aarjaneiro/planetmath huggingface dataset:
Dirichlet kernel

Dirichlet kernel

The

Dirichlet

D

n

of order

n

is defined as

D

n

⁢

(

t

)

=

∑

k

=

-

n

n

e

i

⁢

k

⁢

t

.

It can be represented as

D

n

⁢

(

t

)

=

sin

⁡

(

n

+

1

2

)

⁢

t

sin

⁡

t

2

.

Proof:

It is

∑

k

=

-

n

n

e

i

⁢

k

⁢

t

=

e

-

i

⁢

n

⁢

t

⁢

1

-

e

i

⁢

(

2

⁢

n

+

1

)

⁢

t

1

-

e

i

⁢

t

=

e

i

⁢

(

n

+

1

2

)

⁢

t

-

e

-

i

⁢

(

n

+

1

2

)

⁢

t

e

i

⁢

t

2

-

e

-

i

⁢

t

2

=

sin

⁡

(

n

+

1

2

)

⁢

t

sin

⁡

t

2

.

□

The

Dirichlet kernel

arises in the

analysis

of

periodic functions

because for any

function

f

of period

2

⁢

π

, the

convolution

of

D

N

and

f

results in the Fourier-series approximation of order

n

:

(

D

N

*

f

)

⁢

(

x

)

=

1

2

⁢

π

⁢

∫

-

π

π

f

⁢

(

y

)

⁢

D

n

⁢

(

x

-

y

)

⁢

𝑑

y

=

∑

k

=

-

n

n

f

^

⁢

(

k

)

⁢

e

i

⁢

k

⁢

x

.

Title

Dirichlet kernel

Canonical name

DirichletKernel

Date of creation

2013-03-22 14:11:53

Last modified on

2013-03-22 14:11:53

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

10

Author

mathwizard (128)

Entry type

Definition

Classification

msc 26A30

Related topic

ExampleOfTelescopingSum

Generated on Fri Feb 9 19:51:12 2018 by

LaTeXML
example of a strictly increasing quasisymmetric singular function

example of a strictly increasing quasisymmetric singular function

An example of a

strictly increasing

quasisymmetric function that also a

purely singular function

can be defined as:

f

⁢

(

x

)

=

lim

k

→

∞

⁡

∫

0

x

∏

i

=

1

k

(

1

+

λ

⁢

cos

⁡

n

i

⁢

s

)

⁢

d

⁢

s

,

where

0

<

λ

<

1

and carefully picked

n

i

.
We can pick the

n

i

such that

n

i

+

1

is strictly

greater

then

∑

j

=

1

i

n

j

. However if we pick the

λ

and

n

i

more carefully, we can construct

functions

with the quasisymmetricity
constant as close to 1 as we want. That is, we can construct functions such that

1

M

≤

f

⁢

(

x

+

t

)

-

f

⁢

(

x

)

f

⁢

(

x

)

-

f

⁢

(

x

-

t

)

≤

M

for all

x

and

t

where

M

is as close to 1 as we want. If

M

=

1

note that
the function must be a

straight line

.

It is also possible from this to construct a

quasiconformal mapping

of the

upper half plane

to itself by extending this function to the whole

real line

and then using the

Beurling-Ahlfors quasiconformal extension

. Then we’d have a quasiconformal mapping such that its boundary correspondence would be a purely singular function.

For more detailed explanation, and
proof (it is too long to reproduce here) see bibliography.

Bibliography

•

A. Beurling, L. V. Ahlfors. .

Acta Math.

, 96:125-142,
1956.

•

J. Lebl.

. . Also available at

http://www.jirka.org/thesis.pdf

http://www.jirka.org/thesis.pdf

Title

example of a strictly increasing

quasisymmetric

singular function

Canonical name

ExampleOfAStrictlyIncreasingQuasisymmetricSingularFunction

Date of creation

2013-03-22 14:10:37

Last modified on

2013-03-22 14:10:37

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Example

Classification

msc 26A30

Generated on Fri Feb 9 19:51:13 2018 by

LaTeXML
Fejer kernel

Fejer kernel

The

Fejer kernel

F

n

of order

n

is defined as

F

n

⁢

(

t

)

=

1

n

⁢

∑

k

=

0

n

-

1

D

k

⁢

(

t

)

,

where

D

n

is the

Dirichlet kernel

of order

n

. The Fejer kernel can be written as

F

n

⁢

(

t

)

=

1

n

⁢

(

sin

⁡

n

⁢

t

2

sin

⁡

t

2

)

2

.

