

=

(

s

,

f

⁢

(

s

)

)

, being larger than
the slope of the

segment

C

⁢

B

¯

,

C

=

(

u

,

f

⁢

(

u

)

)

. Since

t

is
between

s

and

u

, and

f

is

continuous

, this implies

f

⁢

(

t

)

>

h

⁢

(

x

)

=

f

⁢

(

u

)

-

f

⁢

(

s

)

u

-

s

⁢

(

x

-

s

)

+

f

⁢

(

s

)

,

(3)

s

<

x

<

u

. This contradicts convexity of

f

on

(

a

,

b

)

. Hence,
(

2

) is false and (

1

) follows.

Note that we have tacitly use the fact that

x

=

λ

⁢

u

+

(

1

-

λ

)

⁢

s

and

h

⁢

(

x

)

=

λ

⁢

f

⁢

(

u

)

+

(

1

-

λ

)

⁢

f

⁢

(

s

)

for some

λ

.

Title

proof of

f

⁢

(

t

)

-

f

⁢

(

s

)

t

-

s

≤

f

⁢

(

u

)

-

f

⁢

(

s

)

u

-

s

≤

f

⁢

(

u

)

-

f

⁢

(

t

)

u

-

t

for convex

f

Canonical name

ProofOffracftfstsleqfracfufsusleqfracfuftutForConvexF

Date of creation

2013-03-22 18:25:34

Last modified on

2013-03-22 18:25:34

Owner

yesitis (13730)

Last modified by

yesitis (13730)

Numerical id

6

Author

yesitis (13730)

Entry type

Proof

Classification

msc 26A51

Generated on Fri Feb 9 19:55:16 2018 by

LaTeXML
{f⁢(t)-f⁢(s)}/{t-s}≤{f⁢(u)-f⁢(s)}/{u-s}≤{f⁢(u)-f⁢(t)}/{u-t} for convex f

f

⁢

(

t

)

-

f

⁢

(

s

)

t

-

s

≤

f

⁢

(

u

)

-

f

⁢

(

s

)

u

-

s

≤

f

⁢

(

u

)

-

f

⁢

(

t

)

u

-

t

for convex

f

If

f

is convex in

(

a

,

b

)

and if

a

<

s

<

t

<

u

<

b

, then

f

⁢

(

t

)

-

f

⁢

(

s

)

t

-

s

≤

f

⁢

(

u

)

-

f

⁢

(

s

)

u

-

s

≤

f

⁢

(

u

)

-

f

⁢

(

t

)

u

-

t

.

(1)

Title

f

⁢

(

t

)

-

f

⁢

(

s

)

t

-

s

≤

f

⁢

(

u

)

-

f

⁢

(

s

)

u

-

s

≤

f

⁢

(

u

)

-

f

⁢

(

t

)

u

-

t

for convex

f

Canonical name

fracftfstsleqfracfufsusleqfracfuftutForConvexF

Date of creation

2013-03-22 18:25:32

Last modified on

2013-03-22 18:25:32

Owner

yesitis (13730)

Last modified by

yesitis (13730)

Numerical id

4

Author

yesitis (13730)

Entry type

Theorem

Classification

msc 26A51

Generated on Fri Feb 9 19:55:17 2018 by

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Appell sequence

Appell sequence

The

sequence

of

polynomials

⟨

P

0

⁢

(

x

)

,

P

1

⁢

(

x

)

,

P

2

⁢

(

x

)

,

…

⟩

(1)

with

P

n

(

x

)

:=

a

x

n

(

n

=

0

,

1

,

2

,

…

)

is a

geometric sequence

and has trivially the

properties

P

n

′

(

x

)

=

n

P

n

-

1

(

x

)

(

n

=

0

,

1

,

2

,

…

)

(2)

and

P

n

⁢

(

x

+

y

)

=

∑

k

=

0

n

(

n

k

)

⁢

P

k

⁢

(

x

)

⁢

y

n

-

k

(3)

(see the

binomial theorem

).  There are also other polynomial sequences (1) having these properties, for example the sequences of the

Bernoulli polynomials

, the

Euler polynomials

and the

Hermite polynomials

.  Such sequences are called

Appell sequences

and their members are sometimes characterised as

generalised

monomials

, because of resemblance to the geometric sequence.

Given the first member

P

0

⁢

(

x

)

, which must be a nonzero constant polynomial, of any Appell sequence (1), the other members are determined recursively by

P

n

⁢

(

x

)

=

∫

0

x

P

n

-

1

⁢

(

t

)

⁢

𝑑

t

+

C

n

(4)

as one gives the values of the

constants of integration

C

n

; thus the number sequence

⟨

C

0

,

C

1

,

C

2

,

…

⟩

determines the Appell sequence uniquely.  So the choice

C

1

=

C

2

=

…

:=

0

yields a geometric sequence and the choice

C

n

:=

B

n

for

n

=

0

,

1

,

2

,

…

the Bernoulli polynomials (

http://planetmath.org/BernoulliPolynomialsAndNumbers

).

