aarjaneiro/planetmath huggingface dataset:
Proof of Dulac’s Criteria

Proof of Dulac’s Criteria

Consider the the planar system

x

˙

=

f

⁢

(

x

)

, where

f

=

(

X

,

Y

)

t

and

x

=

(

x

,

y

)

t

. Consider the vector field

(

-

ρ

⁢

Y

,

ρ

⁢

X

)

. Suppose that there is a periodic orbit

contained

in

E

associated to the planar system. Let

γ

be that periodic orbit. We have:

∫

γ

(

-

ρ

⁢

Y

,

ρ

⁢

X

)

⁢

𝑑

s

=

∫

0

τ

(

-

ρ

⁢

Y

⁢

(

x

,

y

)

,

ρ

⁢

X

⁢

(

x

,

y

)

)

⋅

(

x

˙

,

y

˙

)

⁢

𝑑

t

=

∫

0

τ

-

ρ

⁢

Y

⁢

(

x

,

y

)

⁢

X

⁢

(

x

,

y

)

+

ρ

⁢

X

⁢

(

x

,

y

)

⁢

Y

⁢

(

x

,

y

)

=

0

On the other hand, the region within

E

that is limited by

γ

is

simply connected

because

E

is simply connected. Let

E

~

be the region limited by

γ

. Then, by Green’s

theorem

, we have:

∫

γ

+

(

-

ρ

⁢

Y

,

ρ

⁢

X

)

⁢

𝑑

s

=

∫

∫

E

~

∂

∂

⁡

x

⁢

(

ρ

⁢

X

)

-

∂

∂

⁡

y

⁢

(

-

ρ

⁢

Y

)

⁢

d

⁢

x

⁢

d

⁢

y

=

∫

∫

E

~

∂

∂

⁡

x

⁢

(

ρ

⁢

X

)

+

∂

∂

⁡

y

⁢

(

ρ

⁢

Y

)

⁢

d

⁢

x

⁢

d

⁢

y

Because

E

~

has positive area and the integrand

function

has

constant

signal, then this integral is different from zero. This is a

contradiction

. So there are no periodic orbits.

\qed

Title

Proof of Dulac’s Criteria

Canonical name

ProofOfDulacsCriteria

Date of creation

2013-03-11 19:17:10

Last modified on

2013-03-11 19:17:10

Owner

Filipe (28191)

Last modified by

(0)

Numerical id

9

Author

Filipe (0)

Entry type

Proof

Classification

msc 34C25

Synonym

Related topic

Defines

Generated on Fri Feb 9 20:42:10 2018 by

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van der Pol equation

van der Pol equation

In 1920 the Dutch physicist Balthasar van der Pol studied a

differential equation

that describes the circuit of a vacuum tube.
It has been used
to model other phenomenon such as the human heartbeat by
Johannes van der Mark

[

C

]

.

The

van der Pol equation

equation is a case of a

Lienard system

and is expressed as
a

second order ordinary differential equation

d

2

⁢

x

d

⁢

t

2

-

μ

⁢

(

1

-

x

2

)

⁢

d

⁢

x

d

⁢

t

+

x

=

0

or a first order planar

ordinary differential equation

x

˙

=

y

+

μ

⁢

(

x

-

x

3

)

y

˙

=

-

x

where

μ

is a real

parameter

.
The parameter

μ

is usually considered to be

positive

since
the the term

-

μ

⁢

(

1

-

x

2

)

adds to the model a nonlinear damping.

[

C

]

Properties

:

•

If

μ

=

0

then the origin is a center. In fact, if

μ

=

0

then

d

2

⁢

x

d

⁢

t

2

+

x

=

0

and if we suppose that the

initial condition

are

(

x

0

,

x

˙

0

)

then the solution to the system is

x

⁢

(

t

)

=

x

0

⁢

cos

⁡

t

+

x

˙

0

⁢

sin

⁡

t

.

All
solutions except the origin are periodic and circles. See

phase portrait

below.

•

If

μ

>

0

the system has a unique

limit cycle

, and the limit
cycle is attractive. This follows directly from Lienard’s

theorem

.