(1)

Proof:

Since

D

n

⁢

(

t

)

=

sin

⁡

(

(

n

+

1

2

)

⁢

t

)

sin

⁡

t

2

we have

sin

⁡

t

2

⁢

D

n

⁢

(

t

)

=

sin

⁡

(

(

n

+

1

2

)

⁢

t

)

.

Therefore

n

⁢

sin

2

⁡

t

2

⁢

F

n

⁢

(

t

)

=

∑

k

=

0

n

-

1

sin

⁡

(

(

k

+

1

2

)

⁢

t

)

⁢

sin

⁡

t

2

=

1

2

∑

k

=

0

n

-

1

(

cos

k

t

-

cos

(

(

k

+

1

)

t

)

=

1

2

⁢

(

1

-

cos

⁡

n

⁢

t

)

=

sin

2

⁡

n

⁢

t

2

.

From this follows equation (

1

).

Figure 1:

Graphs of some Fejer kernels

Title

Fejer kernel

Canonical name

FejerKernel

Date of creation

2013-03-22 14:11:56

Last modified on

2013-03-22 14:11:56

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

8

Author

mathwizard (128)

Entry type

Definition

Classification

msc 26A30

Related topic

DiracSequence

Generated on Fri Feb 9 19:51:16 2018 by

LaTeXML
Landau kernel

Landau kernel

For

k

∈

ℕ

the

Landau kernel

L

k

⁢

(

t

)

is defined as

L

k

=

{

1

c

k

⁢

(

1

-

t

2

)

k

if

⁢

t

∈

[

-

1

,

1

]

0

otherwise

with

c

k

:=

∫

-

1

1

(

1

-

t

2

)

k

⁢

𝑑

t

.

L

k

is nonnegative and

continuous

on

ℝ

. Due to the choice of

c

k

we have

∫

-

∞

∞

L

k

⁢

(

t

)

⁢

𝑑

t

=

1

.

Also we have for all

positive

, real

r

:

∫

ℝ

\

[

-

r

,

r

]

L

k

⁢

(

t

)

⁢

𝑑

t

≤

2

c

k

⁢

∫

r

1

(

1

-

t

2

)

k

⁢

𝑑

t

≤

(

k

+

1

)

⁢

(

1

-

r

2

)

k

.

Therefore

(

L

k

)

k

∈

ℕ

is a

Dirac sequence

.

Title

Landau kernel

Canonical

name

LandauKernel

Date of creation

2013-03-22 14:11:38

Last modified on

2013-03-22 14:11:38

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

7

Author

mathwizard (128)

Entry type

Definition

Classification

msc 26A30

Generated on Fri Feb 9 19:51:18 2018 by

LaTeXML
singular function

singular function

Definition

.

A

monotone

, non-constant,

function

f

:

[

a

,

b

]

→

ℝ

is said to be a

singular function

(or a

purely singular function

) if

f

′

⁢

(

x

)

=

0

almost everywhere.

It is easy to see that a singular function cannot be

absolutely continuous

(

http://planetmath.org/AbsolutelyContinuousFunction2

):
If an

absolutely continuous function

f

:

[

a

,

b

]

→

ℝ

satisfies

f

′

⁢

(

x

)

=

0

almost everywhere, then it must be constant.

An example of such a function is the famous

Cantor function

.
While this is not a

strictly increasing function

, there also do exist singular functions which are in fact strictly increasing, and even more amazingly functions that are

quasisymmetric

(see attached example).

Theorem

.

Any

monotone increasing

function can be written as a sum of an absolutely continuous function and a singular function.

