aarjaneiro/planetmath huggingface dataset:
RMS Value of the Fourier Series

RMS Value of the Fourier Series

RMS Value of the

Fourier Series

Swapnil Sunil Jain
December 28, 2006

RMS Value of the Fourier Series

If a function

f

⁢

(

t

)

is given by its Fourier series i.e.

f

⁢

(

t

)

=

a

0

2

+

∑

k

=

1

∞

[

a

k

⁢

cos

⁡

(

k

⁢

ω

⁢

t

)

+

b

k

⁢

sin

⁡

(

k

⁢

ω

⁢

t

)

]

The RMS value

F

rms

of

f

⁢

(

t

)

is

F

rms

=

a

0

2

4

+

1

2

⁢

∑

k

=

1

∞

a

k

2

+

b

k

2

Proof:

The RMS value of a function

f

⁢

(

t

)

is, by

definition

, given by

F

rms

=

1

T

⁢

∫

t

0

t

0

+

T

[

f

⁢

(

t

)

]

2

⁢

𝑑

t

Then,

F

rms

=

1

T

⁢

∫

t

0

t

0

+

T

[

f

⁢

(

t

)

]

2

⁢

𝑑

t

=

1

T

⁢

∫

t

0

t

0

+

T

f

⁢

(

t

)

⋅

f

⁢

(

t

)

⁢

𝑑

t

=

1

T

⁢

∫

t

0

t

0

+

T

f

⁢

(

t

)

⋅

(

a

0

2

+

∑

k

=

1

∞

[

a

k

⁢

cos

⁡

(

k

⁢

ω

⁢

t

)

+

b

k

⁢

sin

⁡

(

k

⁢

ω

⁢

t

)

]

)

⁢

𝑑

t

=

1

T

⁢

∫

t

0

t

0

+

T

(

a

0

⁢

f

⁢

(

t

)

2

+

∑

k

=

1

∞

[

a

k

⁢

f

⁢

(

t

)

⁢

cos

⁡

(

k

⁢

ω

⁢

t

)

+

b

k

⁢

f

⁢

(

t

)

⁢

sin

⁡

(

k

⁢

ω

⁢

t

)

]

)

⁢

𝑑

t

=

1

T

⁢

(

∫

t

0

t

0

+

T

a

0

⁢

f

⁢

(

t

)

2

⁢

𝑑

t

+

∑

k

=

1

∞

[

∫

t

0

t

0

+

T

a

k

⁢

f

⁢

(

t

)

⁢

cos

⁡

(

k

⁢

ω

⁢

t

)

⁢

𝑑

t

+

∫

t

0

t

0

+

T

b

k

⁢

f

⁢

(

t

)

⁢

sin

⁡

(

k

⁢

ω

⁢

t

)

⁢

𝑑

t

]

)

=

1

T

⁢

(

a

0

2

⁢

∫

t

0

t

0

+

T

f

⁢

(

t

)

⁢

𝑑

t

+

∑

k

=

1

∞

[

a

k

⁢

∫

t

0

t

0

+

T

f

⁢

(

t

)

⁢

cos

⁡

(

k

⁢

ω

⁢

t

)

⁢

𝑑

t

+

b

k

⁢

∫

t

0

t

0

+

T

f

⁢

(

t

)

⁢

sin

⁡

(

k

⁢

ω

⁢

t

)

⁢

𝑑

t

]

)

=

1

T

⁢

(

a

0

2

⁢

(

a

0

⁢

T

2

)

+

∑

k

=

1

∞

[

a

k

⁢

(

a

k

⁢

T

2

)

+

b

k

⁢

(

b

k

⁢

T

2

)

]

)

=

a

0

2

4

+

∑

k

=

1

∞

a

k

2

2

+

b

k

2

2

=

a

0

2

4

+

1

2

⁢

∑

k

=

1

∞

a

k

2

+

b

k

2

Title

RMS Value of the Fourier Series

Canonical

name

RMSValueOfTheFourierSeries1

Date of creation

2013-03-11 19:30:56

Last modified on

2013-03-11 19:30:56

Owner

swapnizzle (13346)

Last modified by

(0)

Numerical id

1

Author

swapnizzle (0)

Entry type

Definition

Generated on Fri Feb 9 21:17:29 2018 by

LaTeXML
ratio test of d’Alembert

ratio test of d’Alembert

A lighter version of the

ratio test

is the

Ratio test of d’Alembert.

