aarjaneiro/planetmath huggingface dataset:
application of Cauchy–Schwarz inequality

application of Cauchy–Schwarz inequality

In determining the

perimetre of ellipse

one encounters the

elliptic integral

∫

0

π

2

1

-

ε

2

⁢

sin

2

⁡

t

⁢

𝑑

t

,

where the parametre

ε

is the

eccentricity

of the

ellipse

(

0

≦

ε

<

1

).  A good

upper bound

for the

integral

is obtained by utilising the

http://planetmath.org/node/1628

Cauchy–

Schwarz inequality

|

∫

a

b

f

⁢

g

|

≦

∫

a

b

f

2

⁢

∫

a

b

g

2

choosing in it

f

⁢

(

t

)

:=

1

and

g

⁢

(

t

)

:=

1

-

ε

2

⁢

sin

2

⁡

t

.  Then we get

0

<

∫

0

π

2

1

-

ε

2

⁢

sin

2

⁡

t

⁢

𝑑

t

≦

∫

0

π

2

1

2

⁢

𝑑

t

⁢

∫

0

π

2

(

1

-

ε

2

⁢

sin

2

⁡

t

)

⁢

𝑑

t

=

π

2

⁢

∫

0

π

2

(

1

-

ε

2

⋅

1

-

cos

⁡

2

⁢

t

2

)

⁢

𝑑

t

=

π

2

⁢

1

-

ε

2

2

.

Thus we have the estimation

∫

0

π

2

1

-

ε

2

⁢

sin

2

⁡

t

⁢

𝑑

t

≦

π

2

⁢

1

-

ε

2

2

.

It is the better

approximation

for the perimetre of ellipse the smaller is its eccentricity, i.e. the closer the ellipse is to circle.  The accuracy is

O

⁢

(

ε

4

)

Title

application of Cauchy–Schwarz inequality

Canonical name

ApplicationOfCauchySchwarzInequality

Date of creation

2013-03-22 18:59:42

Last modified on

2013-03-22 18:59:42

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Application

Classification

msc 26A42

Classification

msc 26A06

Synonym

application of Cauchy-Schwarz inequality

Generated on Fri Feb 9 19:52:51 2018 by

LaTeXML
continuous functions are Riemann integrable

continuous functions are Riemann integrable

Let

f

:

[

a

,

b

]

→

ℝ

be a

continuous function

. Then

f

is

Riemann integrable

.

Title

continuous functions are Riemann integrable

Canonical name

ContinuousFunctionsAreRiemannIntegrable

Date of creation

2013-03-22 13:45:32

Last modified on

2013-03-22 13:45:32

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

5

Author

paolini (1187)

Entry type

Theorem

Classification

msc 26A42

Generated on Fri Feb 9 19:52:56 2018 by

LaTeXML
continuous functions of several variables are Riemann summable

continuous functions of several variables are Riemann summable

Theorem 1

.

Continuous functions

defined on

compact subsets

of

R

n

are

Riemann integrable

.

Proof.

Let

D

⊂

ℝ

n

be a compact subset of

ℝ

n

and let

f

:

D

→

ℝ

be a continuous function.
Since

f

is defined on a compact set,

f

is

uniformly continuous

i.e. given

ϵ

>

0

there exists

δ

>

0

such that

|

x

-

y

|

≤

δ

⇒

|

f

⁢

(

x

)

-

f

⁢

(

y

)

|

≤

ϵ

.
Let

R

>

0

be large enough so that

D

⊂

(

-

R

,

R

)

n

(such an

R

exists because

D

is

bounded

).
Let

P

be a

polyrectangle

such that

D

⊂

∪

P

⊂

(

-

R

,

R

)

n

and such that every

rectangle

R

in

P

has

diameter

which is less then

δ

. So one has

sup

R

⁡

f

⁢

(

x

)

-

inf

R

⁡

f

⁢

(

x

)

≤

ϵ

and hence

S

*

⁢

(

f

,

P

)

-

S

*

⁢

(

f

,

P

)

≤

ϵ

⁢

∑

Q

∈

P

meas

⁢

(

Q

)

≤

ϵ

⁢

meas

⁢

(

P

)

≤

ϵ

⁢

meas

⁢

[

-

R

,

R

]

n

=

ϵ

⁢

2

n

⁢

R

n

.