References

1

H. L. Royden.

. Prentice-Hall, Englewood Cliffs, New Jersey, 1988

Title

singular function

Canonical name

SingularFunction

Date of creation

2013-03-22 14:08:05

Last modified on

2013-03-22 14:08:05

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

11

Author

jirka (4157)

Entry type

Definition

Classification

msc 26A30

Synonym

purely singular function

Related topic

AbsolutelyContinuousFunction2

Related topic

CantorFunction

Related topic

CantorSet

Related topic

AbsolutelyContinuousFunction2

Defines

singular function

Generated on Fri Feb 9 19:51:21 2018 by

LaTeXML
fractional integration

fractional integration

The basic idea of ”Riemann-Liouville” type

fractional integration

comes from the following observation:

Given any integrable function

f

:

ℝ

↦

ℝ

in one

variable

, we have the following Cauchy

Integration Formula

:

D

-

n

⁢

(

f

)

⁢

(

x

)

=

∫

t

n

=

0

x

𝑑

t

n

⁢

…

⁢

∫

t

1

=

0

t

2

f

⁢

(

t

1

)

⁢

𝑑

t

1

=

1

(

n

-

1

)

!

⁢

∫

t

=

0

x

f

⁢

(

t

)

⁢

(

x

-

t

)

n

-

1

⁢

𝑑

t

when switching the index from integer

n

to non-integer

α

gives the ideas of the following

definitions

:

Definition 1:

Left-Hand Riemann-Liouville Integration

I

L

α

⁢

(

f

)

⁢

(

s

,

t

)

=

1

Γ

⁢

(

α

)

⁢

∫

u

=

s

t

f

⁢

(

u

)

⁢

(

t

-

u

)

α

-

1

⁢

𝑑

u

=

∫

u

=

s

t

f

⁢

(

u

)

⁢

𝑑

g

t

α

⁢

(

u

)

where

g

t

α

⁢

(

u

)

=

t

α

-

(

t

-

u

)

α

Γ

⁢

(

α

+

1

)

Definition 2:

Right-Hand Riemann-Liouville Integration

I

R

α

⁢

(

f

)

⁢

(

s

,

t

)

=

1

Γ

⁢

(

α

)

⁢

∫

u

=

s

t

f

⁢

(

u

)

⁢

(

u

-

s

)

α

-

1

⁢

𝑑

u

=

∫

u

=

s

t

f

⁢

(

u

)

⁢

𝑑

h

t

α

⁢

(

u

)

where

h

t

α

⁢

(

u

)

=

s

α

+

(

u

-

s

)

α

Γ

⁢

(

α

+

1

)

Definition 3:

Riesz Potential

I

C

α

⁢

(

f

)

⁢

(

s

,

t

;

p

)

=

1

Γ

⁢

(

α

)

⁢

∫

u

=

s

t

f

⁢

(

u

)

⁢

|

u

-

p

|

α

-

1

⁢

𝑑

u

=

∫

u

=

s

t

f

⁢

(

u

)

⁢

𝑑

r

p

α

⁢

(

u

)

where

r

p

α

⁢

(

u

)

=

p

α

+

sign

⁢

(

u

-

p

)

⁢

|

u

-

p

|

α

Γ

⁢

(

α

+

1

)

,

sign

⁢

(

x

)

=

1

for

x

>

0

,

sign

⁢

(

x

)

=

0

for

x

=

0

,

sign

⁢

(

x

)

=

-

1

for

x

<

0

and

Γ

⁢

(

x

)

is the

gamma function

of

x

Title

fractional integration

Canonical

name

FractionalIntegration

Date of creation

2013-03-22 16:17:47

Last modified on

2013-03-22 16:17:47

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

17

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 26A33

Synonym

fractional integral

Generated on Fri Feb 9 19:51:23 2018 by

LaTeXML
a special case of partial integration

a special case of partial integration

In determining the

antiderivative

of a

transcendental

(

http://planetmath.org/AlgebraicFunction

)

function

U

whose

derivative

U

′

is

algebraic

(

http://planetmath.org/AlgebraicFunction

), the result can be obtained when choosing in the formula

∫

U

⁢

V

′

⁢

𝑑

x

=

U

⁢

V

-

∫

V

⁢

U

′

⁢

𝑑

x

of

integration by parts

V

′

≡

1

;  then one has

∫

U

⁢

𝑑

x

=

∫

U

⋅

1

⁢

𝑑

x

=

U

⋅

x

-

∫

x

⋅

U

′

⁢

𝑑

x

.