Let

a

1

+

a

2

+

…

be a series with

positive

terms.

1

∘

.  If there exists a number

q

such that

0

<

q

<

1

and

a

n

+

1

a

n

≤

q

for all

⁢

n

≥

n

0

,

(1)

then the series

converges

.

2

∘

.  If there exists a number

n

0

such that

a

n

+

1

a

n

≥

1

for all

⁢

n

≥

n

0

,

(2)

then the series diverges.

Proof.

1

∘

. By the condition (1), we have

a

n

+

1

≤

a

n

⁢

q

;  thus we get the estimations

a

n

0

+

1

≤

a

n

0

⁢

q

,

a

n

0

+

2

≤

a

n

0

+

1

⁢

q

≤

a

n

0

⁢

q

2

,

⋯

⋯

⋯

a

n

0

+

p

≤

a

n

0

+

p

-

1

⁢

q

≤

…

≤

a

n

0

⁢

q

p

,

⋯

⋯

⋯

Because

a

n

0

⁢

q

+

a

n

0

⁢

q

2

+

…

+

a

n

0

⁢

q

p

+

…

is a

convergent

geometric series

, those

inequalities

and the

comparison test

imply that the series

a

n

0

+

1

+

a

n

0

+

2

+

…

+

a

n

0

+

p

+

…

and as well the whole series

a

1

+

a

2

+

…

is convergent.

2

∘

.  The condition (2) yields

a

n

0

+

1

≥

a

n

0

,

a

n

0

+

2

≥

a

n

0

+

1

≥

a

n

0

,

…

and since

a

n

0

is positive, the limit of

a

n

as

n

tends to infinity cannot be 0.  Hence the given series does not fulfil the necessary condition of

convergence

.

Example.

If the variable

x

in the

power series

∑

n

=

0

∞

n

!

⁢

x

n

is distinct from zero, we have

|

(

n

+

1

)

!

⁢

x

n

+

1

|

|

n

!

⁢

x

n

|

=

(

n

+

1

)

⁢

|

x

|

≥

1

for all

⁢

n

≥

n

0

.

Then the series does not converge absolutely (

http://planetmath.org/AbsoluteConvergence

).  The known theorem of Abel says that the series diverges for all

x

≠

0

.  It means that the

radius of convergence

is 0.

References

1

Л. Д. Кудрявцев:

Математический анализ. I том

.  Издательство
‘‘Высшая школа’’. Москва (1970).

Title

ratio test of d’Alembert

Canonical name

RatioTestOfDAlembert

Date of creation

2013-03-22 19:12:28

Last modified on

2013-03-22 19:12:28

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Theorem

Classification

msc 40A05

Related topic

FiniteChangesInConvergentSeries

Generated on Fri Feb 9 21:17:35 2018 by

LaTeXML
Semilinear biharmonic problem

Semilinear biharmonic problem

Semilinear biharmonic problem

A semilinear biharmonic problem with so-called Navier

boundary conditions

is

(

B

⁢

H

)

⁢

{

Δ

2

⁢

v

=

f

⁢

(

x

,

v

)

x

∈

Ω

v

=

Δ

⁢

v

=

0

x

∈

∂

⁡

Ω

where

Ω

⊂

ℝ

N

(

N

≥

1

)

is an open bounded domain,

f

⁢

(

x

,

v

)

∈

𝒞

1

⁢

(

Ω

¯

×

ℝ

;

ℝ

)

in

v

∈

ℝ

.