Letting

ϵ

→

0

one concludes that

S

*

⁢

(

f

)

=

S

*

⁢

(

f

)

.
∎

Title

continuous functions of several variables are Riemann summable

Canonical name

ContinuousFunctionsOfSeveralVariablesAreRiemannSummable

Date of creation

2013-03-22 15:07:56

Last modified on

2013-03-22 15:07:56

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

9

Author

paolini (1187)

Entry type

Theorem

Classification

msc 26A42

Generated on Fri Feb 9 19:52:58 2018 by

LaTeXML
determination of Fourier coefficients

determination of Fourier coefficients

Suppose that the

real function

f

may be presented as sum of the

Fourier series

:

f

⁢

(

x

)

=

a

0

2

+

∑

m

=

0

∞

(

a

m

⁢

cos

⁡

m

⁢

x

+

b

m

⁢

sin

⁡

m

⁢

x

)

(1)

Therefore,

f

is periodic with period

2

⁢

π

.  For expressing the Fourier coefficients

a

m

and

b

m

with the

function

itself, we first multiply the series (1) by

cos

⁡

n

⁢

x

(

n

∈

ℤ

) and

integrate

from

-

π

to

π

.  Supposing that we can integrate termwise, we may write

∫

-

π

π

f

⁢

(

x

)

⁢

cos

⁡

n

⁢

x

⁢

d

⁢

x

=

a

0

2

⁢

∫

-

π

π

cos

⁡

n

⁢

x

⁢

d

⁢

x

+

∑

m

=

0

∞

(

a

m

⁢

∫

-

π

π

cos

⁡

m

⁢

x

⁢

cos

⁡

n

⁢

x

⁢

d

⁢

x

+

b

m

⁢

∫

-

π

π

sin

⁡

m

⁢

x

⁢

cos

⁡

n

⁢

x

⁢

d

⁢

x

)

.

(2)

When

n

=

0

,  the equation (2) reads

∫

-

π

π

f

⁢

(

x

)

⁢

𝑑

x

=

a

0

2

⋅

2

⁢

π

=

π

⁢

a

0

,

(3)

since in the sum of the right hand side, only the first addend is distinct from zero.

When

n

is a positive integer, we use the

product formulas

of the trigonometric identities, getting

∫

-

π

π

cos

⁡

m

⁢

x

⁢

cos

⁡

n

⁢

x

⁢

d

⁢

x

=

1

2

⁢

∫

-

π

π

[

cos

⁡

(

m

-

n

)

⁢

x

+

cos

⁡

(

m

+

n

)

⁢

x

]

⁢

𝑑

x

,

∫

-

π

π

sin

⁡

m

⁢

x

⁢

cos

⁡

n

⁢

x

⁢

d

⁢

x

=

1

2

⁢

∫

-

π

π

[

sin

⁡

(

m

-

n

)

⁢

x

+

sin

⁡

(

m

+

n

)

⁢

x

]

⁢

𝑑

x

.

The latter

expression

vanishes

always, since the sine is an

odd function

.  If

m

≠

n

,  the former equals zero because the

antiderivative

consists of sine terms which vanish at multiples of

π

; only in the case

m

=

n

we obtain from it a non-zero result

π

.  Then (2) reads

∫

-

π

π

f

⁢

(

x

)

⁢

cos

⁡

n

⁢

x

⁢

d

⁢

x

=

π

⁢

a

n

(4)

to which we can include as a special case the equation (3).