The functions

U

in question are mainly the

logarithm

(

http://planetmath.org/NaturalLogarithm2

), the

cyclometric functions

and the

area functions

.

Examples.

1.

∫

ln

⁡

x

⁢

d

⁢

x

=

x

⁢

ln

⁡

x

-

∫

x

⋅

1

x

⁢

𝑑

x

=

x

⁢

ln

⁡

x

-

x

+

C

2.

∫

arcsin

⁡

x

⁢

d

⁢

x

=

x

⁢

arcsin

⁡

x

-

∫

x

⋅

1

1

-

x

2

⁢

𝑑

x

=

x

⁢

arcsin

⁡

x

+

1

2

⁢

∫

-

2

⁢

x

1

-

x

2

⁢

𝑑

x

=

x

⁢

arcsin

⁡

x

+

1

-

x

2

+

C

3.

∫

arctan

⁡

x

⁢

d

⁢

x

=

x

⁢

arctan

⁡

x

-

∫

x

⋅

1

1

+

x

2

⁢

𝑑

x

=

x

⁢

arctan

⁡

x

-

1

2

⁢

∫

2

⁢

x

1

+

x

2

⁢

𝑑

x

=

x

⁢

arctan

⁡

x

-

1

2

⁢

ln

⁡

(

1

+

x

2

)

+

C

=

x

⁢

arctan

⁡

x

-

ln

⁡

1

+

x

2

+

C

4.

∫

arcosh

⁡

x

⁢

d

⁢

x

=

x

⁢

arcosh

⁡

x

-

∫

x

⋅

1

x

2

-

1

⁢

𝑑

x

=

x

⁢

arcosh

⁡

x

-

x

2

-

1

+

C

The choice

V

′

≡

1

works as well in such cases as

∫

(

ln

⁡

x

)

2

⁢

𝑑

x

and

∫

ln

⁡

(

ln

⁡

x

)

⁢

𝑑

x

,  giving respectively

x

⁢

(

(

ln

⁡

x

)

2

-

2

⁢

ln

⁡

x

+

2

)

+

C

and

x

⁢

ln

⁡

(

ln

⁡

x

)

-

Li

⁡

x

+

C

(see

logarithmic integral

). Also

∫

(

arcsin

⁡

x

)

2

⁢

𝑑

x

, requiring two integrations by parts, and giving the result

x

⁢

(

arcsin

⁡

x

)

2

+

2

⁢

1

-

x

2

⁢

arcsin

⁡

x

-

2

⁢

x

+

C

.

Title

a

special case of partial integration

Canonical name

ASpecialCaseOfPartialIntegration

Date of creation

2013-03-22 17:38:35

Last modified on

2013-03-22 17:38:35

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Feature

Classification

msc 26A36

Related topic

IntegralTables

Generated on Fri Feb 9 19:51:39 2018 by

LaTeXML
antiderivative

antiderivative

Let

I

be an

open interval

of

ℝ

and

f

:

I

⟶

ℝ

a real function.

A

function

F

:

I

⟶

ℝ

is called an

antiderivative

or a

primitive

of

f

if

F

is

differentiable

and its

derivative

is equal to

f

, i.e.

F

′

⁢

(

x

)

=

f

⁢

(

x

)

for all

⁢

x

∈

I

.

Note that there are an

infinite

number of antiderivatives for any function

f

since any

constant

can be added or subtracted from any valid antiderivative to yield another equally valid antiderivative.