Numerical results for the case

Ω

=

(

-

2

,

2

)

×

(

-

2

,

2

)

and

f

⁢

(

x

,

v

)

=

v

p

: p=0.1, 0.7, 1.3, 7. Note that

u

=

-

Δ

⁢

v

.

Title

Semilinear biharmonic problem

Canonical name

SemilinearBiharmonicProblem1

Date of creation

2013-03-11 19:28:39

Last modified on

2013-03-11 19:28:39

Owner

linor (11198)

Last modified by

(0)

Numerical id

1

Author

linor (0)

Entry type

Definition

Generated on Fri Feb 9 21:17:43 2018 by

LaTeXML
sequence determining convergence of series

sequence determining convergence of series

Theorem.

Let

a

1

+

a

2

+

…

be any series of real

a

n

.  If the positive numbers

r

1

,

r

2

,

…

are such that

lim

n

→

∞

⁡

a

n

r

n

=

L

≠

0

,

(1)

then the series

converges

simultaneously with the series

r

1

+

r

2

+

…

Proof.

In the case that the limit (1) is positive, the supposition implies that there is an integer

n

0

such that

0.5

⁢

L

<

a

n

r

n

<

1.5

⁢

L

for

⁢

n

≧

n

0

.

(2)

Therefore

0

<

0.5

⁢

L

⁢

r

n

<

a

n

<

1.5

⁢

L

⁢

r

n

for all

⁢

n

≧

n

0

,

and since the series

∑

n

=

1

∞

0.5

⁢

L

⁢

r

n

and

∑

n

=

1

∞

1.5

⁢

L

⁢

r

n

converge simultaneously with the series

r

1

+

r

2

+

…

, the

comparison test

guarantees that the same concerns the given series

a

1

+

a

2

+

…

The case where (1) is negative, whence we have

lim

n

→

∞

⁡

-

a

n

r

n

=

-

L

>

0

,

may be handled as above.

Note.

For the case

L

=

0

, see the

limit comparison test

.

Title

sequence determining convergence of series

Canonical name

SequenceDeterminingConvergenceOfSeries

Date of creation

2013-03-22 19:06:54

Last modified on

2013-03-22 19:06:54

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Definition

Classification

msc 40A05

Related topic

LimitComparisonTest

Generated on Fri Feb 9 21:17:46 2018 by

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sequentially compact

sequentially compact

A

topological space

X

is

sequentially compact

if every

sequence

in

X

has a convergent subsequence.

Every sequentially compact space is

countably compact

.

Conversely

, every

first countable

countably compact space is sequentially compact. The

ordinal space

W

⁢

(

2

⁢

ω

1

)

is sequentially compact but not first countable, since

ω

1

has not

countable

local basis.

Next,

compactness

and sequential compactness are not

compatible

. In other words, neither one implies the other. Here’s an example of a

compact space

that is not sequentially compact. Let

X

=

I

I

, where

I

is the closed unit interval (with the

usual topology

), and

X

is equipped with the

product topology

. Then

X

is

compact

(since

I

is, together with

Tychonoff theorem

). However,

X

is not sequentially compact. To see this, let

f

n

:

I

→

I

be the

function

such that for any

r

∈

I

,

f

⁢

(

r

)

is the

n

-th digit of

r

in its binary expansion. But the sequence

f

1

,

…

,

f

n

,

…

has no convergent subsequences: if

f

n

1

,

…

,

f

n

k

,

…

is a subsequence, let

r

∈

I

such that its binary expansion has its

k

-th digit

0

iff

k

is odd, and

1

otherwise. Then

f

n

1

⁢

(

r

)

,

…

,

f

n

k

⁢

(

r

)

,

…

is the sequence

0

,

1

,

0

,

1

,

…

, and is clearly not convergent. The ordinal space

Ω

0

:=

W

⁢

(

ω

1

)

is an example of a sequentially compact space that is not compact, since the cover

{

W

⁢

(

α

)

∣

α

∈

Ω

0

}

has no finite

subcover

.