The properties (2) and (3) are

equivalent

(

http://planetmath.org/Equivalent3

).  The

implication

(

2

)

⇒

(

3

)

may be shown by

induction

(

http://planetmath.org/Induction

) on

n

.  The reverse
implication is gotten by using the

definition

of

derivative

:

P

n

′

⁢

(

x

)

=

lim

Δ

⁢

x

→

0

⁡

P

n

⁢

(

x

+

Δ

⁢

x

)

-

P

n

⁢

(

x

)

Δ

⁢

x

=

lim

Δ

⁢

x

→

0

⁡

P

0

⁢

(

x

)

⁢

Δ

⁢

x

n

+

(

n

1

)

⁢

P

1

⁢

(

x

)

⁢

Δ

⁢

x

n

-

1

+

…

+

(

n

n

-

1

)

⁢

P

n

-

1

⁢

(

x

)

⁢

Δ

⁢

x

Δ

⁢

x

=

(

n

n

-

1

)

⁢

P

n

-

1

⁢

(

x

)

=

n

⁢

P

n

-

1

⁢

(

x

)

.

See also

http://en.wikipedia.org/wiki/Appell_polynomials

Wiki.

Title

Appell sequence

Canonical

name

AppellSequence

Date of creation

2014-05-23 17:08:17

Last modified on

2014-05-23 17:08:17

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

15

Author

pahio (2872)

Entry type

Definition

Classification

msc 26A99

Classification

msc 12-00

Classification

msc 11C08

Classification

msc 11B83

Classification

msc 11B68

Related topic

BinomialCoefficient

Related topic

HermitePolynomials

Related topic

HermiteNumbers

Defines

generalized monomials

Generated on Fri Feb 9 19:55:20 2018 by

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differential equations for x^x

differential equations for

x

x

In this entry, we will derive

differential equations

satisfied by the

function

x

x

.

1

1

In this entry, we restrict

x

, and hence

x

x

to be strictly positive
real numbers, hence it is justified to divide by these quantities.

We begin by computing its

derivative

. To do this, we write

x

x

=

e

x

⁢

log

⁡

x

and
apply the

chain rule

:

d

d

⁢

x

⁢

x

x

=

d

d

⁢

x

⁢

e

x

⁢

log

⁡

x

=

e

x

⁢

log

⁡

x

⁢

(

1

+

log

⁡

x

)

=

x

x

⁢

(

1

+

log

⁡

x

)

Set

y

=

x

x

. Then we have

y

′

/

y

=

1

+

log

⁡

x

. Taking
another derivative, we have

d

d

⁢

x

(

y

′

y

)

=

1

x

⋅

Applying the

quotient rule

and simplifying, this becomes

y

⁢

y

′′

-

(

y

′

)

2

-

y

2

/

x

=

0

.

It is also possible to derive an equation in which

x

does not appear.
We start by noting that, if

z

=

1

/

x

, then

z

′

+

z

2

=

0

. If, as
above,

y

=

x

x

, we have

(

d

/

d

⁢

x

)

⁢

(

y

′

/

y

)

=

z

. Combining equations,

d

2

d

⁢

x

2

⁢

(

y

′

y

)

+

(

d

d

⁢

x

⁢

(

y

′

y

)

)

2

=

0

;

applying the quotient rule and simplifying,

y

3

⁢

y

′′′

-

y

2

⁢

(

y

′′

)

2

+

2

⁢

y

⁢

(

y

′

)

2

⁢

y

′′

-

3

⁢

y

2

⁢

y

′

⁢

y

′′

-

(

y

′

)

4

+

2

⁢

y

⁢

(

y

′

)

3

=

0

.

Title

differential equations for

x

x

Canonical name

DifferentialEquationsForXx

Date of creation

2013-03-22 17:24:37

Last modified on

2013-03-22 17:24:37

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Derivation

Classification

msc 26A99

Generated on Fri Feb 9 19:55:22 2018 by

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elementary function

elementary function

An

elementary function

is a

real function

(of one variable) that can be constructed by a finite number of elementary operations (

addition

,

subtraction

,

multiplication

and

division

) and

compositions

from

constant functions

, the

identity function

(

x

↦

x

),

algebraic functions

,

exponential functions

,

logarithm functions

,

trigonometric functions

and

cyclometric functions

.

Examples

•

Consequently, the

polynomial functions

, the

absolute value

|

x

|

=

x

2

,  the

triangular-wave function

arcsin

⁡

(

sin

⁡

x

)

, the

power function

x

π

=

e

π

⁢

ln

⁡

x

and the

function

x

x

=

e

x

⁢

ln

⁡

x

are elementary functions (N.B., the real power functions entail that

x

>

0

).

•

ζ

⁢

(

x

)

:=

∑

n

=

1

∞

1

n

x

and

Li

⁡

x

:=

∫

2

x

d

⁢

t

ln

⁡

t

are not elementary functions — it may be shown that they can not be expressed is such a way which is required in the definition.

Title

elementary function

Canonical name

ElementaryFunction

Date of creation

2013-03-22 14:46:29

Last modified on

2013-03-22 14:46:29

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

18

Author

pahio (2872)

Entry type

Definition

Classification

msc 26A99

Related topic

RiemannZetaFunction

Related topic

LogarithmicIntegral

Related topic

AlgebraicFunction

Related topic

TableOfMittagLefflerPartialFractionExpansions

Generated on Fri Feb 9 19:55:24 2018 by

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