aarjaneiro/planetmath huggingface dataset:
table of values for ERROR(\symoperators)⁢c⁢o⁢s(x)

table of values for

\symoperators

⁢

c

⁢

o

⁢

s

(

x

)

n

∘

cos

⁡

(

n

∘

)

0

1.000000000

1

0.9998476952

2

0.9993908270

3

0.9986295348

4

0.9975640503

5

0.9961946981

6

0.9945218954

7

0.9925461516

8

0.9902680687

9

0.9876883406

10

0.9848077530

11

0.9816271834

12

0.9781476007

13

0.9743700648

14

0.9702957263

15

0.9659258263

16

0.9612616959

17

0.9563047560

18

0.9510565163

19

0.9455185756

20

0.9396926208

21

0.9335804265

22

0.9271838546

23

0.9205048535

24

0.9135454576

25

0.9063077870

26

0.8987940463

27

0.8910065242

28

0.8829475929

29

0.8746197071

30

0.8660254038

31

0.8571673007

32

0.8480480962

33

0.8386705679

34

0.8290375726

35

0.8191520443

36

0.8090169944

37

0.7986355100

38

0.7880107536

39

0.7771459615

40

0.7660444431

41

0.7547095802

42

0.7431448255

43

0.7313537016

44

0.7193398003

45

0.7071067812

46

0.6946583705

47

0.6819983601

48

0.6691306064

49

0.6560590290

50

0.6427876097

51

0.6293203910

52

0.6156614753

53

0.6018150232

54

0.5877852523

55

0.5735764364

56

0.5591929035

57

0.5446390350

58

0.5299192642

59

0.5150380749

60

0.5000000000

61

0.4848096202

62

0.4694715628

63

0.4539904997

64

0.4383711468

65

0.4226182617

66

0.4067366431

67

0.3907311285

68

0.3746065934

69

0.3583679495

70

0.3420201433

71

0.3255681545

72

0.3090169944

73

0.2923717047

74

0.2756373558

75

0.2588190451

76

0.2419218956

77

0.2249510543

78

0.2079116908

79

0.1908089954

80

0.1736481777

81

0.1564344650

82

0.1391731010

83

0.1218693434

84

0.1045284633

85

0.08715574275

86

0.06975647374

87

0.05233595624

88

0.03489949670

89

0.01745240644

90

0

n

∘

cos

⁡

(

n

∘

)

88

0.03489949670

88.05

0.0340274

88.1

0.0331552

88.15

0.032283

88.2

0.0314108

88.25

0.0305385

88.3

0.0296662

88.35

0.028794

88.4

0.0279216

88.45

0.0270493

88.5

0.0261769

88.55

0.0253046

88.6

0.0244322

88.65

0.0235598

88.7

0.0226873

88.75

0.0218149

88.8

0.0209424

88.85

0.0200699

88.9

0.0191974

88.95

0.0183249

89.

0.0174524

89.05

0.0165799

89.1

0.0157073

89.15

0.0148348

89.2

0.0139622

89.25

0.0130896

89.3

0.012217

89.35

0.0113444

89.4

0.0104718

89.45

0.00959916

89.5

0.00872654

89.55

0.0078539

89.6

0.00698126

89.65

0.00610861

89.7

0.00523596

89.75

0.00436331

89.8

0.00349065

89.85

0.00261799

89.9

0.00174533

89.95

0.000872665

90.

0

Title

table of values for

\symoperators

⁢

c

⁢

o

⁢

s

(

x

)

Canonical name

TableOfValuesForcosx

Date of creation

2013-03-22 15:05:08

Last modified on

2013-03-22 15:05:08

Owner

drini (3)

Last modified by

drini (3)

Numerical id

5

Author

drini (3)

Entry type

Example

Classification

msc 43-XX

Classification

msc 51-XX

Classification

msc 42-XX

Classification

msc 33Bxx

Generated on Fri Feb 9 21:31:00 2018 by

LaTeXML
table of values for ERROR(\symoperators)⁢s⁢i⁢n(x)

table of values for

\symoperators

⁢

s

⁢

i

⁢

n

(

x

)

n

∘

sin

⁡

(

n

∘

)