When

X

is a

metric space

, the following are

equivalent

:

•

X

is sequentially compact.

•

X

is

limit point compact

.

•

X

is compact.

•

X

is

totally bounded

and

complete

.

Title

sequentially compact

Canonical name

SequentiallyCompact

Date of creation

2013-03-22 12:50:05

Last modified on

2013-03-22 12:50:05

Owner

mps (409)

Last modified by

mps (409)

Numerical id

12

Author

mps (409)

Entry type

Definition

Classification

msc 40A05

Classification

msc 54D30

Synonym

sequential compactness

Related topic

Compact

Related topic

LimitPointCompact

Related topic

BolzanoWeierstrassTheorem

Related topic

Net

Generated on Fri Feb 9 21:17:48 2018 by

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silver ratio

silver ratio

The

silver ratio

is the sum of 1 and the

square root of 2

, represented by the Greek letter delta with a subscript S. That is,

δ

S

=

1

+

2

, with an approximate value of 2.4142135623730950488 (see A014176 in Sloane’s OEIS). Its

continued fraction

is

2

+

1

2

+

1

2

+

1

2

+

⋱

,

which suggests that the

Pell numbers

P

n

can be used as

convergents

. Similarly, the

n

th power of the silver ratio for

n

>

0

is

P

n

⁢

δ

S

+

P

n

-

1

.

Title

silver ratio

Canonical name

SilverRatio

Date of creation

2013-03-22 16:42:10

Last modified on

2013-03-22 16:42:10

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 40A05

Generated on Fri Feb 9 21:17:49 2018 by

LaTeXML
silver rectangle

silver rectangle

A

silver rectangle

is a rectangle with either the silver ratio for its width-to-length ratio or the ratio

1

:

2

. Like the

golden rectangle

, both silver rectangles have the property that if we remove from them a square, their ratio remains the same.

Figure 1:

Removing the square on the left (demarcated by the dotted line) leaves a smaller silver rectangle on the right.

Title

silver rectangle

Canonical name

SilverRectangle

Date of creation

2013-03-22 16:42:12

Last modified on

2013-03-22 16:42:12

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 40A05

Generated on Fri Feb 9 21:17:51 2018 by

LaTeXML
slower convergent series

slower convergent series

Theorem

.

If

a

1

+

a

2

+

a

3

+

⋯

(1)

is a converging series with positive , then one can always form another converging series

g

1

+

g

2

+

g

3

+

⋯

such that

lim

n

→

∞

⁡

g

n

a

n

=

∞

(2)

Proof.

Let

S

be the sum of (1),

S

n

=

a

1

+

a

2

+

⋯

+

a

n

the

n

th

partial sum

of (1) and

R

n

+

1

=

S

-

S

n

=

a

n

+

1

+

a

n

+

2

+

⋯

the corresponding

remainder term

.  Then we have

a

n

=

R

n

-

R

n

+

1

=

(

R

n

+

R

n

+

1

)

⁢

(

R

n

-

R

n

+

1

)

.

We set

g

n

:=

a

n

R

n

+

R

n

+

1

=

R

n

-

R

n

+

1

∀

n

=

1

,

2

,

3

,

…

Then the series

g

1

+

g

2

+

g

3

+

⋯

fulfils the requirements in the theorem.  Its

g

n

are positive.  Further, it

converges

because its

n

th

partial sum is equal to

R

1

-

R

n

+

1

which tends to the limit

R

1

=

S

as

n

→

∞

since

R

n

+

1

→

0

;  this implies also (2).

Title

slower convergent series

Canonical name

SlowerConvergentSeries

Date of creation

2013-03-22 15:08:24

Last modified on

2013-03-22 15:08:24

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Theorem

Classification

msc 40A05

Related topic

SlowerDivergentSeries

Related topic

NonExistenceOfUniversalSeriesConvergenceCriterion

Generated on Fri Feb 9 21:17:54 2018 by

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slower divergent series

slower divergent series

Theorem

.