To account for this, we express the

general antiderivative

, or

indefinite integral

, as follows:

∫

f

⁢

(

x

)

⁢

𝑑

x

=

F

+

C

where

C

is an arbitrary constant called the

constant of integration

. The

d

⁢

x

portion means “with respect to

x

”, because after all, our functions

F

and

f

are functions of

x

.

There is no loss in generality with this notation since in fact

all

antiderivatives of

f

take this form as the following

theorem

demonstrates:

Theorem.

Let

F

,

G

be two antiderivatives of a given function

f

defined on an open interval

I

. Then

F

-

G

=

𝑐𝑜𝑛𝑠𝑡

.

Proof.

Since

F

′

⁢

(

x

)

=

f

⁢

(

x

)

and

G

′

⁢

(

x

)

=

f

⁢

(

x

)

,  we have

F

′

⁢

(

x

)

-

G

′

⁢

(

x

)

=

0

on the whole

I

.  Thus, by the

fundamental theorem of integral calculus

,

F

⁢

(

x

)

-

G

⁢

(

x

)

=

const

.

□

This is no longer true if the domain of the function

f

is not an open interval (is not

connected

). For that scenario, the following more general result holds:

Theorem.

Let

U

⊂

R

be an

open set

(not necessarily an interval). Suppose

F

,

G

are antiderivatives of a given function

f

:

U

⟶

R

. Then

F

-

G

is constant in each

connected component

of

U

(each interval in

U

).

For example, consider the function

f

:

ℝ

∖

{

0

}

⟶

ℝ

given by

f

⁢

(

x

)

=

1

x

. Notice that the domain of

f

is not an interval, but the union of the

disjoint

intervals

(

-

∞

,

0

)

and

(

0

,

+

∞

)

. Then, all the antiderivatives of

f

take the form

{

log

⁡

(

-

x

)

+

C

1

,

if

⁢

x

<

0

log

⁡

(

x

)

+

C

2

,

if

⁢

x

>

0

0.1

Remarks

•

For complex functions, the definition of antiderivative is exactly the same and the above results also hold (one just needs to consider “connected open subsets” instead of “open intervals”).

Title

antiderivative

Canonical name

Antiderivative

Date of creation

2013-03-22 12:14:55

Last modified on

2013-03-22 12:14:55

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

21

Author

asteroid (17536)

Entry type

Definition

Classification

msc 26A36

Synonym

general antiderivative

Synonym

indefinite integral

Synonym

primitive

Related topic

AntiderivativeOfComplexFunction

Defines

constant of integration

Generated on Fri Feb 9 19:51:41 2018 by

LaTeXML
general formulas for integration

general formulas for integration

1.

∫

f

′

⁢

(

x

)

⁢

𝑑

x

=

f

⁢

(

x

)

+

C

2.

∫

λ

⁢

𝑑

x

=

λ

⁢

x

+

C

3.

∫

λ

⁢

f

⁢

(

x

)

⁢

𝑑

x

=

λ

⁢

∫

f

⁢

(

x

)

⁢

𝑑

x

4.

∫

(

f

⁢

(

x

)

+

g

⁢

(

x

)

)

⁢

𝑑

x

=

∫

f

⁢

(

x

)

⁢

𝑑

x

+

∫

g

⁢

(

x

)

⁢

𝑑

x

5.

∫

f

⁢

(

x

)

⁢

g

′

⁢

(

x

)

⁢

𝑑

x

=

f

⁢

(

x

)

⁢

g

⁢

(

x

)

-

∫

g

⁢

(

x

)

⁢

f

′

⁢

(

x

)

⁢

𝑑

x

6.

∫

g

⁢

(

f

⁢

(

x

)

)

⁢

f

′

⁢

(

x

)

⁢

𝑑

x

=

G

⁢

(

f

⁢

(

x

)

)

+

C

if

G

′

⁢

(

t

)

=

g

⁢

(

t

)

7.

∫

[

f

⁢

(

x

)

]

r

⁢

f

′

⁢

(

x

)

⁢

𝑑

x

=

1

r

+

1

⁢

[

f

⁢

(

x

)

]

r

+

1

+

C

for

r

≠

-

1

8.