[

P

]

•

The system is sometimes given under the form

X

˙

=

-

Y

Y

˙

=

X

+

μ

⁢

(

1

-

X

2

)

⁢

Y

which equivalent to the previous planar system
under the change of coordinate

(

X

,

Y

)

=

(

3

⁢

x

,

-

3

⁢

(

y

+

μ

⁢

(

x

-

x

3

)

)

)

.

[

C

]

Example:

The geometric representation of the phase portrait is done
by taking initial condition from
an equally spaced grid and calculating the solution for positive and
negative time.

For the parameter

μ

=

1

, the system has
an attractive limit cycle and the origin is a repulsive focus.

Phase portrait when

μ

=

1

.

When the parameter

μ

=

0

the origin is a center.

Phase portrait when

μ

=

0

.

For the parameter

μ

=

-

1

, the system has a repulsive limit cycle and the origin is
an attractive focus.

Phase portrait when

μ

=

-

1

.

References

C

Chicone, Carmen

,

Ordinary Differential Equations with Applications

,
Springer, New York, 1999.

P

Perko, Lawrence

,

Differential Equations and

Dynamical Systems

,
Springer, New York, 2001.

Title

van der Pol equation

Canonical

name

VanDerPolEquation

Date of creation

2013-03-22 16:06:42

Last modified on

2013-03-22 16:06:42

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

13

Author

Daume (40)

Entry type

Definition

Classification

msc 34C25

Classification

msc 34C07

Classification

msc 34-00

Synonym

van der Pol oscillator

Generated on Fri Feb 9 20:42:13 2018 by

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attractor

attractor

Let

x

˙

=

f

⁢

(

x

)

be a system of

autonomous

ordinary differential equation

in

ℝ

n

defined by a vector field

f

:

ℝ

n

→

ℝ

n

.
A set

A

is said to be an

attracting set

[

GH

,

P

]

if

1.

A

is closed and invariant,

2.

there exists an open neighborhood

U

of

A

such that all solution
with initial solution in

U

will eventually enter

A

(

x

⁢

(

t

)

→

A

) as

t

→

∞

.

Additionally, if

A

contains a dense orbit then

A

is said to be an

attractor

[

GH

,

P

]

.

Conversely, a set

R

is said to be a

repelling set

[

GH

]

if

R

satisfy the condition 1. and 2. where

t

→

∞

is replaced by

t

→

-

∞

. Similarly, if

R

contains a dense orbit then

R

is said to be a

repellor

[

GH

]

.

References

GH

Guckenheimer, John & Holmes, Philip

,

Nonlinear Oscillations,

Dynamical Systems

,
and

Bifurcations

of Vector Fields

,
Springer, New York, 1983.

P

Perko, Lawrence

,

Differential Equations

and Dynamical Systems

,
Springer, New York, 2001.

Title

attractor

Canonical name

Attractor

Date of creation

2013-03-22 15:17:34

Last modified on

2013-03-22 15:17:34

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

4

Author

Daume (40)

Entry type

Definition

Classification

msc 34C99

Defines

attracting set

Defines

repelling set

Defines

repellor

Generated on Fri Feb 9 20:42:15 2018 by

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equilibrium point

equilibrium point

Consider an autonomous differential equation

x

˙

=

f

⁢

(

x

)

.

(1)

An

equilibrium point

x

0

of (

1

) is such that

f

⁢

(

x

0

)

=

0

.

Conversely

a

regular point

of (

1

) is such that

f

⁢

(

x

0

)

≠

0

.

If the linearization

D

⁢

f

⁢

(

x

0

)

has no eigenvalue with zero real
part,

x

0

is said to be a hyperbolic equilibrium, whereas if
there exists an eigenvalue with zero real part, the equilibrium
point is nonhyperbolic.

An equilibrium point

x

0

is said to be

stable

if for
every

neighborhood

x

0

,

U

there exists a neighborhood
of

x

0

,

U

′

⊂

U

such that every solution of (

1

)
with

initial condition

in

U

′

(i.e.

x

⁢

(

0

)

∈

U

′

)

,

satisfies

x

⁢

(

t

)

∈

U

for all

t

≥

0

.