If

a

1

+

a

2

+

a

3

+

⋯

(1)

is a diverging series with positive , then one can always form another diverging series

s

1

+

s

2

+

s

3

+

⋯

with positive such that

lim

n

→

∞

⁡

s

n

a

n

=

0

.

(2)

Proof.

Let

S

n

=

a

1

+

a

2

+

⋯

+

a

n

be the

n

th

partial sum

of (1).  Then we have

a

n

=

S

n

-

S

n

-

1

=

(

S

n

+

S

n

-

1

)

⁢

(

S

n

-

S

n

-

1

)

.

We set

s

1

:=

S

1

and

s

n

:=

a

n

S

n

+

S

n

-

1

=

S

n

-

S

n

-

1

for

n

=

2

,

3

,

4

,

…

Then the of the series

∑

n

=

1

∞

s

n

=

S

1

+

∑

n

=

1

∞

(

S

n

+

1

-

S

n

)

apparently are positive.  This series is however

divergent

, because the sum of its

n

first is equal to

S

n

which grows without bound along with

n

since (1)

diverges

.  For this reason we also get the result (2).

Remark.

Niels Henrik Abel

has presented a simpler example on such series

s

1

+

s

2

+

s

3

+

⋯

:

1

+

a

2

a

1

+

a

2

+

a

3

a

1

+

a

2

+

a

3

+

a

4

a

1

+

a

2

+

a

3

+

a

4

+

⋯

Title

slower divergent series

Canonical name

SlowerDivergentSeries

Date of creation

2013-03-22 15:08:27

Last modified on

2013-03-22 15:08:27

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Theorem

Classification

msc 40A05

Related topic

SlowerConvergentSeries

Related topic

NonExistenceOfUniversalSeriesConvergenceCriterion

Generated on Fri Feb 9 21:17:57 2018 by

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Stolz-Cesaro theorem

Stolz-Cesaro theorem

Let

(

a

n

)

n

≥

1

and

(

b

n

)

n

≥

1

be two

sequences

of

real numbers

. If

b

n

is positive,

strictly increasing

and

unbounded

and the following limit exists:

lim

n

→

∞

⁡

a

n

+

1

-

a

n

b

n

+

1

-

b

n

=

l

Then the limit:

lim

n

→

∞

⁡

a

n

b

n

also exists and it is equal to

l

.

Remark.

This

theorem

is also valid if

a

n

and

b

n

are

monotone

, tending to

0

.

Title

Stolz-Cesaro theorem

Canonical name

StolzCesaroTheorem

Date of creation

2013-03-22 13:17:16

Last modified on

2013-03-22 13:17:16

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 40A05

Related topic

CesaroMean

Related topic

ExampleUsingStolzCesaroTheorem

Related topic

KroneckersLemma

Generated on Fri Feb 9 21:17:58 2018 by

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Subject of the month club

Subject of the month club

Subject of the month club

Introduction

Proposed subjects

1.

calculus on manifolds. the

calculus

of forms,

vector fields

,
flows, Lie group, Lie bracket, integrability.

2.

Functional analysis

3.

Topological vector spaces

Title

Subject of the month club

Canonical

name

SubjectOfTheMonthClub1

Date of creation

2013-03-11 19:22:01

Last modified on

2013-03-11 19:22:01

Owner

matte (1858)

Last modified by

(0)

Numerical id

1

Author

matte (0)

Entry type

Definition

Generated on Fri Feb 9 21:18:00 2018 by

LaTeXML
summation by parts

summation by parts

The following corollaries apply Abel’s lemma to allow estimation of certain bounded sums:

Corollary 1

(

Summation by parts

)

Let

{

a

i

}

,

{

b

i

}

be

sequences

of

complex numbers

. Suppose the

partial sums

of the

a

i

are

bounded

in magnitude by

h

, that

∑

0

∞

|

b

i

-

b

i

+

1

|

converges

, and that

lim

i

→

∞

⁡

b

i

=

0

. Then

∑

0

∞

a

i

⁢

b

i

converges, and

|

∑

0

∞

a

i

⁢

b

i

|

≤

h

⁢

∑

0

∞

|

b

i

-

b

i

+

1

|

Proof.