∫

f

′

⁢

(

x

)

f

⁢

(

x

)

⁢

𝑑

x

=

ln

⁡

|

f

⁢

(

x

)

|

+

C

9.

∫

e

f

⁢

(

x

)

⁢

f

′

⁢

(

x

)

⁢

𝑑

x

=

e

f

⁢

(

x

)

+

C

10.

∫

f

⁢

(

x

)

(

f

⁢

(

x

)

+

a

)

⁢

(

f

⁢

(

x

)

+

b

)

⁢

𝑑

x

=

a

a

-

b

⁢

∫

d

⁢

x

f

⁢

(

x

)

+

a

-

b

a

-

b

⁢

∫

d

⁢

x

f

⁢

(

x

)

+

b

11.

∫

sin

⁡

(

ω

⁢

x

+

φ

)

⁢

𝑑

x

=

-

cos

⁡

(

ω

⁢

x

+

φ

)

ω

+

C

12.

∫

cos

⁡

(

ω

⁢

x

+

φ

)

⁢

𝑑

x

=

sin

⁡

(

ω

⁢

x

+

φ

)

ω

+

C

13.

∫

sinh

⁡

(

ω

⁢

x

+

φ

)

⁢

𝑑

x

=

cosh

⁡

(

ω

⁢

x

+

φ

)

ω

+

C

14.

∫

cosh

⁡

(

ω

⁢

x

+

φ

)

⁢

𝑑

x

=

sinh

⁡

(

ω

⁢

x

+

φ

)

ω

+

C

15.

∫

a

⁢

x

+

b

⁢

𝑑

x

=

2

3

⁢

a

⁢

(

a

⁢

x

+

b

)

⁢

a

⁢

x

+

b

+

C

16.

∫

a

⁢

x

2

+

b

⁢

𝑑

x

=

x

2

⁢

a

⁢

x

2

+

b

+

b

2

⁢

a

⁢

ln

⁡

(

x

⁢

a

+

a

⁢

x

2

+

b

)

+

C

17.

∫

sin

n

⁡

x

⁢

cos

m

⁡

x

⁢

d

⁢

x

=

-

sin

n

-

1

⁡

x

⁢

cos

m

+

1

⁡

x

m

+

n

+

n

-

1

m

+

n

⁢

∫

sin

n

-

2

⁡

x

⁢

cos

m

⁡

x

⁢

d

⁢

x

18.

∫

sin

n

⁡

x

⁢

cos

m

⁡

x

⁢

d

⁢

x

=

sin

n

+

1

⁡

x

⁢

cos

m

-

1

⁡

x

m

+

n

+

m

-

1

m

+

n

⁢

∫

sin

n

⁡

x

⁢

cos

m

-

2

⁡

x

⁢

d

⁢

x

Some series-formed

antiderivatives

:

∫

f

⁢

(

x

)

⁢

𝑑

x

=

C

+

f

⁢

(

0

)

⁢

x

+

f

′

⁢

(

0

)

2

!

⁢

x

2

+

f

′′

⁢

(

0

)

3

!

⁢

x

3

+

…

∫

f

(

x

)

d

x

=

C

+

x

f

(

x

)

-

x

2

2

!

f

′

(

x

)

+

x

3

3

!

f

′′

(

x

)

-

+

…

∫

U

V

d

x

=

U

V

(

-

1

)

-

U

′

V

(

-

2

)

+

U

′′

V

(

-

3

)

-

+

…

=

∑

n

=

0

∞

(

-

1

)

n

U

(

n

)

V

(

-

n

-

1

)

The

derivatives

with negative order (

http://

planetmath

.org/HigherOrderDerivatives

) that

V

has been integrated repeatedly.