0

0

1

0.01745240644

2

0.03489949670

3

0.05233595624

4

0.06975647374

5

0.08715574275

6

0.1045284633

7

0.1218693434

8

0.1391731010

9

0.1564344650

10

0.1736481777

11

0.1908089954

12

0.2079116908

13

0.2249510543

14

0.2419218956

15

0.2588190451

16

0.2756373558

17

0.2923717047

18

0.3090169944

19

0.3255681545

20

0.3420201433

21

0.3583679495

22

0.3746065934

23

0.3907311285

24

0.4067366431

25

0.4226182617

26

0.4383711468

27

0.4539904997

28

0.4694715628

29

0.4848096202

30

0.5000000000

31

0.5150380749

32

0.5299192642

33

0.5446390350

34

0.5591929035

35

0.5735764364

36

0.5877852523

37

0.6018150232

38

0.6156614753

39

0.6293203910

40

0.6427876097

41

0.6560590290

42

0.6691306064

43

0.6819983601

44

0.6946583705

45

0.7071067812

46

0.7193398003

47

0.7313537016

48

0.7431448255

49

0.7547095802

50

0.7660444431

51

0.7771459615

52

0.7880107536

53

0.7986355100

54

0.8090169944

55

0.8191520443

56

0.8290375726

57

0.8386705679

58

0.8480480962

59

0.8571673007

60

0.8660254038

61

0.8746197071

62

0.8829475929

63

0.8910065242

64

0.8987940463

65

0.9063077870

66

0.9135454576

67

0.9205048535

68

0.9271838546

69

0.9335804265

70

0.9396926208

71

0.9455185756

72

0.9510565163

73

0.9563047560

74

0.9612616959

75

0.9659258263

76

0.9702957263

77

0.9743700648

78

0.9781476007

79

0.9816271834

80

0.9848077530

81

0.9876883406

82

0.9902680687

83

0.9925461516

84

0.9945218954

85

0.9961946981

86

0.9975640503

87

0.9986295348

88

0.9993908270

89

0.9998476952

90

1.000000000

n

∘

sin

⁡

(

n

∘

)

0

0

0.05

0.000872665

0.1

0.00174533

0.15

0.00261799

0.2

0.00349065

0.25

0.00436331

0.3

0.00523596

0.35

0.00610861

0.4

0.00698126

0.45

0.0078539

0.5

0.00872654

0.55

0.00959916

0.6

0.0104718

0.65

0.0113444

0.7

0.012217

0.75

0.0130896

0.8

0.0139622

0.85

0.0148348

0.9

0.0157073

0.95

0.0165799

1.

0.0174524

1.05

0.0183249

1.1

0.0191974

1.15

0.0200699

1.2

0.0209424

1.25

0.0218149

1.3

0.0226873

1.35

0.0235598

1.4

0.0244322

1.45

0.0253046

1.5

0.0261769

1.55

0.0270493

1.6

0.0279216

1.65

0.028794

1.7

0.0296662

1.75

0.0305385

1.8

0.0314108

1.85

0.032283

1.9

0.0331552

1.95

0.0340274

2.

0.0348995

Title

table of values for

\symoperators

⁢

s

⁢

i

⁢

n

(

x

)

Canonical name

TableOfValuesForsinx

Date of creation

2013-03-22 15:05:04

Last modified on

2013-03-22 15:05:04

Owner

drini (3)

Last modified by

drini (3)

Numerical id

4

Author

drini (3)

Entry type

Example

Classification

msc 43-XX

Classification

msc 51-XX

Classification

msc 42-XX

Classification

msc 33Bxx

Generated on Fri Feb 9 21:31:02 2018 by

LaTeXML
amenable group

amenable group

Let

G

be a

locally compact group

and

L

∞

⁢

(

G

)

be the

Banach space

of all

essentially bounded functions

G

→

ℝ

with respect to the

Haar measure

.

Definition 1

.

A

linear functional

on

L

∞

⁢

(

G

)

is called a

mean

if it maps the

constant function

f

⁢

(

g

)

=

1

to 1 and non-negative

functions

to non-negative numbers.

Definition 2

.

Let

L

g

be the

left action

of

g

∈

G

on

f

∈

L

∞

⁢

(

G

)

,
i.e.