By Abel’s lemma,

∑

i

=

0

N

a

i

⁢

b

i

=

∑

i

=

0

N

-

1

A

i

⁢

(

b

i

-

b

i

+

1

)

+

A

N

⁢

b

N

so that

|

∑

i

=

0

N

a

i

⁢

b

i

|

=

|

∑

i

=

0

N

-

1

A

i

⁢

(

b

i

-

b

i

+

1

)

+

A

N

⁢

b

N

|

≤

∑

i

=

0

N

-

1

|

A

i

⁢

(

b

i

-

b

i

+

1

)

|

+

|

A

N

⁢

b

N

|

≤

h

⁢

∑

i

=

0

N

-

1

|

b

i

-

b

i

+

1

|

+

h

⁢

|

b

N

|

The condition that the

b

i

→

0

is easily seen to imply that the sequence

|

∑

i

=

0

N

a

i

⁢

b

i

|

is Cauchy hence

convergent

, so that

|

∑

i

=

0

∞

a

i

⁢

b

i

|

≤

h

⁢

∑

i

=

0

∞

|

b

i

-

b

i

+

1

|

since

b

N

→

0

.

Corollary 2

(Summation by parts for real sequences)

Let

{

a

i

}

be a sequence of complex numbers. Suppose the partial sums are bounded in magnitude by

h

. Let

{

b

i

}

be a sequence of decreasing positive real numbers such that

lim

i

→

∞

⁡

b

i

=

0

. Then

∑

1

∞

a

i

⁢

b

i

converges, and

|

∑

1

∞

a

i

⁢

b

i

|

≤

h

⁢

b

1

.

Proof.

This follows immediately from the above, since

|

b

i

-

b

i

+

1

|

=

b

i

-

b

i

+

1

.

Title

summation by parts

Canonical name

SummationByParts

Date of creation

2013-03-22 16:28:10

Last modified on

2013-03-22 16:28:10

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

8

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 40A05

Classification

msc 40D05

Synonym

partial summation

Generated on Fri Feb 9 21:18:13 2018 by

LaTeXML
Tabulation Examples

Tabulation Examples

Tabulation Examples

This documentation serves to illustrate the use of the various packages aiming at creating tables in LaTeX by giving examples of actual code and the corresponding results. First note that, it is possible to create tables with the list of standard packages provided by the preamble section of a generic

PlanetMath

entry. However, the following package

\usepackage{tabls}

is recommended, in particular when the contents of the cells in a table are of different sizes.

Elementary Examples

1.

\begin{tabular}{c}
a\\
bc\\
def\\
ghij\\
klmno
\end{tabular}

Title

Tabulation Examples

Canonical

name

TabulationExamples1

Date of creation

2013-03-11 19:36:07

Last modified on

2013-03-11 19:36:07

Owner

CWoo (3771)

Last modified by

(0)

Numerical id

1

Author

CWoo (0)

Entry type

Definition

\@unrecurse

Generated on Fri Feb 9 21:21:09 2018 by

LaTeXML
telescoping sum

telescoping sum

A

telescoping sum

is a sum in which cancellation occurs between subsequent terms, allowing the sum to be expressed using only the initial and final terms.

Formally a telescoping sum is or can be rewritten in the form

S

=

∑

n

=

α

β

(

a

n

-

a

n

+

1

)

=

a

α

-

a

β

+

1

where

a

n

is a

sequence

.

Example:

Define

S

⁢

(

N

)

=

∑

n

=

1

N

1

n

⁢

(

n

+

1

)

. Note that by

partial fractions of expressions

:

1

n

⁢

(

n

+

1

)

=

1

n

-

1

n

+

1

and thus

a

n

=

1

n

in this example.