By multiplying (1) by

sin

⁡

n

⁢

x

and integrating termwise, one obtains similarly

∫

-

π

π

f

⁢

(

x

)

⁢

sin

⁡

n

⁢

x

⁢

d

⁢

x

=

π

⁢

b

n

.

(5)

The equations (4) and (5) imply the formulas

a

n

=

1

π

∫

-

π

π

f

(

x

)

cos

n

x

d

x

(

n

=

0

,

1

,

2

,

…

)

and

b

n

=

1

π

∫

-

π

π

f

(

x

)

sin

n

x

d

x

(

n

=

1

,

2

,

3

,

…

)

for finding the values of the Fourier coefficients of

f

.

Title

determination of Fourier coefficients

Canonical

name

DeterminationOfFourierCoefficients

Date of creation

2013-03-22 18:22:47

Last modified on

2013-03-22 18:22:47

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Derivation

Classification

msc 26A42

Classification

msc 42A16

Synonym

calculation of Fourier coefficients

Related topic

UniquenessOfFourierExpansion

Related topic

FourierSineAndCosineSeries

Related topic

OrthogonalityOfChebyshevPolynomials

Generated on Fri Feb 9 19:53:01 2018 by

LaTeXML
facts about Riemann–Stieltjes integral

facts about Riemann–Stieltjes integral

•

If the

integrator

g

of the

http://planetmath.org/node/3187

Riemann–

Stieltjes integral

∫

a

b

f

⁢

(

x

)

⁢

𝑑

g

⁢

(

x

)

is the

identity function

, then the

integral

reduces to the

Riemann integral

∫

a

b

f

⁢

(

x

)

⁢

𝑑

x

.

•

If the

integrand

of the Riemann–Stieltjes integral is a

constant function

, one has

∫

a

b

c

⁢

𝑑

g

⁢

(

x

)

=

c

⋅

(

g

⁢

(

b

)

-

g

⁢

(

a

)

)

.

•

If the integrand

f

is

continuous

and the integrator

g

monotonically nondecreasing on the

interval

[

a

,

b

]

,  then there exists a number

ξ

on the interval such that

∫

a

b

f

⁢

(

x

)

⁢

𝑑

g

⁢

(

x

)

=

f

⁢

(

ξ

)

⁢

(

g

⁢

(

b

)

-

g

⁢

(

a

)

)

.

Cf. the

integral mean value theorem

.

•

If

f

is continuous,

g

monotonically nondecreasing and

differentiable

on the interval

[

a

,

b

]

,  then

d

d

⁢

x

⁢

∫

a

x

f

⁢

(

t

)

⁢

𝑑

g

⁢

(

t

)

=

f

⁢

(

x

)

⁢

g

′

⁢

(

x

)

for

⁢

a

<

x

<

b

.

Title

facts about Riemann–Stieltjes integral

Canonical

name

FactsAboutRiemannStieltjesIntegral

Date of creation

2013-03-22 18:55:03

Last modified on

2013-03-22 18:55:03

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Topic

Classification

msc 26A42

Synonym

properties

of Riemann–Stieltjes integral

Related topic

PropertiesOfRiemannStieltjesIntegral

Generated on Fri Feb 9 19:53:12 2018 by

LaTeXML
fundamental theorem of calculus

fundamental theorem of calculus

Let

f

:

[

a

,

b

]

→

𝐑

be a

continuous function

, let

c

∈

[

a

,

b

]

be given
and consider the integral function

F

defined on

[

a

,

b

]

as

F

⁢

(

x

)

=

∫

c

x

f

⁢

(

t

)

⁢

𝑑

t

.

Then

F

is an

antiderivative

of

f

that is,

F

is

differentiable

in

[

a

,

b

]

and

F

′

⁢

(

x

)

=

f

⁢

(

x

)

∀

x

∈

[

a

,

b

]

.