(

L

g

⁢

f

)

⁢

(

h

)

=

f

⁢

(

g

-

1

⁢

h

)

.
Then, a mean

μ

is said to be

left

invariant

if

μ

⁢

(

L

g

⁢

f

)

=

μ

⁢

(

f

)

for all

g

∈

G

and

f

∈

L

∞

⁢

(

G

)

.
Similarly,

right invariant

if

μ

⁢

(

R

g

⁢

f

)

=

μ

⁢

(

f

)

,
where

R

g

is the right action

(

R

g

⁢

f

)

⁢

(

h

)

=

f

⁢

(

h

⁢

g

)

.

Definition 3

.

A locally compact group

G

is

amenable

if there is a left (or right) invariant mean on

L

∞

⁢

(

G

)

.

Example 1

(Amenable groups)

All

finite groups

and all

abelian groups

are amenable.
Compact groups are amenable as the Haar measure is an (unique) invariant mean.

Example 2

(Non-amenable groups)

If a group contains a free (

non-abelian

)

subgroup

on two

generators

then it is not amenable.

Title

amenable group

Canonical

name

AmenableGroup

Date of creation

2013-03-22 13:09:26

Last modified on

2013-03-22 13:09:26

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

9

Author

mhale (572)

Entry type

Definition

Classification

msc 43A07

Related topic

LpSpace

Defines

amenable

Defines

mean

Generated on Fri Feb 9 21:31:04 2018 by

LaTeXML
Ornstein-Weiss lemma

Ornstein-Weiss lemma

Let

G

be a group.
For a fixed

K

⊆

G

,
define the

K

-boundary

of

U

⊆

G

as

∂

K

⁡

U

=

{

g

∈

G

∣

K

⁢

g

∩

U

,

K

⁢

g

∩

(

G

∖

U

)

≠

∅

}

.

(1)

Let

𝒫

⁢

ℱ

⁢

(

G

)

be the set of finite subsets of

G

.
Call a

Følner net

for

G

a net

𝒳

=

{

X

i

}

i

∈

ℐ

⊆

𝒫

⁢

ℱ

⁢

(

G

)

,

ℐ

being a directed set,
such that for every

finite

K

⊆

G

,

lim

i

∈

ℐ

⁡

|

∂

K

⁡

X

i

|

|

X

i

|

=

0

,

(2)

where the limit is taken in the sense of directed sets.
Recall that

G

has a Følner net if and only if

G

is amenable.

Theorem 1

(Ornstein-Weiss lemma)

Let

G

be an amenable group and

F

:

P

⁢

F

⁢

(

G

)

→

R

a subadditive, right-invariant function, that is:

1.

For any two finite subsets

U

,

V

of

G

,

F

⁢

(

U

∪

V

)

≤

F

⁢

(

U

)

+

F

⁢

(

V

)

.

(3)

2.

For any

g

∈

G

and finite

U

⊆

G

,

F

⁢

(

U

⁢

g

)

=

F

⁢

(

U

)

.

(4)

Then for any Følner net

X

=

{

X

i

}

i

∈

I

on

G

, the limit

L

=

lim

i

∈

ℐ

⁡

F

⁢

(

X

i

)

|

X

i

|

(5)

exists, and does not depend on the choice of

X

.

The Ornstein-Weiss lemma allows to prove variants of Birkhoff’s

ergodic theorem

for actions of amenable groups,
rather than only those generated by an

invertible

, measure

invariant

map.
Moreover, it shares several similarities
with Fekete’s lemma on subadditive functions over the positive integers,
although it is not a

complete

counterpart.
In fact, putting

X

n

=

{

1

,

…

,

n

}

determines a Følner sequence on

ℤ

;
however, if

f

:

ℕ

→

[

0

,

∞

)

is subadditive,
then

F

⁢

(

U

)

=

f

⁢

(

|

U

|

)

is right-invariant, but not necessarily subadditive.
(Counterexample:

f

⁢

(

n

)

=

n

⁢

mod

⁢

2

,

U

=

{

1

,

2

}

,

V

=

{

2

,

3

}

.)

References

1

Gromov, M. (1999)

Topological invariants

of

dynamical systems

and spaces of holomorphic maps, I.

Math. Phys. Anal. Geom.

2

, 323–415.