S

⁢

(

N

)

=

∑

n

=

1

N

(

1

n

-

1

n

+

1

)

=

(

1

-

1

2

)

+

⋯

+

(

1

n

-

1

n

+

1

)

+

(

1

n

+

1

-

1

n

+

2

)

+

⋯

+

(

1

N

-

1

N

+

1

)

=

1

+

(

-

1

2

+

1

2

)

+

⋯

+

(

-

1

n

+

1

+

1

n

+

1

)

+

⋯

-

1

N

+

1

=

1

-

1

N

+

1

Title

telescoping sum

Canonical name

TelescopingSum

Date of creation

2013-03-22 14:25:18

Last modified on

2013-03-22 14:25:18

Owner

cvalente (11260)

Last modified by

cvalente (11260)

Numerical id

8

Author

cvalente (11260)

Entry type

Definition

Classification

msc 40A05

Defines

telescope

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

LaTeXML
test collaboration 1234

test collaboration 1234

test collaboration 1234

test test

sentence 2!

sentence 3!

Title

test collaboration 1234

Canonical

name

TestCollaboration12341

Date of creation

2013-03-11 19:24:21

Last modified on

2013-03-11 19:24:21

Owner

akrowne (2)

Last modified by

(0)

Numerical id

1

Author

akrowne (0)

Entry type

Definition

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

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The Dirac Delta Function

The Dirac Delta Function

The

Dirac Delta Function

Swapnil Sunil Jain
December 27, 2006

The Dirac Delta Function

Definition

The Dirac delta function can be defined as

δ

⁢

(

t

)

≜

lim

ϵ

→

0

⁡

1

ϵ

⁢

Π

⁢

(

t

ϵ

)

where

Π

⁢

(

t

)

is the pulse function. The delta function can also be defined as

δ

⁢

(

t

)

=

{

0

for

⁢

t

≠

0

∞

for

⁢

t

=

0

,

s.t

⁢

∫

-

∞

+

∞

δ

⁢

(

t

)

⁢

𝑑

t

=

1

or it can also be defined as an operator s.t.

δ

⁢

(

t

)

⁢

[

f

⁢

(

t

)

]

≜

lim

ϵ

→

0

⁡

1

ϵ

⁢

∫

-

∞

+

∞

Π

⁢

(

t

ϵ

)

⁢

f

⁢

(

t

)

⁢

𝑑

t

Properties

1.

⁢

∫

-

∞

∞

f

⁢

(

t

)

⁢

δ

⁢

(

t

-

a

)

⁢

𝑑

t

=

f

⁢

(

a

)

Proof:

∫

-

∞

∞

f

⁢

(

t

)

⁢

δ

⁢

(

t

-

a

)

⁢

𝑑

t

=

lim

ϵ

→

0

⁡

(

∫

-

∞

a

-

ϵ

f

⁢

(

t

)

⁢

δ

⁢

(

t

-

a

)

⁢

𝑑

t

+

∫

1

-

ϵ

a

+

ϵ

f

⁢

(

t

)

⁢

δ

⁢

(

t

-

a

)

⁢

𝑑

t

+

∫

a

+

ϵ

∞

f

⁢

(

t

)

⁢

δ

⁢

(

t

-

a

)

⁢

𝑑

t

)

=

lim

ϵ

→

0

⁡

(

∫

a

-

ϵ

a

+

ϵ

f

⁢

(

t

)

⁢

δ

⁢

(

t

-

a

)

⁢

𝑑

t

)

=

f

⁢

(

a

)

⁢

lim

ϵ

→

0

⁡

(

∫

a

-

ϵ

a

+

ϵ

δ

⁢

(

t

-

a

)

⁢

𝑑

t

)

=

f

⁢

(

a

)

⁢

(

1

)

=

f

⁢

(

a

)

2.

⁢

∫

-

∞

∞

f

⁢

(

t

)

⁢

δ

⁢

(

t

)

⁢

𝑑

t

=

f

⁢

(

0

)

Proof: Readily seen if we set

a

=

0

in

Property

#1.

3.