Consequently an equilibrium point

x

0

is said to be

unstable

if it is not stable.

Moreover an equilibrium point

x

0

is said to be

asymptotically stable

if it is stable and there exists

U

′′

such that every solution of (

1

) with initial
condition in

U

′′

(i.e.

x

⁢

(

0

)

∈

U

′′

)

satisfies

lim

t

→

∞

⁡

x

⁢

(

t

)

=

x

0

.

Title

equilibrium point

Canonical

name

EquilibriumPoint

Date of creation

2013-03-22 13:18:34

Last modified on

2013-03-22 13:18:34

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

10

Author

Daume (40)

Entry type

Definition

Classification

msc 34C99

Synonym

steady state solution

Synonym

fixed point

Synonym

singular point

Defines

hyperbolic equilibrium

Defines

nonhyperbolic equilibrium

Defines

stable

Defines

unstable

Defines

asymptotically stable

Generated on Fri Feb 9 20:42:17 2018 by

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Hartman-Grobman theorem

Hartman-Grobman theorem

Consider the

differential equation

x

′

=

f

⁢

(

x

)

(1)

where

f

is a

C

1

vector field. Assume that

x

0

is a

hyperbolic equilibrium

of

f

. Denote

Φ

t

⁢

(

x

)

the flow of (

1

) through

x

at time

t

. Then there exists a

homeomorphism

φ

⁢

(

x

)

=

x

+

h

⁢

(

x

)

with

h

bouded, such that

φ

∘

e

t

⁢

D

⁢

f

⁢

(

x

0

)

=

Φ

t

∘

φ

is a sufficiently small neighboorhood of

x

0

.

This fundamental theorem in the qualitative analysis of nonlinear differential equations states that, in a small

neighborhood

of

x

0

, the flow of the nonlinear equation (

1

) is qualitatively similar to that of the linear system

x

′

=

D

⁢

f

⁢

(

x

0

)

⁢

x

.

Title

Hartman-Grobman theorem

Canonical name

HartmanGrobmanTheorem

Date of creation

2013-03-22 13:18:37

Last modified on

2013-03-22 13:18:37

Owner

jarino (552)

Last modified by

jarino (552)

Numerical id

4

Author

jarino (552)

Entry type

Theorem

Classification

msc 34C99

Generated on Fri Feb 9 20:42:19 2018 by

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nullcline

nullcline

Let

x

˙

1

=

f

1

⁢

(

x

1

,

…

,

x

n

)

⋮

x

˙

n

=

f

n

⁢

(

x

1

,

…

,

x

n

)

be a system of first order

ordinary differential equation

. The

x

j

nullcline

is the set of points which satisfy

f

j

⁢

(

x

1

,

…

,

x

n

)

=

0

. Note that at an intersection point of all the nullclines implies that

0

=

f

1

⁢

(

x

1

,

…

,

x

n

)

⋮

0

=

f

n

⁢

(

x

1

,

…

,

x

n

)

.

Hence the intersection point of all the nullclines is an

equilibrium point

of the system.

example:

•

see some qualitative analysis of

FitzHugh-Nagumo equation

using nullclines

Title

nullcline

Canonical

name

Nullcline

Date of creation

2013-03-22 14:27:14

Last modified on

2013-03-22 14:27:14

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

5

Author

Daume (40)

Entry type

Definition

Classification

msc 34C99

Generated on Fri Feb 9 20:42:21 2018 by

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stable manifold theorem

stable manifold theorem

Let

E

be an

open subset

of

ℝ

n

containing the origin, let

f

∈

C

1

⁢

(

E

)

, and let

ϕ

t

be the flow of the nonlinear system

x

′

=

f

⁢

(

x

)

.