2

Krieger, F. (2007)
Le lemme d’Ornstein-Weiss d’après Gromov.
In B. Hasselblatt (ed.),

Dynamics,

Ergodic Theory

, and Geometry

.
Cambridge University Press.

3

Krieger, F.
The Ornstein-Weiss lemma for discrete amenable groups.
Preprint.

4

Ornstein, D.S. and Weiss, B. (1987)

Entropy

and isomorphism theorems for actions of amenable groups.

J. Anal. Math.

48

, 1–141.

Title

Ornstein-Weiss lemma

Canonical name

OrnsteinWeissLemma

Date of creation

2013-03-22 19:20:24

Last modified on

2013-03-22 19:20:24

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

5

Author

Ziosilvio (18733)

Entry type

Theorem

Classification

msc 43A07

Generated on Fri Feb 9 21:31:07 2018 by

LaTeXML
Bargmann-Fock space

Bargmann-Fock space

The Bargmann-Fock space (or simply Fock space) is the

Hilbert space

of

entire functions

,

ℱ

2

⁢

(

ℂ

)

s.t.

∫

ℂ

|

F

⁢

(

z

)

|

2

⁢

e

-

π

⁢

|

z

|

2

⁢

𝑑

x

⁢

𝑑

y

<

∞

with associated

inner product

∫

ℂ

F

⁢

(

z

)

⁢

G

⁢

(

z

)

¯

⁢

e

-

π

⁢

|

z

|

2

⁢

𝑑

x

⁢

𝑑

y

where

z

=

x

+

i

⁢

y

References

1

V. Bargmann, “Remarks on a Hilbert Space of

Analytic Function

”

Proceedings of the National Academy of Sciences of the United States of America

48

(1962): 199 - 204

2

V. Bargmann & I. T. Todorov, “

Spaces of analytic functions

on a

complex

cone as carriers for the

symmetric tensor

representations of SO(n)”

Journal of Mathematical Physics

18

6 (1977): 1141 - 1148

Title

Bargmann-Fock space

Canonical name

BargmannFockSpace

Date of creation

2013-03-22 16:42:44

Last modified on

2013-03-22 16:42:44

Owner

ErlendA (6587)

Last modified by

ErlendA (6587)

Numerical id

14

Author

ErlendA (6587)

Entry type

Definition

Classification

msc 43A15

Synonym

Fock space

Defines

Fock space

Generated on Fri Feb 9 21:31:09 2018 by

LaTeXML
Bargmann transform

Bargmann transform

The

Bargmann transform

of a function,

f

, is a

linear map

B

:

X

⁢

(

ℝ

)

→

Y

⁢

(

ℂ

)

defined by

B

⁢

f

⁢

(

z

)

=

2

⁢

∫

ℝ

f

⁢

(

t

)

⁢

e

2

⁢

π

⁢

t

⁢

z

-

π

⁢

t

2

-

π

2

⁢

z

2

⁢

𝑑

t

Theorem

.

The Bargmann transform on

L

2

⁢

(

R

)

,

B

:

L

2

⁢

(

R

)

→

F

2

⁢

(

C

)

, is a

unitary transformation

. Here

F

2

⁢

(

C

)

is the

Fock space

.

References

1

Karlheinz GrÃÂ¶chenig, ”Foundations of Time-Frequency Analysis,”

BirkhhÃÂ¤user

(2000)

Title

Bargmann transform

Canonical name

BargmannTransform

Date of creation

2013-03-22 16:44:45

Last modified on

2013-03-22 16:44:45

Owner

ErlendA (6587)

Last modified by

ErlendA (6587)

Numerical id

7

Author

ErlendA (6587)

Entry type

Definition

Classification

msc 43A15

Defines

Bargmann transform

Generated on Fri Feb 9 21:31:11 2018 by

LaTeXML
Pontryagin duality

Pontryagin duality

1

Pontryagin dual

Let

G

be a

locally compact

abelian group

(

http://planetmath.org/TopologicalGroup

) and

𝕋

the 1-torus (

http://planetmath.org/NTorus

), i.e. the unit circle in

ℂ

.

Definition -

A

continuous

homomorphism

G

⟶

𝕋

is called a

character

of

G

. The set of all characters is called the

Pontryagin dual

of

G

and is denoted by

G

^

.