The previous

relation

rewritten as

d

d

⁢

x

⁢

∫

c

x

f

⁢

(

t

)

⁢

𝑑

t

=

f

⁢

(

x

)

shows that the

differentiation

operator

d

d

⁢

x

is the

inverse

of the integration operator

∫

c

x

. This

formula

is sometimes called Newton-Leibniz formula.

On the other hand if

f

:

[

a

,

b

]

→

𝐑

is a continuous function
and

G

:

[

a

,

b

]

→

𝐑

is any antiderivative of

f

, i.e.

G

′

⁢

(

x

)

=

f

⁢

(

x

)

for all

x

∈

[

a

,

b

]

, then

∫

a

b

f

⁢

(

t

)

⁢

𝑑

t

=

G

⁢

(

b

)

-

G

⁢

(

a

)

.

(1)

This shows that up to a constant, the integration operator is the inverse of the

derivative

operator:

∫

a

x

D

⁢

G

=

G

-

G

⁢

(

a

)

.

Notes

Equation (

1

) is sometimes called “Barrow’s rule” or “Barrow’s formula”.

Title

fundamental theorem of calculus

Canonical name

FundamentalTheoremOfCalculus

Date of creation

2013-03-22 14:13:27

Last modified on

2013-03-22 14:13:27

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

13

Author

paolini (1187)

Entry type

Theorem

Classification

msc 26A42

Synonym

Newton-Leibniz

Synonym

Barrow’s rule

Synonym

Barrow’s formula

Related topic

FundamentalTheoremOfCalculus

Related topic

FundamentalTheoremOfCalculusForKurzweilHenstockIntegral

Related topic

FundamentalTheoremOfCalculusForRiemannIntegration

Related topic

LaplaceTransformOfFracftt

Related topic

LimitsOfNaturalLogarithm

Related topic

FundamentalTheoremOfIntegralCalculus

Generated on Fri Feb 9 19:53:14 2018 by

LaTeXML
fundamental theorem of calculus for Kurzweil-Henstock integral

fundamental theorem of calculus for Kurzweil-Henstock integral

Let the

∫

symbol denote the

Kurzweil-Henstock integral

. We can then give the most general version of the

fundamental theorem of calculus

.

Theorem

.

Let

F

:

[

a

,

b

]

→

R

and suppose the

derivative

F

′

⁢

(

x

)

exists for all

x

∈

[

a

,

b

]

. Then

∫

a

b

F

′

⁢

(

x

)

⁢

𝑑

x

=

F

⁢

(

b

)

-

F

⁢

(

a

)

.

The reader should note the subtle

difference

from the standard version. Here we do not assume anything about

F

′

except that it exists. For the standard version we usually assume that

F

′

is

continuous

, and if we use the

Lebesgue integral

we must assume that

F

′

is

Lebesgue integrable

. Part of this

theorem

is that

F

′

is Kurzweil-Henstock

integrable

, hence no extra

assumptions

are necessary.

An example of a

function

where the standard version has problems is the function

F

⁢

(

x

)

:=

{

x

2

⁢

sin

⁡

1

x

2

if

x

≠

0

0

if

x

=

0

.

F

is

differentiable

everywhere, but

F

′

⁢

(

x

)

=

{

2

⁢

x

⁢

sin

⁡

1

x

2

-

2

x

⁢

cos

⁡

1

x

2

if

x

≠

0

0

if

x

=

0

.

Which is not continuous and in fact

unbounded

on any

interval

containing zero.

Title

fundamental theorem of calculus for Kurzweil-Henstock integral

Canonical

name

FundamentalTheoremOfCalculusForKurzweilHenstockIntegral

Date of creation

2013-03-22 16:44:27

Last modified on

2013-03-22 16:44:27

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

4

Author

jirka (4157)

Entry type

Theorem

Classification

msc 26A42

Related topic

FundamentalTheoremOfCalculusClassicalVersion

Generated on Fri Feb 9 19:53:16 2018 by

LaTeXML
fundamental theorem of calculus for Riemann integration

fundamental theorem of calculus for Riemann integration

In this entry we discuss the

fundamental theorems of calculus for Riemann integration

.

