Appel W. Mathematics for Physics and Physicists

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Deﬁnitions and ex amples of distributions 187

−a

The δ distribution δ

−

−a

δ The X distribution

7.2.c Support of a distribution

Two distributions S a nd T are e qual if 〈S , ϕ〉 = 〈T , ϕ〉 for any ϕ ∈ D. The y

are equal on an open subset Ω in RR

if 〈S, ϕ〉= 〈T , ϕ〉 for any ϕ ∈ D such

that the support of ϕ is contained in Ω (i.e., for any ϕ ∈ D which is zero

outside Ω).

Example 7.17 Let H be th e Heaviside function. The regular distributions 1 and H are equal

on ]0,+∞[. Indeed, for any function ϕ ∈ D wh ich vanishes on R

−

, we have

〈1, ϕ〉 =

+∞

−∞

1 ·ϕ(x) dx =

+∞

ϕ(x) dx = 〈H ,ϕ〉.

Similarly, X and δ are equal on ]−

Consider now all t he open subsets Ω on wh ich a given distribution T is

zero (i. e. , is equal to the zero distribut ion on Ω). The union of t hese open sets

is itself an open set . One can then show (but this is not obvious and we will

omit it here, see [80] for instance) that this is the largest open set on which T

is zero. Its complement, which is therefore closed, is called the support of the

distribution T , and is denoted supp(T ).

Example 7.18 The suppor t of the Dirac distribution centered at 0 is supp(δ) = {0}, and more

generally, for a ∈ R, we have supp(δ

) = {a}.

The support of t h e Dirac comb is supp(X) = Z.

7.2.d Other examples

PROPOSITION 7.19 Let ϕ ∈ D(R) be a test function. The quantity

−η

−∞

ϕ(x)

dx +

+∞

+η

ϕ(x)

has a ﬁnite limit as [η → 0

]. M oreover, the ma p

ϕ 7−→ lim

η→0

|x|>η

ϕ(x)

dx (7.2)

is linear and continuous on D(R).

Proof. Because of its l ength, the proof of this theorem is given in Appendix D,

page 600.

188 Distributions I

DEFINITION 7.20 The Cauchy principal value of 1/ x is the distr ibution on

R deﬁ ned by its action on any ϕ g iven by



, ϕ

= pv

ϕ(x)

def

= lim

η→0

|x|>η

ϕ(x)

dx.

It s hould be emphasized that this integral is deﬁned symmetrically around the

origin. It is this symmetry which provides the ex istence of the limit d eﬁning

the distribution. The distribution pv 1/x can also be seen as the derivative

of the regular distrib ut ion associated to the locally integrable function x 7→

log |x| (see Exercise 8.4 on page 241).

Other properties and examples of physical applications of the Cauchy

principal value will be given in the next chapter. In addition, a generalization

of the method leading to the deﬁnition of a distribution from the function

x 7→ 1/x (which is not locally integrable) is given in Exercises 8.5 and 8.6 on

page 241.

7.3

Elementary properties. Oper ations

7.3.a Operations on distributions

We wish to deﬁne certain operations on distributions, such as translations,

scaling, derivation, and so on. For this purpose, the method will be consis-

tently the same:

Consider how those operations are deﬁned for a locally integrable func-

tion; write them, in the la ngua ge of distributions, for the associated

regular distribution; then generalize this to an arbitrary distribution.

For instance, for any locally integrable function f on R

and any a ∈ R

we deﬁne the translate f

of the function f by

∀x ∈ R f

(x)

def

= f (x −a).

In other words, the graph (in R

n+1

) of the function f has been translated by

the vector a:

Elementary properties. Operations 189

f f

The regular distribution associated to this function, which is also denoted

, th erefore satisﬁes

〈f

, ϕ〉 =

f (x − a) ϕ(x) dx =

f ( y) ϕ( y + a) dy =



f , ϕ

−a



This can be generalized to an arbitrary distribution T in the following man-

ner:

DEFINITION 7.21 The translate of the distrib ut ion T by a, denoted T

T (x −a), is deﬁned by

〈T

, ϕ〉

def



T , ϕ

−a



for any ϕ ∈ D(R

which is also written



T (x −a), ϕ(x)



def



T (x), ϕ(x + a)



The reader ca n easily check that the number



T , ϕ

−a



exists and that the map

ϕ 7→



T , ϕ

−a



is indeed linear and continuous.