Under

pointwise

multiplication

G

^

is also an abelian group. Since

G

^

is a group of

functions

we can make it a

topological group

under the

compact-open topology

(

topology

of convergence on

compact sets

).

2

Examples

•

ℤ

^

≅

𝕋

, via

n

↦

z

n

with

z

∈

𝕋

.

•

𝕋

^

≅

ℤ

, via

z

↦

z

n

with

n

∈

ℤ

.

•

ℝ

^

≅

ℝ

, via

t

↦

e

i

⁢

s

⁢

t

with

s

∈

ℝ

.

3

Properties

The following are some important of the dual group:

Theorem

-

Let

G

be a locally compact abelian group. We have that

•

G

^

is also locally compact.

•

G

^

is

second countable

if and only if

G

is second countable.

•

G

^

is

compact

if and only if

G

is

discrete

.

•

G

^

is discrete if and only if

G

is compact.

•

(

⊕

i

∈

J

G

i

)

^

≅

⊕

i

∈

J

G

i

^

for any

finite set

J

. This

isomorphism

is natural.

4

Pontryagin duality

Let

f

:

G

⟶

H

be a continuous homomorphism of locally compact abelian groups. We can associate to it a

canonical map

f

^

:

H

^

⟶

G

^

defined by

f

^

⁢

(

ϕ

)

⁢

(

s

)

:=

ϕ

⁢

(

f

⁢

(

s

)

)

,

ϕ

∈

H

^

,

s

∈

G

This canonical construction preserves

identity mappings

and compositions, i.e. the dualization process

^

is a

functor

:

Theorem -

The dualization

^

:

𝐋𝐜𝐀

⟶

𝐋𝐜𝐀

is a

contravariant functor

from the

category

of locally compact abelian groups to itself.

5

Isomorphism with the second dual

Although in general there is not a canonical identification of

G

with its dual

G

^

, there is a

natural isomorphism

between

G

and its dual’s dual

G

^

^

:

Theorem -

The map

G

⟶

G

^

^

defined by

s

↦

s

^

^

, where

s

^

^

⁢

(

ϕ

)

:=

ϕ

⁢

(

s

)

, is a natural isomorphism between

G

and

G

^

^

.

6

Applications

The study of dual groups allows one to visualize

Fourier series

,

Fourier transforms

and

discrete Fourier transforms

from a more abstract and unified view-point, providing the for a general definition of Fourier transform. Thus, dual groups and

Pontryagin duality

are the of the of abstract

abelian

harmonic analysis

.

Title

Pontryagin duality

Canonical name

PontryaginDuality

Date of creation

2013-03-22 17:42:42

Last modified on

2013-03-22 17:42:42

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

7

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 43A40

Classification

msc 22B05

Classification

msc 22D35

Synonym

Pontrjagin duality

Synonym

Pontriagin duality

Related topic

DualityInMathematics

Defines

Pontryagin dual

Defines

Pontrjagin dual

Defines

Pontriagin dual

Defines

dual of an abelian group

Defines

character

Generated on Fri Feb 9 21:31:13 2018 by

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irreducible unitary representations of compact groups are finite-dimensional

irreducible unitary representations of compact groups are finite-dimensional

Theorem -

If

π

∈

r

⁢

e

⁢

p

⁢

(

G

,

H

)

is a

unitary representation

of a compact topological group

G

in a

Hilbert space

H

, then

π

has a finite-dimensional

subrepresentation

(

http://planetmath.org/TopologicalGroupRepresentation

).

Corollary 1 -

If

π

is irreducible (

http://planetmath.org/TopologicalGroupRepresentation

), then

H

must be finite-dimensional.

Corollary 2 -

π

has an .

Title

irreducible unitary representations of compact groups are finite-dimensional

Canonical name

IrreducibleUnitaryRepresentationsOfCompactGroupsAreFinitedimensional

Date of creation

2013-03-22 18:02:44

Last modified on

2013-03-22 18:02:44

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

13

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 43A65

Classification

msc 22C05

Classification

msc 22A25

Synonym

unitary representation of a compact group has a finite-dimensional subrepresentation

Related topic

UnitaryRepresentation

Defines

unitary representation of compact group has an irreducible subrepresentation

Defines

unitary group

of a complex Hilbert space

Generated on Fri Feb 9 21:31:15 2018 by

LaTeXML
representations of compact groups are equivalent to unitary representations

representations of compact groups are equivalent to unitary representations

\PMlinkescapephrase

representation

\PMlinkescapephrase

representations

\PMlinkescapephrase

equivalent

Theorem

-

Let

G

be a compact topological group. If

(

π

,

V

)

is a finite-dimensional representation (

http://planetmath.org/TopologicalGroupRepresentation

) of

G

in a

normed vector space

V

, then

π

is equivalent (

http://planetmath.org/TopologicalGroupRepresentation

) to a

unitary representation

.