-

Let

f

be a

Riemann integrable

function

on an

interval

[

a

,

b

]

and

F

defined in

[

a

,

b

]

by

F

⁢

(

x

)

=

∫

a

x

f

⁢

(

t

)

⁢

𝑑

t

+

k

, where

k

∈

ℝ

is a

constant

. Then

F

is

continuous

in

[

a

,

b

]

and

F

′

=

f

almost everywhere (

http://planetmath.org/MeasureZeroInMathbbRn

).

-

Let

F

be a

continuous function

in an interval

[

a

,

b

]

and

f

a Riemann integrable function such that

F

′

⁢

(

x

)

=

f

⁢

(

x

)

except at most in a finite number of points

x

. Then

F

⁢

(

x

)

-

F

⁢

(

a

)

=

∫

a

x

f

⁢

(

t

)

⁢

𝑑

t

.

0.1

Observations

Notice that the second fundamental theorem is not a

converse

of the first. In the first we conclude that

F

′

=

f

except in a set of

measure zero

(

http://planetmath.org/MeasureZeroInMathbbRn

), while in the second we assume that

F

′

=

f

except in a finite number of points. In fact, the two

theorems

can never be the converse of each other as the following example shows:

Example :

Let

F

be the

devil staircase

function, defined on

[

0

,

1

]

. We have that

•

F

is continuous in

[

0

,

1

]

,

•

F

′

=

0

except in a set of (this set must be

contained

in the

Cantor set

),

•

f

:=

0

is clearly a Riemann integrable function and

∫

0

x

0

⁢

𝑑

t

=

0

.

Thus,

F

⁢

(

x

)

≠

∫

0

x

F

′

⁢

(

t

)

⁢

𝑑

t

.

This leads to the question: what kind functions

F

can be expressed as

F

⁢

(

x

)

=

F

⁢

(

a

)

+

∫

a

x

g

⁢

(

t

)

⁢

𝑑

t

, for some function

g

? The answer to this question lies in the

concept

of

absolute continuity

(

http://planetmath.org/AbsolutelyContinuousFunction2

) (a which the devil staircase does not possess), but for that a more general of integration must be developed (the Lebesgue integration (

http://planetmath.org/Integral2

)).

Title

fundamental theorem of calculus for Riemann integration

Canonical name

FundamentalTheoremOfCalculusForRiemannIntegration

Date of creation

2013-03-22 17:57:32

Last modified on

2013-03-22 17:57:32

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

10

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 26A42

Related topic

FundamentalTheoremOfCalculusClassicalVersion

Related topic

FundamentalTheoremOfCalculus

Defines

first fundamental theorem of calculus

(Riemann integral)

Defines

second fundamental theorem of calculus

(Riemann integral)

Generated on Fri Feb 9 19:53:18 2018 by

LaTeXML
generalized Riemann integral

generalized Riemann integral

A

gauge

δ

is a

function

which assigns to every

real number

x

an

interval

δ

⁢

(

x

)

such that

x

∈

δ

⁢

(

x

)

.

Given a gauge

δ

, a

partition

U

i

i

=

1

n

of an interval

[

a

,

b

]

is said to be

δ

-fine if, for every point

x

∈

[

a

,

b

]

, the set

U

i

containing

x

is a subset of

δ

⁢

(

x

)

A function

f

:

[

a

,

b

]

→

ℝ

is said to be

generalized Riemann integrable

on

[

a

,

b

]

if there exists a number

L

∈

ℝ

such that for every

ϵ

>

0

there exists a gauge

δ

ϵ

on

[

a

,

b

]

such that if

𝒫

˙

is any

δ

ϵ

-fine partition of

[

a

,

b

]

, then

|

S

⁢

(

f

;

𝒫

˙

)

-

L

|

<

ϵ

,

where

S

⁢

(

f

;

𝒫

˙

)

is any

Riemann sum

for

f

using the partition

𝒫

˙

. The

collection

of all generalized Riemann integrable functions is usually denoted by

ℛ

*

⁢

[

a

,

b

]

.

