aarjaneiro/planetmath huggingface dataset:
This is my sample file

This is my sample file

Abstract

This paper is about blah blah …

This is my sample file

sdsd

This is the title of my paper

1

Introduction

Introduce stuff.

This is another paragraph.

2

Background

2.1

Item A

Background on thing A.

2.2

Item B

Background on thing B.

3

Experiment

Describe an experiment. Cite something

[

1

]

.

4

Results

Present results. Refere to Figure

LABEL:FigChart

.

5

Conclusion

The end.

References

1

This is an item in the bibliography.

Title

This is my sample file

Canonical

name

ThisIsMySampleFile1

Date of creation

2013-03-11 19:21:17

Last modified on

2013-03-11 19:21:17

Owner

chrisranjana (5956)

Last modified by

(0)

Numerical id

1

Author

chrisranjana (0)

Entry type

Definition

Generated on Fri Feb 9 21:21:39 2018 by

LaTeXML
unconditional convergence

unconditional convergence

A series

∑

n

=

1

∞

x

n

in a Banach space

X

is

unconditionally convergent

if for every

permutation

σ

:

ℕ

→

ℕ

the series

∑

n

=

1

∞

x

σ

⁢

(

n

)

converges

.

Alternatively, for every chain of finite subsets

S

1

⊆

S

2

⊆

⋯

of

ℕ

, the

partial sums

∑

k

∈

S

1

x

k

,

⁢

∑

k

∈

S

2

x

k

,

,

…

converges. The trick to see this equivalence is to realize two facts: 1. every

subsequence

of a

convergent sequence

is

convergent

, and 2. every chain

{

S

i

}

can be enlarged to a maximal chain

{

T

i

}

, such that

|

T

i

|

=

i

. Then the series

indexed by

{

S

i

}

is a subseries indexed by

{

T

i

}

, which is a subseries of a permutation of the original

convergent series

.

Yet a third

equivalent

(

http://planetmath.org/Equivalent3

) definition is given as follows: A series is unconditionally convergent if
for every

sequence

(

ε

n

)

n

=

1

∞

, with

ε

n

∈

{

±

1

}

, the
series

∑

n

=

1

∞

ε

n

⁢

x

n

converges.

Every absolutely convergent series is unconditionally convergent, the converse implication does not hold in general.

When

X

=

ℝ

n

then by a famous

theorem

of Riemann

(

∑

x

n

)

is unconditionally convergent if and only if it is absolutely convergent.

References

1

K. Knopp:

Theory and application of

infinite series

.

2

K. Knopp:

Infinite sequences and series

.

3

P. Wojtaszczyk:

Banach spaces for analysts

.

4

Ch. Heil:

http://www.math.gatech.edu/ heil/papers/bases.pdf

A basis theory primer.

Title

unconditional convergence

Canonical name

UnconditionalConvergence

Date of creation

2013-03-22 15:29:57

Last modified on

2013-03-22 15:29:57

Owner

kompik (10588)

Last modified by

kompik (10588)

Numerical id

11

Author

kompik (10588)

Entry type

Definition

Classification

msc 40A05

Synonym

unconditionally convergent

Related topic

AbsoluteConvergence

Related topic

ConditionallyConvergentSeriesOfRealNumbersCanBeRearrangedToConvergeToAnyNumber

Generated on Fri Feb 9 21:21:46 2018 by

LaTeXML
uniqueness of limit of sequence

uniqueness of limit of sequence

If a number

sequence

has a limit, then the limit is uniquely determined.

Proof.

For an indirect proof (

http://planetmath.org/ReductioAdAbsurdum

), suppose that a sequence

a

1

,

a

2

,

a

3

,

…

has two distinct limits

a

and

b

.  Thus we must have both

|

a

n

-

a

|

<

|

a

-

b

|

2

for all

⁢

n

>

some

⁢

n

1

and

|

a

n

-

b

|

<

|

a

-

b

|

2

for all

⁢

n

>

some

⁢

n

2

But when

n

exceeds the

greater

of

n

1

and

n

2

, we can write

|

a

-

b

|

=

|

a

-

a

n

+

a

n

-

b

|

≦

|

a

-

a

n

|

+

|

a

n

-

b

|

<

|

a

-

b

|

2

+

|

a

-

b

|

2

=

|

a

-

b

|

.