Proof:

Let

⟨

⋅

,

⋅

⟩

denote an

inner product

in

V

. Define a new inner product in the

vector space

V

by

⟨

v

,

w

⟩

1

:=

∫

G

⟨

π

⁢

(

s

)

⁢

v

,

π

⁢

(

s

)

⁢

w

⟩

⁢

𝑑

μ

⁢

(

s

)

where

μ

is a

Haar measure

in

G

. It is easy to see that

⟨

⋅

,

⋅

⟩

1

defines indeed an inner product, noting that

⟨

π

⁢

(

⋅

)

⁢

v

,

π

⁢

(

⋅

)

⁢

w

⟩

is a

continuous function

in

G

.

Now we claim that, for every

s

∈

G

,

π

⁢

(

s

)

is a

unitary operator

for this new inner product. This is true since

⟨

π

⁢

(

t

)

⁢

v

,

π

⁢

(

t

)

⁢

w

⟩

1

=

∫

G

⟨

π

⁢

(

s

)

⁢

π

⁢

(

t

)

⁢

v

,

π

⁢

(

s

)

⁢

π

⁢

(

t

)

⁢

w

⟩

⁢

𝑑

μ

⁢

(

s

)

=

∫

G

⟨

π

⁢

(

s

⁢

t

)

⁢

v

,

π

⁢

(

s

⁢

t

)

⁢

w

⟩

⁢

𝑑

μ

⁢

(

s

)

=

∫

G

⟨

π

⁢

(

s

)

⁢

v

,

π

⁢

(

s

)

⁢

w

⟩

⁢

𝑑

μ

⁢

(

s

)

=

⟨

v

,

w

⟩

1

Denote by

V

′

the space

V

endowed the inner product

⟨

⋅

,

⋅

⟩

1

. As we have seen,

(

π

,

V

′

)

is a unitary representation of

G

in

V

′

. Of course,

(

π

,

V

′

)

and

(

π

,

V

)

are equivalent representations, since

π

=

id

-

1

⁢

π

⁢

id

where

id

:

V

⟶

V

′

is the

identity mapping

. Thus,

(

π

,

V

)

is equivalent to a unitary representation.

□

Title

representations of compact groups are equivalent to unitary representations

Canonical name

RepresentationsOfCompactGroupsAreEquivalentToUnitaryRepresentations

Date of creation

2013-03-22 18:02:28

Last modified on

2013-03-22 18:02:28

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

9

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 43A65

Classification

msc 22A25

Classification

msc 22C05

Generated on Fri Feb 9 21:31:17 2018 by

LaTeXML
delay theorem

delay theorem

Theorem

.

If

f

⁢

(

t

)

≡

0

for

t

<

0

and

ℒ

⁢

{

f

⁢

(

t

)

}

:=

F

⁢

(

s

)

,  one has

ℒ

⁢

{

f

⁢

(

t

-

t

0

)

}

=

e

-

t

0

⁢

s

⁢

F

⁢

(

s

)

.

Proof.

Since

f

⁢

(

t

-

t

0

)

≡

0

for

t

<

t

0

,  the

definition

of

Laplace transform

at first gives

ℒ

⁢

{

f

⁢

(

t

-

t

0

)

}

=

∫

t

0

∞

e

-

s

⁢

t

⁢

f

⁢

(

t

-

t

0

)

⁢

𝑑

t

.

The substitution (

http://planetmath.org/SubstitutionForIntegration

)

t

-

t

0

:=

u

yields

ℒ

⁢

{

f

⁢

(

t

-

t

0

)

}

=

∫

0

∞

e

-

s

⁢

(

u

+

t

0

)

⁢

f

⁢

(

u

)

⁢

𝑑

u

=

e

-

t

0

⁢

s

⁢

∫

0

∞

e

-

s

⁢

u

⁢

f

⁢

(

u

)

⁢

𝑑

u

=

e

-

t

0

⁢

s

⁢

F

⁢

(

s

)

.