Suppose that

f

⁢

(

x

0

)

=

0

and that

D

⁢

f

⁢

(

x

0

)

has

k

eigenvalues with negative real part and

n

-

k

eigenvalues with positive real part. Then there exists a

k

-dimensional

differentiable manifold

S

tangent to the stable subspace

E

S

of the linear system

x

′

=

D

⁢

f

⁢

(

x

)

⁢

x

at

x

0

such that for all

t

≥

0

,

ϕ

t

⁢

(

S

)

⊂

S

and for all

y

∈

S

,

lim

t

→

∞

⁡

ϕ

t

⁢

(

y

)

=

x

0

and there exists an

n

-

k

dimensional differentiable manifold

U

tangent to the

unstable

subspace

E

U

of

x

′

=

D

⁢

f

⁢

(

x

)

⁢

x

at

x

0

such that for all

t

≤

0

,

ϕ

t

⁢

(

U

)

⊂

U

and for all

y

∈

U

,

lim

t

→

-

∞

⁡

ϕ

t

⁢

(

y

)

=

x

0

.

Title

stable manifold theorem

Canonical name

StableManifoldTheorem

Date of creation

2013-03-22 12:57:17

Last modified on

2013-03-22 12:57:17

Owner

jarino (552)

Last modified by

jarino (552)

Numerical id

4

Author

jarino (552)

Entry type

Theorem

Classification

msc 34C99

Generated on Fri Feb 9 20:42:32 2018 by

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Lyapunov stable

Lyapunov stable

A

fixed point

x

*

is

Lyapunov stable

if trajectories of nearby points

x

remain close for future time. More formally the fixed point

x

*

is Lyapunov stable, if for any

ϵ

>

0

, there is a

δ

>

0

such that for all

t

≥

0

and for all

x

≠

x

*

it is verified

d

⁢

(

x

*

,

x

)

<

δ

⇒

d

⁢

(

x

*

,

x

⁢

(

t

)

)

<

ϵ

.

In particular,

d

⁢

(

x

*

,

x

⁢

(

0

)

)

=

0

.

Title

Lyapunov stable

Canonical

name

LyapunovStable

Date of creation

2013-03-22 13:06:29

Last modified on

2013-03-22 13:06:29

Owner

armbrusterb (897)

Last modified by

armbrusterb (897)

Numerical id

10

Author

armbrusterb (897)

Entry type

Definition

Classification

msc 34D20

Synonym

Lyapunov stability

Synonym

Liapunov stable

Synonym

Liapunov stability

Related topic

AsymptoticallyStable

Related topic

AttractingFixedPoint

Related topic

StableFixedPoint

Related topic

NeutrallyStableFixedPoint

Related topic

UnstableFixedPoint

Generated on Fri Feb 9 20:42:33 2018 by

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neutrally stable fixed point

neutrally stable fixed point

A

fixed point

is considered neutrally stable if is

Liapunov stable

but not attracting. A center is an example of such a fixed point.

Title

neutrally stable fixed point

Canonical

name

NeutrallyStableFixedPoint

Date of creation

2013-03-22 13:06:35

Last modified on

2013-03-22 13:06:35

Owner

armbrusterb (897)

Last modified by

armbrusterb (897)

Numerical id

6

Author

armbrusterb (897)

Entry type

Definition

Classification

msc 34D20

Classification

msc 34K20

Related topic

AttractingFixedPoint

Related topic

LiapunovStable

Related topic

StableFixedPoint

Related topic

UnstableFixedPoint

Generated on Fri Feb 9 20:42:35 2018 by

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stable fixed point

stable fixed point

Let

X

be a vector field on a manifold

M

. A

fixed point

of

X

is said to be stable if it is both attracting (

http://planetmath.org/AttractingFixedPoint

) and

Lyapunov stable

.

Title

stable fixed point

Canonical

name

StableFixedPoint

Date of creation

2013-03-22 13:06:32

Last modified on

2013-03-22 13:06:32

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 34D20

Synonym

asymptotically stable fixed point

Related topic

AttractingFixedPoint

Related topic

LiapunovStable

Related topic

NeutrallyStableFixedPoint

Related topic

UnstableFixedPoint

Generated on Fri Feb 9 20:42:37 2018 by

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stable matrix

stable matrix

A

square matrix

is said to be a

stable matrix

if every

eigenvalue

of has

negative

real part. The matrix is called

positive
stable

if every eigenvalue has positive real part.