This

inequality

an impossibility, whence the antithesis made in the begin is wrong and the assertion is .

Title

uniqueness of limit of sequence

Canonical name

UniquenessOfLimitOfSequence

Date of creation

2013-03-22 19:00:23

Last modified on

2013-03-22 19:00:23

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

4

Author

pahio (2872)

Entry type

Theorem

Classification

msc 40A05

Generated on Fri Feb 9 21:21:59 2018 by

LaTeXML
values of {F_n}/{F_{n-1}} for 1<n<26

values of

F

n

F

n

-

1

for

1

<

n

<

26

The

golden ratio

ϕ

is an

irrational number

, approximately 1.6180339887498948482045868. Dividing one

Fibonacci number

by the previous gives better and better approximations to the golden ratio the bigger the numbers get. One

division

gets a little under, then the next a little over, as the table below shows. 144 divided by 89 is probably good enough for most practical purposes.

F

n

F

n

-

1

F

n

F

n

-

1

to 20

decimal places

Approximation error

1

1

1.00000000000000000000

-0.61803398874989484820

2

1

2.00000000000000000000

0.38196601125010515179

3

2

1.50000000000000000000

-0.11803398874989484820

5

3

1.66666666666666666666

0.04863267791677181846

8

5

1.60000000000000000000

-0.01803398874989484820

13

8

1.62500000000000000000

0.00696601125010515179

21

13

1.61538461538461538461

-0.00264937336527946358

34

21

1.61904761904761904761

0.00101363029772419941

55

34

1.61764705882352941176

-0.00038692992636543643

89

55

1.61818181818181818181

0.00014782943192333361

144

89

1.61797752808988764044

-0.00005646066000720775

233

144

1.61805555555555555555

0.00002156680566070735

377

233

1.61802575107296137339

-0.00000823767693347481

610

377

1.61803713527851458885

0.00000314652861974065

987

610

1.61803278688524590163

-1.20186464894656524257

1597

987

1.61803444782168186423

0.00000045907178701603

2584

1597

1.61803381340012523481

-0.00000017534976961338

4181

2584

1.61803405572755417956

0.00000066977659331361

6765

4181

1.61803396316670652953

-0.00000002558318831866

10946

6765

1.61803399852180339985

0.00000000977190855164

17711

10946

1.61803398501735793897

-0.00000000373253690923

28657

17711

1.61803399017559708655

0.00000000142570223835

46368

28657

1.61803398820532505147

-0.00000000054456979673

75025

46368

1.61803398895790200138

0.00000000020800715317

Title

values of

F

n

F

n

-

1

for

1

<

n

<

26

Canonical

name

ValuesOffracFnFn1For1N26

Date of creation

2013-03-22 17:23:54

Last modified on

2013-03-22 17:23:54

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 40A05

Classification

msc 11B39

Generated on Fri Feb 9 21:22:11 2018 by

LaTeXML
Works on Polynomials

Works on Polynomials

Title

Works on Polynomials

Canonical

name

WorksOnPolynomials1

Date of creation

2013-03-11 19:36:13

Last modified on

2013-03-11 19:36:13

Owner

Karem_B (19867)

Last modified by

(0)

Numerical id

1

Author

Karem_B (0)

Entry type

Definition

Generated on Fri Feb 9 21:22:30 2018 by

LaTeXML
Xypic Examples

Xypic Examples

Xypic Examples

This documentation serves to illustrate the use of the xypic package, by giving examples of actual code and the corresponding results. Before proceeding to add graphs, it is important to remember to add

\usepackage{xypic}

to the preamble first!