Corollary.

For any

f

⁢

(

t

)

and the

Heaviside step function

H

⁢

(

t

)

, one has

ℒ

⁢

{

f

⁢

(

t

-

a

)

⁢

H

⁢

(

t

-

a

)

}

=

e

-

a

⁢

s

⁢

F

⁢

(

s

)

.

Title

delay theorem

Canonical

name

DelayTheorem

Date of creation

2013-03-22 18:02:46

Last modified on

2013-03-22 18:02:46

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Theorem

Classification

msc 44A10

Synonym

delay theorem of Laplace transform

Related topic

HeavisideStepFunction

Related topic

TelegraphEquation

Generated on Fri Feb 9 21:31:36 2018 by

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differentiation of Laplace transform with respect to parameter

differentiation of Laplace transform with respect to parameter

We use the curved from the Laplace-transformed

functions

to the corresponding initial functions.

If

f

⁢

(

t

,

x

)

↶

F

⁢

(

s

,

x

)

,

then one can differentiate both functions with respect to the parametre

x

:

f

x

′

⁢

(

t

,

x

)

↶

F

x

′

⁢

(

s

,

x

)

(1)

(1) may be written also as

ℒ

⁢

{

∂

∂

⁡

x

⁢

f

⁢

(

t

,

x

)

}

=

∂

∂

⁡

x

⁢

ℒ

⁢

{

f

⁢

(

t

,

x

)

}

.

(2)

Proof.

We differentiate partially both sides of the defining
equation

F

⁢

(

s

,

x

)

:=

∫

0

∞

e

-

s

⁢

t

⁢

f

⁢

(

t

,

x

)

⁢

𝑑

t

,

on the right hand side
under the integration sign (

http://planetmath.org/differentiationundertheintegralsign

), getting

F

x

′

⁢

(

s

,

x

)

=

∫

0

∞

e

-

s

⁢

t

⁢

f

x

′

⁢

(

t

,

x

)

⁢

𝑑

x

,

(3)

which means same as (1).  Q.E.D.

Example.

If the rule

s

s

2

-

a

2

↷

cosh

⁡

a

⁢

t

is differentiated with respect to

a

, the result is

2

⁢

a

⁢

s

(

s

2

-

a

2

)

2

↷

t

⁢

sinh

⁡

a

⁢

t

.

References

1

K. Väisälä:

Laplace-muunnos

.  Handout Nr. 163. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1968).

Title

differentiation of Laplace transform

with respect to

parameter

Canonical name

DifferentiationOfLaplaceTransformWithRespectToParameter

Date of creation

2014-03-09 12:52:11

Last modified on

2014-03-09 12:52:11

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

2

Author

pahio (2872)

Entry type

Theorem

Classification

msc 44A10

Generated on Fri Feb 9 21:31:39 2018 by

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Fekete’s subadditive lemma

Fekete’s subadditive lemma

Let

(

a

n

)

n

be a

subadditive

sequence

in

[

-

∞

,

∞

)

. Then, the following limit exists in

[

-

∞

,

∞

)

and equals the

infimum

of the same sequence:

lim

n

⁡

a

n

n

=

inf

n

⁡

a

n

n

Although the lemma is usually stated for subadditive sequences, an analogue

conclusion

is valid for

superadditive

sequences. In that case, for

(

a

n

)

n

a subadditive sequence in

(

-

∞

,

∞

]

, one has:

lim

n

⁡

a

n

n

=

sup

n

⁡

a

n

n

The proof of the superadditive case is obtained by taking the symmetric sequence

(

-

a

n

)

n

and applying the subadditive version of the

theorem

.

Title

Fekete’s subadditive lemma

Canonical name

FeketesSubadditiveLemma

Date of creation

2014-03-19 22:15:10

Last modified on

2014-03-19 22:15:10

Owner

Filipe (28191)

Last modified by

Filipe (28191)

Numerical id

4

Author

Filipe (28191)

Entry type

Theorem

Generated on Fri Feb 9 21:31:42 2018 by

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