Motivation:

In the following system of

linear differential
equations

,

𝐱

′

⁢

(

t

)

=

M

⁢

𝐱

⁢

(

t

)

it is easy to see that the point

𝐱

=

𝟎

is an

equilibrium point

. The trajectory

𝐱

⁢

(

t

)

will converge to

𝟎

for every initial value

𝐱

⁢

(

0

)

if and only if
the matrix

M

is a stable matrix.

Title

stable matrix

Canonical

name

StableMatrix

Date of creation

2013-03-22 15:27:40

Last modified on

2013-03-22 15:27:40

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 34D23

Classification

msc 15A57

Defines

positive stable

Generated on Fri Feb 9 20:42:38 2018 by

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Gelfand spectral radius theorem

Gelfand spectral radius theorem

For every

self-consistent matrix norm

,

|

|

⋅

|

|

, and every

square matrix

𝐀

we can write

ρ

⁢

(

𝐀

)

=

lim

n

→

∞

⁡

||

𝐀

n

||

1

n

.

Note:

ρ

⁢

(

𝐀

)

denotes the

spectral radius

of

𝐀

.

This

theorem

also generalizes to

infinite

dimensions

and plays an important role in the theory of operator algebras. If

𝒜

is a

Banach algebra

with norm

|

|

⋅

|

|

and

A

∈

𝒜

, then we have

ρ

⁢

(

𝐀

)

=

lim

n

→

∞

⁡

||

𝐀

n

||

1

n

.

It is worth pointing out that the self-consistency condition which was imposed on the

matrix norm

is part of the definition of a Banach algebra. A common case of the infinite-dimensional

generalization

occurs when

𝒜

is the algebra of

bounded operators

on a

Hilbert space

— the

operators

may be regarded as an infinite-dimensional generalization of the square matrices.

Title

Gelfand spectral radius theorem

Canonical name

GelfandSpectralRadiusTheorem

Date of creation

2013-03-22 13:39:19

Last modified on

2013-03-22 13:39:19

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

9

Author

Andrea Ambrosio (7332)

Entry type

Theorem

Classification

msc 34L05

Synonym

spectral radius formula

Related topic

SelfConsistentMatrixNorm

Generated on Fri Feb 9 20:42:40 2018 by

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proof of Gelfand spectral radius theorem

proof of Gelfand spectral radius theorem

For any

ϵ

>

0

, consider the matrix

A

~

=

(

ρ

⁢

(

A

)

+

ϵ

)

-

1

⁢

A

Then, obviously,

ρ

⁢

(

A

~

)

=

ρ

⁢

(

A

)

ρ

⁢

(

A

)

+

ϵ

<

1

and, by a well-known result on convergence of

matrix powers

,

lim

k

→

∞

⁡

A

~

k

=

0

.

That means, by

sequence

limit

definition

, a

natural number

N

1

∈

𝐍

exists such that

∀

k

≥

N

1

⇒

∥

A

~

k

∥

<

1

which in turn means:

∀

k

≥

N

1

⇒

∥

A

k

∥

<

(

ρ

⁢

(

A

)

+

ϵ

)

k

or

∀

k

≥

N

1

⇒

∥

A

k

∥

1

/

k

<

(

ρ

⁢

(

A

)

+

ϵ

)

.

Let’s now consider the matrix

A

ˇ

=

(

ρ

⁢

(

A

)

-

ϵ

)

-

1

⁢

A

Then, obviously,

ρ

⁢

(

A

ˇ

)

=

ρ

⁢

(

A

)

ρ

⁢

(

A

)

-

ϵ

>

1

and so, by the same convergence

theorem

,

∥

A

ˇ

k

∥

is not bounded.
This means a natural number

N

2

∈

𝐍

exists such that

∀

k

≥

N

2

⇒

∥

A

ˇ

k

∥

>

1

which in turn means:

∀

k

≥

N

2

⇒

∥

A

k

∥

>

(

ρ

⁢

(

A

)

-

ϵ

)

k

or

∀

k

≥

N

2

⇒

∥

A

k

∥

1

/

k

>

(

ρ

⁢

(

A

)

-

ϵ

)

.