0.1

Basic Examples

1.

\begin{center}
$\xymatrix{a\ar[r] & b}$
\end{center}

Generated on Fri Feb 9 21:22:35 2018 by

LaTeXML
zero sequence

zero sequence

Let a field

k

be equipped with a

rank one valuation

|

.

|

.  A sequence

⟨

a

1

,

a

2

,

…

⟩

(1)

of elements of

k

is called a

zero sequence

or a

null sequence

, if

lim

n

→

∞

⁡

a

n

=

0

in the metric

induced

by

|

.

|

.

If

k

together with the metric induced by its

valuation

|

.

|

is a

complete ultrametric field

, it’s clear that its sequence
(1) has a limit (in

k

) as soon as the sequence

⟨

a

2

-

a

1

,

a

3

-

a

2

,

a

4

-

a

3

,

…

⟩

is a zero sequence.

If

k

is not

complete

with respect to its valuation

|

.

|

, its

completion

(

http://planetmath.org/Completion

) can be made as follows.  The

Cauchy sequences

(1) form an

integral domain

D

when the
operations “

+

” and “

⋅

” are defined componentwise.  The
subset

P

of

D

formed by the zero sequences is a

maximal ideal

, whence the

quotient ring

D

/

P

is a field

K

.  Moreover,

k

may be isomorphically embedded into

K

and
the valuation

|

.

|

may be uniquely extended to a valuation of

K

.  The field

K

then is complete with respect to

|

.

|

and

k

is

dense in

K

.

Title

zero sequence

Canonical name

ZeroSequence

Date of creation

2015-07-10 21:03:45

Last modified on

2015-07-10 21:03:45

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Definition

Classification

msc 40A05

Synonym

null sequence

Generated on Fri Feb 9 21:22:37 2018 by

LaTeXML
a sufficient condition for convergence of integral

a sufficient condition for convergence of integral

Suppose that the real function

f

is

positive

and

continuous

on the

interval

[

a

,

∞

)

.  A sufficient condition for the

convergence

(

http://planetmath.org/ConvergentIntegral

) of the

improper integral

∫

a

∞

f

⁢

(

x

)

⁢

𝑑

x

(1)

is that

lim

x

→

∞

⁡

f

⁢

(

x

+

1

)

f

⁢

(

x

)

=

q

<

1

.

(2)

Proof.

Assume that the condition (2) is in .  For an indirect proof (

http://planetmath.org/ReductioAdAbsurdum

), make the antithesis that the

integral

(

http://planetmath.org/RiemannIntegral

) (1)

diverges

(

http://planetmath.org/DivergentIntegral

).

Because of the positiveness, we have

∫

a

∞

f

⁢

(

x

)

⁢

𝑑

x

=

∞

.  We can use l’Hôpital’s rule (

http://planetmath.org/LHpitalsRule

):

lim

c

→

∞

⁡

∫

a

c

f

⁢

(

x

+

1

)

⁢

𝑑

x

∫

a

c

f

⁢

(

x

)

⁢

𝑑

x

=

lim

c

→

∞

⁡

f

⁢

(

c

+

1

)

f

⁢

(

c

)

.

Using the

http://planetmath.org/node/11373

substitution

x

+

1

=

t

we get

∫

a

c

f

⁢

(

x

+

1

)

⁢

𝑑

x

=

∫

a

-

1

c

-

1

f

⁢

(

t

)

⁢

𝑑

t

=

∫

a

-

1

a

f

⁢

(

t

)

⁢

𝑑

t

+

∫

a

c

f

⁢

(

t

)

⁢

𝑑

t

-

∫

c

-

1

c

f

⁢

(

t

)

⁢

𝑑

t

,

and dividing this equation by

∫

a

c

f

⁢

(

t

)

⁢

𝑑

t

and taking limits (

http://planetmath.org/ImproperLimits

) yield (

f

is

bounded

!)

1

>

q

=

lim

c

→

∞

⁡

∫

a

c

f

⁢

(

x

+

1

)

⁢

𝑑

x

∫

a

c

f

⁢

(

x

)

⁢

𝑑

x

=

0

+

1

-

0

=

1

.

This contradictory result shows that the antithesis is wrong; thus (1) must be

convergent

(

http://planetmath.org/ConvergentIntegral

).

Note.

The condition (2) is not necessary for the convergence of (1).  This is seen e.g. in the case of the converging of (2) equals 1.