If

f

∈

ℛ

*

⁢

[

a

,

b

]

then the number

L

is uniquely determined, and is called the

generalized Riemann integral

of

f

over

[

a

,

b

]

.

The reason that this is called a generalized Riemann integral is that, in the special case where

δ

⁢

(

x

)

=

[

x

-

y

,

x

+

y

]

for some number

y

, we recover the

Riemann integral

as a special case.

Figure 1:

Riemann sum over a

δ

-fine partition

Title

generalized Riemann integral

Canonical name

GeneralizedRiemannIntegral

Date of creation

2013-03-22 13:40:03

Last modified on

2013-03-22 13:40:03

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

12

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 26A42

Synonym

Kurzweil-Henstock integral

Synonym

gauge integral

Defines

generalized Riemann integrable

Defines

gauge

Generated on Fri Feb 9 19:53:20 2018 by

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integrating t⁢a⁢nx over [0,π/2]

integrating

t

⁢

a

⁢

n

x

over

[

0

,

π

2

]

Note that what is meant by

∫

0

π

2

tan

⁡

x

⁢

d

⁢

x

is actually

lim

t

→

π

2

-

⁡

∫

0

t

tan

⁡

x

⁢

d

⁢

x

, since

tan

⁡

x

is defined on

[

0

,

π

2

)

but not at

π

2

.

∫

0

π

2

tan

⁡

x

⁢

d

⁢

x

=

lim

t

→

π

2

-

⁡

∫

0

t

tan

⁡

x

⁢

d

⁢

x

=

lim

t

→

π

2

-

⁡

ln

⁡

|

sec

⁡

x

|

|

0

t

=

lim

t

→

π

2

-

⁡

ln

⁡

|

sec

⁡

t

|

-

ln

⁡

|

sec

⁡

0

|

=

lim

t

→

π

2

-

⁡

ln

⁡

|

sec

⁡

t

|

=

∞

.

Title

integrating

t

⁢

a

⁢

n

x

over

[

0

,

π

2

]

Canonical name

IntegratingtanXOver0fracpi2

Date of creation

2013-03-22 15:57:47

Last modified on

2013-03-22 15:57:47

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

14

Author

Wkbj79 (1863)

Entry type

Example

Classification

msc 26A42

Classification

msc 45-01

Related topic

ImproperLimits

Related topic

OneSidedLimit

Generated on Fri Feb 9 19:53:25 2018 by

LaTeXML
integration of polynomial

integration of polynomial

Theorem

.

For all nonnegative integers

n

,

∫

x

n

⁢

𝑑

x

=

1

n

+

1

⁢

x

n

+

1

+

C

.

Proof.

It will first be proven that, for any nonnegative integer

n

and any

a

∈

ℝ

,

∫

0

a

x

n

⁢

𝑑

x

=

1

n

+

1

⁢

a

n

+

1

.

If

a

=

0

, the above statement is obvious. If

a

>

0

, the following computation uses the

right hand rule

for computing the

integral

(

http://planetmath.org/RiemannIntegral

); if

a

<

0

, the following computation uses the

left hand rule

for computing the integral:

∫

0

a

x

n

⁢

𝑑

x

=

lim

t

→

∞

⁡

∑

k

=

1

t

(

a

⁢

k

t

)

n

⁢

(

a

t

)

=

a

n

+

1

⁢

lim

t

→

∞

⁡

1

t

n

+

1

⁢

∑

k

=

1

t

k

n

=

a

n

+

1

⁢

lim

t

→

∞

⁡

1

t

n

+

1

⁢

∑

l

=

1

n

+

1

(

n

+

1

r

)