Taking

N

:=

m

⁢

a

⁢

x

⁢

(

N

1

,

N

2

)

and putting it all together, we obtain:

∀

ϵ

>

0

,

∃

N

∈

ℕ

:

∀

k

≥

N

⇒

ρ

⁢

(

A

)

-

ϵ

<

∥

A

k

∥

1

/

k

<

ρ

⁢

(

A

)

+

ϵ

which, by definition, is

lim

k

→

∞

⁡

∥

A

k

∥

1

/

k

=

ρ

⁢

(

A

)

.

□

Actually, in case the norm is

self-consistent

(

http://planetmath.org/SelfConsistentMatrixNorm

), the proof shows more than the thesis; in fact, using the fact that

|

λ

|

≤

ρ

⁢

(

A

)

, we can replace in the limit definition the left lower bound with the

spectral radius

itself and write more precisely:

∀

ϵ

>

0

,

∃

N

∈

ℕ

:

∀

k

≥

N

⇒

ρ

⁢

(

A

)

≤

∥

A

k

∥

1

/

k

<

ρ

⁢

(

A

)

+

ϵ

which, by definition, is

lim

k

→

∞

⁡

∥

A

k

∥

1

/

k

=

ρ

⁢

(

A

)

+

.

Title

proof of Gelfand spectral radius theorem

Canonical

name

ProofOfGelfandSpectralRadiusTheorem

Date of creation

2013-03-22 15:33:55

Last modified on

2013-03-22 15:33:55

Owner

Andrea Ambrosio (7332)

Last modified by

Andrea Ambrosio (7332)

Numerical id

7

Author

Andrea Ambrosio (7332)

Entry type

Proof

Classification

msc 34L05

Generated on Fri Feb 9 20:42:42 2018 by

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Rayleigh quotient

Rayleigh quotient

Definition

.

The

Rayleigh quotient

,

R

A

, to the

Hermitian matrix

A

is defined as

R

𝐀

⁢

(

𝐱

)

=

𝐱

H

⁢

𝐀𝐱

𝐱

H

⁢

𝐱

,

𝐱

≠

𝟎

,

where

x

H

is the

Hermitian conjugate

of

x

.

The importance of this quantity (in fact, the reason Rayleigh first
introduced it) is that its critical values are the

eigenvectors

of

A

and the values of the

quotient

at these special vectors are the
corresponding

eigenvalues

. This observation leads to the

variational
method

for computing the

spectrum

of a

positive matrix

(either exactly or
approximately). Namely, one first minimizes the Rayleigh quotient over the
whole

vector space

. This gives the lowest eigenvalue and corresponding
eigenvector. Next, one restricts attention to the

orthogonal complement

of the eigenvector found in the first step and minimizes over this

subspace

.
That produces the next lowest eigenvalue and corresponding eigenvector. One
can continue this process recursively. At each step, one minimizes the
Rayleigh quotient over the subspace

orthogonal

to all the vectors found in
the preceding steps to find another eigenvalue and its corresponding
eigenvector.

This

concept

of Rayleigh quotient also makes sense in the more general
setting when

A

is a

Hermitian operator

on a Hilbert space. Furthermore,
it is possible to make use of the

Rayleigh-Ritz method

in cases where the

operator

has a discrete spectrum

bounded from below

, such as the Laplace
operator on a compact domain. This method is often employed in practise
because, in physical applications, one is oftentimes interested in only the
lowest eigenvalue or perhaps the first few lowest eigenvalues and not so
concerned with the rest of the spectrum.

Title

Rayleigh quotient

Canonical

name

RayleighQuotient

Date of creation

2013-03-22 13:39:17

Last modified on

2013-03-22 13:39:17

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

9

Author

alozano (2414)

Entry type

Definition

Classification

msc 34L15

Defines

variational method

Generated on Fri Feb 9 20:42:44 2018 by

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construction of Dirac delta function

construction of Dirac delta function

The

Dirac delta function

is notorious in mathematical circles for having no actual as a

function

. However, a little known secret is that in the domain of

nonstandard analysis

, the Dirac delta function admits a completely legitimate construction as an actual function. We give this construction here.