Title

a

sufficient condition for convergence of integral

Canonical name

ASufficientConditionForConvergenceOfIntegral

Date of creation

2013-03-22 19:01:13

Last modified on

2013-03-22 19:01:13

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Theorem

Classification

msc 40A10

Related topic

RatioTest

Generated on Fri Feb 9 21:22:39 2018 by

LaTeXML
absolute convergence of integral and boundedness of derivative

absolute convergence of integral and boundedness of derivative

Theorem.

Assume that we have an

http://planetmath.org/node/11865

absolutely converging

integral

∫

a

∞

f

⁢

(

x

)

⁢

𝑑

x

where the

real function

f

and its

derivative

f

′

are

continuous

and

f

′

additionally bounded on the

interval

[

a

,

∞

)

.  Then

lim

x

→

∞

⁡

f

⁢

(

x

)

=

0

.

(1)

Proof.

If

c

>

a

,  we obtain

∫

a

c

f

(

x

)

f

′

(

x

)

d

x

=

1

2

/

a

c

(

f

(

x

)

)

2

=

(

f

⁢

(

c

)

)

2

-

(

f

⁢

(

a

)

)

2

2

,

from which

(

f

⁢

(

c

)

)

2

=

(

f

⁢

(

a

)

)

2

+

2

⁢

∫

a

c

f

⁢

(

x

)

⁢

f

′

⁢

(

x

)

⁢

𝑑

x

.

(2)

Using the boundedness of

f

′

and the

absolute convergence

, we can estimate upwards the integral

∫

a

c

|

f

⁢

(

x

)

⁢

f

′

⁢

(

x

)

|

⁢

𝑑

x

=

∫

a

c

|

f

⁢

(

x

)

|

⁢

|

f

′

⁢

(

x

)

|

⁢

𝑑

x

≦

M

⁢

∫

a

c

|

f

⁢

(

x

)

|

⁢

𝑑

x

≦

M

⁢

∫

a

∞

|

f

⁢

(

x

)

|

⁢

𝑑

x

∀

c

∈

[

a

,

∞

)

whence

∫

a

∞

|

f

⁢

(

x

)

⁢

f

′

⁢

(

x

)

|

⁢

𝑑

x

is finite and thus

∫

a

∞

f

⁢

(

x

)

⁢

f

′

⁢

(

x

)

⁢

𝑑

x

converges absolutely.  Hence (2) implies

lim

c

→

∞

⁡

(

f

⁢

(

c

)

)

2

=

(

f

⁢

(

a

)

)

2

+

2

⁢

∫

a

∞

f

⁢

(

x

)

⁢

f

′

⁢

(

x

)

⁢

𝑑

x

,

i.e.

lim

x

→

∞

⁡

(

f

⁢

(

x

)

)

2

exists as finite, therefore also

lim

x

→

∞

⁡

|

f

⁢

(

x

)

|

:=

A

.

Antithesis:

A

>

0

.  It implies that there is an

x

0

(

≧

a

)

such that

|

f

⁢

(

x

)

|

≧

A

2

∀

x

≧

x

0

.

If now

b

>

x

0

, then we had

∫

x

0

b

|

f

⁢

(

x

)

|

⁢

𝑑

x

≧

A

2

⁢

(

b

-

x

0

)

⟶

∞

as

⁢

b

→

∞

.

This means that

∫

x

0

∞

|

f

⁢

(

x

)

|

⁢

𝑑

x

and consequently also

∫

a

∞

|

f

⁢

(

x

)

|

⁢

𝑑

x

would be divergent.  Since it is not true, we infer that

A

=

0

,  i.e. that the assertion (1) is true.

Title

absolute convergence of integral and boundedness of derivative

Canonical name

AbsoluteConvergenceOfIntegralAndBoundednessOfDerivative

Date of creation

2013-03-22 19:01:28

Last modified on

2013-03-22 19:01:28

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Theorem

Classification

msc 40A10

Related topic

NecessaryConditionOfConvergence

Generated on Fri Feb 9 21:22:42 2018 by

LaTeXML
convergence of integrals

convergence of integrals

Similarly as one speaks of

convergence

of series, one can speak of

convergence of integrals

, especially of

Riemann integrals

∫

I

f

⁢

(

t

)

⁢

𝑑

t

.