Title

general formulas for integration

Canonical name

GeneralFormulasForIntegration

Date of creation

2013-03-22 17:39:31

Last modified on

2013-03-22 17:39:31

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

16

Author

pahio (2872)

Entry type

Topic

Classification

msc 26A36

Synonym

integration formulas

Related topic

TableOfDerivatives

Related topic

IntegralTables

Related topic

IntegrationByParts

Related topic

ReductionFormulasForIntegrationOfPowers

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

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integration by parts

integration by parts

When we want to integrate a product of two functions, it is sometimes preferable to simplify the integrand by integrating one of the functions and differentiating the other. This process is called integrating by parts, and is done in the following way, where

u

and

v

are functions of

x

.

∫

u

⋅

v

′

⁢

𝑑

x

=

u

⋅

v

-

∫

v

⋅

u

′

⁢

𝑑

x

This process may be repeated indefinitely, and in some cases it may be used to solve for the original

integral

algebraically. For

definite integrals

, the rule appears as

∫

a

b

u

⁢

(

x

)

⋅

v

′

⁢

(

x

)

⁢

𝑑

x

=

(

u

⁢

(

b

)

⋅

v

⁢

(

b

)

-

u

⁢

(

a

)

⋅

v

⁢

(

a

)

)

-

∫

a

b

v

⁢

(

x

)

⋅

u

′

⁢

(

x

)

⁢

𝑑

x

Proof:

Integration by parts

is simply the

antiderivative

of a

product rule

. Let

G

⁢

(

x

)

=

u

⁢

(

x

)

⋅

v

⁢

(

x

)

. Then,

G

′

⁢

(

x

)

=

u

′

⁢

(

x

)

⁢

v

⁢

(

x

)

+

u

⁢

(

x

)

⁢

v

′

⁢

(

x

)

Therefore,

G

′

⁢

(

x

)

-

v

⁢

(

x

)

⁢

u

′

⁢

(

x

)

=

u

⁢

(

x

)

⁢

v

′

⁢

(

x

)

We can now integrate both sides with respect to

x

to get

G

⁢

(

x

)

-

∫

v

⁢

(

x

)

⁢

u

′

⁢

(

x

)

⁢

𝑑

x

=

∫

u

⁢

(

x

)

⁢

v

′

⁢

(

x

)

⁢

𝑑

x

which is just integration by parts rearranged.

Example:

We integrate the function

f

⁢

(

x

)

=

x

⁢

sin

⁡

x

: Therefore we define

u

⁢

(

x

)

:=

x

and

v

′

⁢

(

x

)

=

sin

⁡

x

. So integration by parts yields us:

∫

x

⁢

sin

⁡

x

⁢

𝑑𝑥

=

-

x

⁢

cos

⁡

x

+

∫

cos

⁡

x

⁢

𝑑𝑥

=

-

x

⁢

cos

⁡

x

+

sin

⁡

x

+

C

,

where

C

is an arbitrary constant.

Title

integration by parts

Canonical name

IntegrationByParts

Date of creation

2013-03-22 12:28:33

Last modified on

2013-03-22 12:28:33

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

11

Author

mathwizard (128)

Entry type

Theorem

Classification

msc 26A36

Related topic

GeneralFormulasForIntegration

Related topic

IntegrationOfSqrtx21

Generated on Fri Feb 9 19:52:03 2018 by

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integration of differential binomial

integration of differential binomial

Theorem.

Let

a

,

b

,

c

,

α

,

β

be given

real numbers

and

α

⁢

β

≠

0

.  The

antiderivative

I

=

∫

x

a

⁢

(

α

+

β

⁢

x

b

)

c

⁢

𝑑

x

is

expressible

by of the

elementary functions

only in the three cases:

(

1

)

⁢

a

+

1

b

+

c

∈

ℤ

,

(

2

)

⁢

a

+

1

b

∈

ℤ

,

(

3

)

⁢

c

∈

ℤ

In accordance with P. L. Chebyshev (1821

-

1894), who has proven this theorem, the expression

x

a

⁢

(

α

+

β

⁢

x

b

)

c

⁢

d

⁢

x

is called a

differential binomial

.