⁢

B

n

+

1

-

l

n

+

1

⁢

(

t

+

1

)

l

by this theorem (

http://planetmath.org/SumOfKthPowersOfTheFirstNPositiveIntegers

),

=

a

n

+

1

⁢

lim

t

→

∞

⁡

1

t

n

+

1

⁢

(

n

+

1

n

+

1

)

⁢

B

n

+

1

-

(

n

+

1

)

n

+

1

⁢

(

t

+

1

)

n

+

1

=

B

0

n

+

1

⁢

a

n

+

1

⁢

lim

t

→

∞

⁡

(

t

+

1

t

)

n

+

1

=

1

n

+

1

⁢

a

n

+

1

Thus, if

a

,

b

∈

ℝ

, then

∫

a

b

x

n

⁢

𝑑

x

=

∫

0

b

x

n

⁢

𝑑

x

-

∫

0

a

x

n

⁢

𝑑

x

=

1

n

+

1

⁢

b

n

+

1

-

1

n

+

1

⁢

a

n

+

1

.

It follows that

∫

x

n

⁢

𝑑

x

=

1

n

+

1

⁢

x

n

+

C

.
∎

Title

integration of polynomial

Canonical name

IntegrationOfPolynomial

Date of creation

2013-03-22 15:57:29

Last modified on

2013-03-22 15:57:29

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

30

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 26A42

Generated on Fri Feb 9 19:53:29 2018 by

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integration under integral sign

integration under integral sign

Let

I

⁢

(

α

)

=

∫

a

b

f

⁢

(

x

,

α

)

⁢

𝑑

x

.

where

f

⁢

(

x

,

α

)

is

continuous

in the rectangle

a

≦

x

≦

b

,

α

1

≦

α

≦

α

2

.

Then

α

↦

I

⁢

(

α

)

is continuous and hence

integrable

(

http://planetmath.org/RiemannIntegrable

) on the

interval

α

1

≦

α

≦

α

2

;  we have

∫

α

1

α

2

I

⁢

(

α

)

⁢

𝑑

α

=

∫

α

1

α

2

(

∫

a

b

f

⁢

(

x

,

α

)

⁢

𝑑

x

)

⁢

𝑑

α

.

This is a

double integral

over a in the

x

⁢

α

-plane, whence one can change the order of integration (

http://planetmath.org/FubinisTheorem

) and accordingly write

∫

α

1

α

2

(

∫

a

b

f

⁢

(

x

,

α

)

⁢

𝑑

x

)

⁢

𝑑

α

=

∫

a

b

(

∫

α

1

α

2

f

⁢

(

x

,

α

)

⁢

𝑑

α

)

⁢

𝑑

x

.

Thus, a

definite integral

depending on a

parametre

may be integrated with respect to this parametre by performing the integration under the

integral sign

.

Example.

For being able to evaluate the improper integral

I

=

∫

0

∞

e

-

a

⁢

x

-

e

-

b

⁢

x

x

d

x

(

a

>

0

,

b

>

0

)

,

we may interprete the integrand as a definite integral:

e

-

a

⁢

x

-

e

-

b

⁢

x

x

=

/

α

=

b

a

⁡

e

-

α

⁢

x

x

=

∫

a

b

e

-

α

⁢

x

⁢

𝑑

α

.