This

integral

is

convergent

, if it exists, and otherwise

divergent

.  One can also speak of

absolute convergence

of integrals

.

Example.

Study the convergence of the integral

∫

1

2

d

⁢

x

(

ln

⁡

x

)

c

(1)

where

c

is a real constant.

According to the

logarithm series

, we may write for

1

<

x

<

b

,  where

b

is sufficiently close to

1

, the estimations

ln

⁡

(

x

-

1

)

=

x

-

1

+

O

⁢

(

(

x

-

1

)

2

)

=

(

x

-

1

)

⁢

[

1

+

O

⁢

(

x

-

1

)

]

⁢

{

≤

2

⁢

(

x

-

1

)

,

≥

1

2

⁢

(

x

-

1

)

.

Let

1

<

a

<

b

.

1

∘

.  For

c

>

1

:

∫

a

b

d

⁢

x

(

ln

⁡

x

)

c

≧

∫

a

b

d

⁢

x

2

c

⁢

(

x

-

1

)

c

=

-

1

2

c

⁢

/

x

=

a

b

⁡

1

(

c

-

1

)

⁢

(

x

-

1

)

c

-

1

=

1

2

c

⁢

(

c

-

1

)

⁢

[

1

(

a

-

1

)

c

-

1

-

1

(

b

-

1

)

c

-

1

]

⟶

∞

as

a

→

1

+

2

∘

.  For

c

=

1

:

∫

a

b

d

⁢

x

ln

⁡

x

≧

∫

a

b

d

⁢

x

2

⁢

(

x

-

1

)

=

1

2

⁢

/

a

b

⁡

ln

⁡

(

x

-

1

)

=

1

2

⁢

[

ln

⁡

(

b

-

1

)

-

ln

⁡

(

a

-

1

)

]

⟶

∞

as

a

→

1

+

3

∘

.  For

c

<

1

:

0

<

∫

a

b

d

⁢

x

(

ln

⁡

x

)

c

≦

∫

a

b

2

c

⁢

d

⁢

x

(

x

-

1

)

c

=

2

c

⁢

/

x

=

a

b

⁡

x

1

-

c

1

-

c

=

2

c

1

-

c

⁢

[

(

b

-

1

)

1

-

c

-

(

a

-

1

)

1

-

c

]

⟶

2

c

1

-

c

⁢

(

b

-

1

)

1

-

c

as

a

→

1

+

Consequently, the integral

∫

a

b

d

⁢

x

(

ln

⁡

x

)

c

, and thus also (1),

converges

if and only if

c

<

1

.

Title

convergence of integrals

Canonical name

ConvergenceOfIntegrals

Date of creation

2013-03-22 18:59:51

Last modified on

2013-03-22 18:59:51

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Example

Classification

msc 40A10

Related topic

UniformConvergenceOfIntegral

Related topic

LogarithmicIntegral2

Related topic

ListOfImproperIntegrals

Related topic

SubstitutionNotation

Related topic

ONotation

Defines

convergent integral

Defines

divergent integral

Generated on Fri Feb 9 21:22:45 2018 by

LaTeXML
example of improper integral

example of improper integral

The integrand of

I

=

∫

0

1

arctan

⁡

x

x

⁢

1

-

x

2

⁢

𝑑

x

(1)

is undefined both at the lower and the

upper limit

.  However, the value of the

improper integral

exists and may be found via the more general integral

I

⁢

(

y

)

=

∫

0

1

arctan

⁡

x

⁢

y

x

⁢

1

-

x

2

⁢

𝑑

x

.

(2)

Denote the integrand of (2) by

f

⁢

(

x

,

y

)

.  For any fixed real value

y

,

f

⁢

(

x

,

y

)

∈

O

⁢

(

1

)

⁢

as

⁢

x

→

0

,

f

⁢

(

x

,

y

)

∈

O

⁢

(

1

1

-

x

2

)

⁢

as

⁢

x

→

1

,

where the Landau big ordo (

http://planetmath.org/formaldefinitionoflandaunotation

) notation has been used.  Accordingly, the

integral

(2)

converges

for every

y

.