It may be worth noting that the differential binomial may be expressed in terms of the

incomplete beta function

and the

hypergeometric function

. Define

y

=

β

⁢

x

b

/

α

. Then we have

I

=

1

b

⁢

α

a

+

1

b

+

c

⁢

β

-

a

+

1

b

⁢

B

y

⁢

(

1

+

a

b

,

c

-

1

)

=

1

1

+

a

⁢

α

a

+

1

b

+

c

⁢

β

-

a

+

1

b

⁢

y

1

+

a

b

⁢

F

⁢

(

a

+

1

b

,

2

-

c

;

1

+

a

+

b

b

;

y

)

Chebyshev’s theorem then follows from the theorem on elementary cases of the hypergeometric function.

Title

integration of differential binomial

Canonical name

IntegrationOfDifferentialBinomial

Date of creation

2013-03-22 14:45:49

Last modified on

2013-03-22 14:45:49

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 26A36

Generated on Fri Feb 9 19:52:05 2018 by

LaTeXML
integration of fraction power expressions

integration of fraction power expressions

The

antiderivatives

of every

expression

containing

fraction powers

can not be expressed by using

elementary functions

. However, there are after making a substitution.

•

∫

R

⁢

(

x

,

x

r

1

,

…

,

x

r

m

)

⁢

𝑑

x

,  where

R

means a

rational function

of its

arguments

.
If the common denominator of the fraction power

exponents

r

j

is

n

, the substitution

x

:=

t

n

,

d

⁢

x

=

n

⁢

t

n

-

1

⁢

d

⁢

t

changes each exponent to an integer and the whole

integrand

to a rational function in the

variable

t

.

Example.

For

∫

x

1

2

x

3

4

+

1

⁢

𝑑

x

the least common multiple of the denominators of

1

2

and

3

4

is 4, whence we make the substitution

x

=

t

4

,

d

⁢

x

=

4

⁢

t

3

⁢

d

⁢

t

.  Then we obtain

∫

x

1

2

x

3

4

+

1

⁢

𝑑

x

=

4

⁢

∫

t

5

⁢

d

⁢

t

t

3

+

1

=

4

⁢

∫

(

t

2

-

t

2

t

3

+

1

)

⁢

𝑑

t

=

4

⁢

(

t

3

3

-

1

3

⁢

ln

⁡

|

t

3

+

1

|

)

+

C

=

4

3

⁢

(

x

3

4

-

ln

⁡

|

x

3

4

+

1

|

)

+

C

.

•

In

∫

R

⁢

(

x

,

(

a

⁢

x

+

b

c

⁢

x

+

d

)

r

1

,

…

,

(

a

⁢

x

+

b

c

⁢

x

+

d

)

r

m

)

⁢

𝑑

x

,  correspondently the substitution

a

⁢

x

+

b

c

⁢

x

+

d

:=

t

n

changes the integrand to a rational function.

Example.

For

∫

x

+

4

x

⁢

𝑑

x

we substitute

x

+

4

=

t

2

,

d

⁢

x

=

2

⁢

t

⁢

d

⁢

t

,  getting

∫

x

+

4

x

⁢

𝑑

x

=

2

⁢

∫

t

2

t

2

-

4

⁢

𝑑

t

=

2

⁢

∫

(

1

+

4

t

2

-

4

)

⁢

𝑑

t

=

2

⁢

t

+

2

⁢

ln

⁡

|

t

-

2

t

+

2

|

+

C

=

2

⁢

x

+

4

+

2

⁢

ln

⁡

|

x

+

4

-

2

x

+

4

+

2

|

+

C

.

References

1

N. Piskunov:

Diferentsiaal- ja integraalarvutus kõrgematele tehnilistele õppeasutustele

. Viies, täiendatud trükk.  Kirjastus “Valgus”, Tallinn (1965).

Title

integration of fraction power expressions

Canonical

name

IntegrationOfFractionPowerExpressions

Date of creation

2013-03-22 17:50:33

Last modified on

2013-03-22 17:50:33

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Application

Classification

msc 26A36

Related topic

FractionPower

Related topic

RationalFunction

Related topic

IntegrationBySubstitution

Related topic

SubstitutionForIntegration

Generated on Fri Feb 9 19:52:08 2018 by

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