⁢

∫

-

∞

∞

f

⁢

(

t

)

⁢

δ

⁢

(

a

⁢

t

)

⁢

𝑑

t

=

1

a

⁢

f

⁢

(

0

)

Proof: Set

u

=

a

⁢

t

, then

∫

-

∞

∞

f

⁢

(

t

)

⁢

δ

⁢

(

a

⁢

t

)

⁢

𝑑

t

=

∫

-

∞

∞

f

⁢

(

u

a

)

⁢

δ

⁢

(

u

)

⁢

d

⁢

u

a

=

1

a

⁢

f

⁢

(

0

)

4.

⁢

∫

-

∞

∞

f

⁢

(

t

)

⁢

δ

⁢

(

a

⁢

t

-

t

0

)

⁢

𝑑

t

=

1

a

⁢

f

⁢

(

t

0

a

)

Proof: Set

u

=

a

⁢

t

-

t

0

, then

∫

-

∞

∞

f

⁢

(

t

)

⁢

δ

⁢

(

a

⁢

t

-

t

0

)

⁢

𝑑

t

=

∫

-

∞

∞

f

⁢

(

u

-

t

0

a

)

⁢

δ

⁢

(

u

)

⁢

d

⁢

u

a

=

1

a

⁢

f

⁢

(

t

0

a

)

5.

⁢

∫

-

∞

∞

f

⁢

(

t

)

⁢

δ

⁢

(

g

⁢

(

t

)

)

⁢

𝑑

t

=

∑

{

x

i

|

g

⁢

(

x

i

)

=

0

}

1

|

g

′

⁢

(

x

i

)

|

⁢

f

⁢

(

x

i

)

6.

⁢

∫

-

∞

∞

f

⁢

(

t

)

⁢

δ

′

⁢

(

t

)

⁢

𝑑

t

=

-

f

′

⁢

(

0

)

Proof: If we let

u

=

f

⁢

(

t

)

and

d

⁢

v

=

δ

′

⁢

(

t

)

⁢

d

⁢

t

, then, using

integration by parts

,

∫

-

∞

∞

f

⁢

(

t

)

⁢

δ

′

⁢

(

t

)

⁢

𝑑

t

=

f

⁢

(

t

)

⁢

δ

⁢

(

t

)

|

-

∞

∞

-

∫

-

∞

∞

f

′

⁢

(

t

)

⁢

δ

⁢

(

t

)

⁢

𝑑

t

=

-

f

′

⁢

(

0

)

Title

The Dirac Delta Function

Canonical

name

TheDiracDeltaFunction1

Date of creation

2013-03-11 19:30:28

Last modified on

2013-03-11 19:30:28

Owner

swapnizzle (13346)

Last modified by

(0)

Numerical id

1

Author

swapnizzle (0)

Entry type

Definition

Generated on Fri Feb 9 21:21:28 2018 by

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the HÃÂ©non equation

the HÃÂ©non equation

the HÃÂ©non equation

The Henon equation and its associated energy

functional

can be written as

{

-

Δ

⁢

u

=

|

x

|

r

⁢

|

u

|

q

-

1

⁢

u

,

x

∈

Ω

u

=

0

,

x

∈

∂

⁡

Ω

,

⁢

J

⁢

(

u

)

=

∫

Ω

[

1

2

⁢

|

∇

⁡

u

|

2

-

1

q

+

1

⁢

|

x

|

r

⁢

|

u

|

q

+

1

]

⁢

𝑑

x

where

u

∈

H

0

1

⁢

(

Ω

)

,

Ω

⊂

ℝ

N

(

N

≥

2

)

is an open bounded domain,

r

≥

0

and

1

<

q

≤

N

+

2

N

-

2

.

Title

the HÃÂ©non equation

Canonical

name

TheHnonEquation

Date of creation

2013-03-11 19:28:19

Last modified on

2013-03-11 19:28:19

Owner

linor (11198)

Last modified by

(0)

Numerical id

1

Author

linor (0)

Entry type

Definition

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

LaTeXML