The

inequality

|

∂

⁡

f

⁢

(

x

,

y

)

∂

⁡

y

|

=

1

(

1

+

x

2

⁢

y

2

)

⁢

1

-

x

2

≦

1

1

-

x

2

and the

convergence

of the integral

∫

0

1

d

⁢

x

1

-

x

2

=

π

2

imply that the integral

∫

0

1

∂

⁡

f

⁢

(

x

,

y

)

∂

⁡

y

⁢

𝑑

x

(3)

http://planetmath.org/node/6277

converges uniformly

on the whole

y

-axis and equals

I

′

⁢

(

y

)

.  For expressing this

derivative

in a

closed form

(

http://planetmath.org/ExpressibleInClosedForm

), one may utilise the changes of variable (

http://planetmath.org/ChangeOfVariableInDefiniteIntegral

)

x

:=

cos

⁡

φ

,

tan

⁡

φ

:=

t

which yield

I

′

⁢

(

y

)

=

∫

0

1

d

⁢

x

(

1

+

x

2

⁢

y

2

)

⁢

1

-

x

2

=

∫

0

π

2

d

⁢

φ

1

+

y

2

⁢

cos

2

⁡

φ

=

∫

0

∞

d

⁢

t

1

+

y

2

+

t

2

=

/

t

=

0

∞

⁡

1

1

+

y

2

⁢

arctan

⁡

t

1

+

y

2

=

π

2

⁢

1

+

y

2

.

Hence,

I

⁢

(

y

)

=

π

2

⁢

∫

0

y

d

⁢

y

1

+

y

2

=

/

0

y

⁡

ln

⁡

(

y

+

1

+

y

2

)

and the integral (1) equals

I

=

I

⁢

(

1

)

=

π

2

⁢

ln

⁡

(

1

+

2

)

,  i.e.

∫

0

1

arctan

⁡

x

x

⁢

1

-

x

2

⁢

𝑑

x

=

π

2

⁢

ln

⁡

(

1

+

2

)

.

(4)

Title

example of improper integral

Canonical name

ExampleOfImproperIntegral

Date of creation

2014-11-07 11:47:42

Last modified on

2014-11-07 11:47:42

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Example

Classification

msc 40A10

Related topic

SubstitutionNotation

Generated on Fri Feb 9 21:22:48 2018 by

LaTeXML
Laplace integrals

Laplace integrals

The improper integrals

∫

-

∞

∞

a

⁢

cos

⁡

x

x

2

+

a

2

⁢

𝑑

x

and

∫

-

∞

∞

x

⁢

sin

⁡

x

x

2

+

a

2

⁢

𝑑

x

,

where

a

is a

positive

, are called

Laplace integrals

.  Both of them have the same value

π

⁢

e

-

a

.

The evaluation of the Laplace integrals can be performed by first determining the

integrals

∫

-

∞

∞

e

i

⁢

x

x

-

i

⁢

a

⁢

𝑑

x

and

∫

-

∞

∞

e

i

⁢

x

x

+

i

⁢

a

⁢

𝑑

x

where one

integrates

along the

real axis

.  Therefore one has to determine the integrals

∮

e

i

⁢

z

z

-

i

⁢

a

⁢

𝑑

z

and

∮

e

i

⁢

z

z

+

i

⁢

a

⁢

𝑑

z

around the perimeter of the half-disk with the arc in the

upper half-plane

, centered in the origin and with the diameter

(

-

R

,

+

R

)

.  The

residue theorem

yields the values

∮

e

i

⁢

z

z

-

i

⁢

a

⁢

𝑑

z

=

2

⁢

i

⁢

π

⁢

e

-

a

and

∮

e

i

⁢

z

z

+

i

⁢

a

⁢

𝑑

z

=

0

.