Accordingly, we can calculate as follows:

I

=

∫

0

∞

(

∫

a

b

e

-

α

⁢

x

⁢

𝑑

α

)

⁢

𝑑

x

=

∫

a

b

(

∫

0

∞

e

-

α

⁢

x

⁢

𝑑

x

)

⁢

𝑑

α

=

∫

a

b

(

/

x

=

0

∞

-

e

-

α

⁢

x

α

)

⁢

𝑑

α

=

∫

a

b

1

α

⁢

𝑑

α

=

/

a

b

⁡

ln

⁡

α

=

ln

⁡

b

a

Title

integration under integral sign

Canonical

name

IntegrationUnderIntegralSign

Date of creation

2013-03-22 18:46:27

Last modified on

2013-03-22 18:46:27

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26A42

Related topic

FubinisTheorem

Related topic

DifferentiationUnderIntegralSign

Related topic

RelativeOfExponentialIntegral

Generated on Fri Feb 9 19:53:31 2018 by

LaTeXML
Laplace transform of convolution

Laplace transform of convolution

Theorem.

If

ℒ

⁢

{

f

1

⁢

(

t

)

}

=

F

1

⁢

(

s

)

and

ℒ

⁢

{

f

2

⁢

(

t

)

}

=

F

2

⁢

(

s

)

,

then

ℒ

⁢

{

∫

0

t

f

1

⁢

(

τ

)

⁢

f

2

⁢

(

t

-

τ

)

⁢

𝑑

τ

}

=

F

1

⁢

(

s

)

⁢

F

2

⁢

(

s

)

.

Proof.

According to the definition of

Laplace transform

, one has

ℒ

⁢

{

∫

0

t

f

1

⁢

(

τ

)

⁢

f

2

⁢

(

t

-

τ

)

⁢

𝑑

τ

}

=

∫

0

∞

e

-

s

⁢

t

⁢

(

∫

0

t

f

1

⁢

(

τ

)

⁢

f

2

⁢

(

t

-

τ

)

⁢

𝑑

τ

)

⁢

𝑑

t

,

where the right hand side is a

double integral

over the angular region bounded by the lines

τ

=

0

and

τ

=

t

in the first quadrant of the

t

⁢

τ

-plane.  Changing the of integration, we write

ℒ

⁢

{

∫

0

t

f

1

⁢

(

τ

)

⁢

f

2

⁢

(

t

-

τ

)

⁢

𝑑

τ

}

=

∫

0

∞

(

f

1

⁢

(

τ

)

⁢

∫

τ

∞

e

-

s

⁢

t

⁢

f

2

⁢

(

t

-

τ

)

⁢

𝑑

t

)

⁢

𝑑

τ

.

Making in the inner

integral

the substitution

t

-

τ

:=

u

,  we obtain

∫

τ

∞

e

-

s

⁢

t

⁢

f

2

⁢

(

t

-

τ

)

⁢

𝑑

t

=

∫

0

∞

e

-

(

u

+

τ

)

⁢

s

⁢

f

2

⁢

(

u

)

⁢

𝑑

u

=

e

-

τ

⁢

s

⁢

∫

0

∞

e

-

s

⁢

u

⁢

f

2

⁢

(

u

)

⁢

𝑑

u

=

e

-

τ

⁢

s

⁢

F

2

⁢

(

s

)

,

whence

ℒ

⁢

{

∫

0

t

f

1

⁢

(

τ

)

⁢

f

2

⁢

(

t

-

τ

)

⁢

𝑑

τ

}

=

∫

0

∞

f

1

⁢

(

τ

)

⁢

e

-

τ

⁢

s

⁢

F

2

⁢

(

s

)

⁢

𝑑

τ

=

F

2

⁢

(

s

)

⁢

∫

0

∞

f

1

⁢

(

τ

)

⁢

e

-

τ

⁢

s

⁢

𝑑

τ

=

F

1

⁢

(

s

)

⁢

F

2

⁢

(

s

)

,

Q.E.D.

Title

Laplace transform of convolution

Canonical name

LaplaceTransformOfConvolution

Date of creation

2013-03-22 18:24:04

Last modified on

2013-03-22 18:24:04

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26A42

Classification

msc 44A10

Synonym

convolution property of Laplace transform

Related topic

Convolution

Generated on Fri Feb 9 19:53:34 2018 by

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