Choose any

positive

infinitesimal

ε

and define the hyperreal valued function

δ

:

*

ℝ

⟶

*

ℝ

by

δ

⁢

(

x

)

:=

{

1

/

ε

-

ε

/

2

<

x

<

ε

/

2

,

0

otherwise.

We verify that the above function

satisfies

the required

properties

of the Dirac delta function. By

definition

,

δ

⁢

(

x

)

=

0

for all nonzero

real numbers

x

. Moreover,

∫

-

∞

∞

δ

⁢

(

x

)

⁢

𝑑

x

=

∫

-

ε

/

2

ε

/

2

1

ε

⁢

𝑑

x

=

1

,

so the

integral

property is satisfied. Finally, for any

continuous

real function

f

:

ℝ

⟶

ℝ

, choose an infinitesimal

z

>

0

such that

|

f

⁢

(

x

)

-

f

⁢

(

0

)

|

<

z

for all

|

x

|

<

ε

/

2

; then

ε

⋅

f

⁢

(

0

)

-

z

ε

<

∫

-

∞

∞

δ

⁢

(

x

)

⁢

f

⁢

(

x

)

⁢

𝑑

x

<

ε

⋅

f

⁢

(

0

)

+

z

ε

which implies that

∫

-

∞

∞

δ

⁢

(

x

)

⁢

f

⁢

(

x

)

⁢

𝑑

x

is within an infinitesimal of

f

⁢

(

0

)

, and thus has real part equal to

f

⁢

(

0

)

.

Title

construction of Dirac delta function

Canonical

name

ConstructionOfDiracDeltaFunction

Date of creation

2013-03-22 12:35:48

Last modified on

2013-03-22 12:35:48

Owner

djao (24)

Last modified by

djao (24)

Numerical id

5

Author

djao (24)

Entry type

Derivation

Classification

msc 34L40

Classification

msc 26E35

Generated on Fri Feb 9 20:42:46 2018 by

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Dirac delta function

Dirac delta function

The Dirac delta “

function

”

δ

⁢

(

x

)

, or

distribution

is not a true function because it is not uniquely defined for all values of the

argument

x

. Similar to the

Kronecker delta

symbol, the notation

δ

⁢

(

x

)

stands for

δ

⁢

(

x

)

=

0

⁢

for

⁢

x

≠

0

,

and

⁢

∫

-

∞

∞

δ

⁢

(

x

)

⁢

𝑑

x

=

1

For any

continuous function

F

:

∫

-

∞

∞

δ

⁢

(

x

)

⁢

F

⁢

(

x

)

⁢

𝑑

x

=

F

⁢

(

0

)

or in

n

dimensions

:

∫

ℝ

n

δ

⁢

(

x

-

s

)

⁢

f

⁢

(

s

)

⁢

d

n

⁢

s

=

f

⁢

(

x

)

δ

⁢

(

x

)

can also be defined as a normalized

Gaussian function

(

normal distribution

) in the limit of zero width.

Notes:

However, the limit of the normalized Gaussian function is still meaningless as a function, but some people still write such a limit as being equal to the Dirac distribution considered above in the first paragraph.

An example of how the Dirac distribution arises in a physical, classical context is available

http://www.rose-hulman.edu/ rickert/Classes/ma222/Wint0102/dirac.pdf

on line.

Remarks:

Distributions play important roles in Dirac’s formulation of quantum mechanics.

References

1

W. Rudin,

Functional Analysis

,
McGraw-Hill Book Company, 1973.

2

L. Hörmander,

The

Analysis

of Linear Partial Differential Operators I,
(Distribution theory and Fourier Analysis)

, 2nd ed, Springer-Verlag, 1990.

3

Originally from The Data Analysis Briefbook
(

http://rkb.home.cern.ch/rkb/titleA.html

http://rkb.home.cern.ch/rkb/titleA.html)

Title

Dirac delta function

Canonical name

DiracDeltaFunction

Date of creation

2013-03-22 12:11:45

Last modified on

2013-03-22 12:11:45

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

16

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 34L40

Synonym

delta function

Related topic

DiracSequence

Related topic

DiracMeasure

Related topic

Distribution4

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