As in the entry example of using residue theorem, the parts of these

contour integrals

along the half-circle tend to zero when

R

→

∞

.  Consequently,

∫

-

∞

∞

e

i

⁢

x

x

-

i

⁢

a

⁢

𝑑

x

=

2

⁢

i

⁢

π

⁢

e

-

a

and

∫

-

∞

∞

e

i

⁢

x

x

+

i

⁢

a

⁢

𝑑

x

=

0

.

These equations imply by adding and subtracting and then taking the real (

http://planetmath.org/RealPart

) and the

imaginary parts

, the

∫

-

∞

∞

a

⁢

cos

⁡

x

x

2

+

a

2

⁢

𝑑

x

=

∫

-

∞

∞

x

⁢

sin

⁡

x

x

2

+

a

2

⁢

𝑑

x

=

π

⁢

e

-

a

.

References

1

R. Nevanlinna & V. Paatero

:

Funktioteoria

.  Kustannusosakeyhtiö Otava. Helsinki (1963).

Title

Laplace integrals

Canonical

name

LaplaceIntegrals

Date of creation

2013-03-22 18:43:17

Last modified on

2013-03-22 18:43:17

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Definition

Classification

msc 40A10

Generated on Fri Feb 9 21:22:53 2018 by

LaTeXML
methods of evaluating improper integrals

methods of evaluating improper integrals

There are some general

methods of evaluating improper integrals

in such cases when one cannot directly use the

antiderivative

of the

integrand

.  Which method is suitable in a certain instance, is dependent on the kind of the

integral

(

http://planetmath.org/DefiniteIntegral

).

•

Differentiation

under the integral sign with respect to a

parametre

in the integrand; one can add a new parametre to a
suitable place.  The differentiated form may then be
integrated directly or from a

differential equation

.  Examples:
a (

http://planetmath.org/twoimproperintegrals

),
b (

http://planetmath.org/generalisationofgaussianintegral

),
c (

http://planetmath.org/relativeofexponentialintegral

),
d (

http://planetmath.org/integralrelatedtoarcsine

),

http://planetmath.org/node/11617

e.

•

Laplace transform (

http://planetmath.org/laplacetransform

).  If the
integrand has, as above, a parametre in a suitable place, the
Laplace transform of the integrand with respect to this
parametre is often simpler to integrate and the new

improper
integral

to evaluate; thereafter one simply
inversely.  Examples:
f (

http://planetmath.org/sineintegralatinfinity

),
g (

http://planetmath.org/usingconvolutiontofindlaplacetransform

),
h (

http://planetmath.org/relativeofcosineintegral

),
i (

http://planetmath.org/laplacetransformoftnft

),

http://planetmath.org/node/10637

j.

•

Cauchy residue theorem

.  The integral may be obtained as
limit of a

contour integral

in the

complex plane

.  Examples:
k (

http://planetmath.org/usingresiduetheoremnearbranchpoint

),
l (

http://planetmath.org/fresnelformulas

),
m (

http://planetmath.org/laplaceintegrals

),

http://planetmath.org/node/11489

n.

•

Expanding the integrand to series.  Example:
o (

http://planetmath.org/applicationoflogarithmseries

).

•

Changing

variable

(

http://planetmath.org/ChangeOfVariableInDefiniteIntegral

) in an improper integral sometimes may recur it to a known improper integral.  Examples:
p (

http://planetmath.org/areaundergaussiancurve

),
q (

http://planetmath.org/exampleofimproperintegral

),
r (

http://planetmath.org/exampleofchangingvariable

).

Title

methods of evaluating improper integrals

Canonical

name

MethodsOfEvaluatingImproperIntegrals

Date of creation

2014-11-07 19:12:16

Last modified on

2014-11-07 19:12:16

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

34

Author

pahio (2872)

Entry type

Application

Classification

msc 40A10

Related topic

IntegrationOfLaplaceTransformWithRespectToParameter

Related topic

ListOfImproperIntegrals

Related topic

IntegralRelatedToArcSine

Related topic

ExampleOfChangingVariable

Generated on Fri Feb 9 21:22:59 2018 by

LaTeXML