Remark 7.22 As in Remark 7.15, the notation T (x − a) is simply synonymous with T

; simi-

larly, with an abuse of language, ϕ(x −a) sometimes designates (for physicists more than for

mathematicians) the function t 7→ ϕ(t −a) and not the value of ϕ at x −a.

Example 7.23 Let a ∈ R. The translate by a of the Cauchy principal value is

x − a

: ϕ 7−→ lim

η→0



a−η

−∞

ϕ(x)

x − a

dx +

+∞

a+η

ϕ(x)

x − a



By analogous reasoning, we deﬁne the “transpose” of a distribution. For

a function f : R

→ C, we denote by

f t he transpose of f , which is the

function x 7→

f (x)

def

= f (−x). If f is locally integrable, then so is

f ; hence

both f and

f deﬁne regular distributions. Moreover, for any ϕ ∈ D(R

), we

have

〈

f , ϕ〉 =

f (x) ϕ(x) dx =

f (−x) ϕ(x) dx

x=−t

f (t) ϕ(−t) dt = 〈f , ˇϕ〉.

This justiﬁes the following deﬁnition:

190 Distributions I

DEFINITION 7.24 (Transpose) Th e transpose of a distribution T on R

, de-

noted

T or T (−x), is deﬁned by

〈

T , ϕ〉

def

= 〈T , ˇϕ〉,

which can also be written:



T (−x), ϕ(x)



def



T (x), ϕ(−x)



for any ϕ ∈ D(R

Similarly, for any a ∈ R

∗

, a change of scale deﬁnes the dilation by a of a

function f by x 7→ f (ax). The regular distribution a ssociated to the dilation

of a function satisﬁes



f (ax), ϕ(x)



f (ax) ϕ(x) dx =

f (x) ϕ





|a|

f (x), ϕ



E

This leads to the following generalization:

DEFINITION 7.25 (Dilat ion) Let a ∈ R

∗

. The dilation of a distribution T by

the factor a, denoted T (ax), is deﬁned by



T ( ax), ϕ(x)



def

|a|

T (x), ϕ



E

for any ϕ ∈ D(R).

Dilations of a distribution on R

are deﬁned in exactly the same manner.

However, the jacobian associated to the change of variable y = ax is |a|

instead of |a|, which gives the dilation relations



T (a x), ϕ( x)



|a|

T ( x), ϕ



E

on R

There is , unfortunately, no general deﬁnition of the product of two dis-

tributions.

Already, if f a nd g are two locally integrable functions (hence

deﬁning regular distributions), their product is not necessarily locally inte-

grable.

However, if ψ is a function of class C

∞

, then for any ϕ ∈ D, the

product ψϕ is a test function (still being C

∞

and with bounded support); if

f is locally integrable, then ψ f is also locally integ rab le. From the point of

view of distribut ions, we ca n write

〈ψ f , ϕ〉 =



ψ(x) f (x)



ϕ(x) dx =

f (x)



ψ(x) ϕ(x)



dx = 〈f , ψϕ〉.

In particular, i t is easy to notice that taking the square of the Dirac distribution raises

arduous questions. However, it turns out to be p o ssible to deﬁne a p roduct i n a larger space

than the space of distributions, the space of generalized distributions of Colombeau [22, 23].

Simply take f (x) = g(x) = 1/

Elementary properties. Operations 191

This leads us to state the following deﬁnition:

DEFINITION 7.26 (Product of a distribution by a CC

∞

function) Let T ∈ D

′

be a distr ibution and let ψ be a function of clas s C

∞

. The product ψT ∈ D

′

is deﬁned by

〈ψT , ϕ〉

def

= 〈T , ψϕ〉 for any ϕ ∈ D.

By applying this deﬁnition to the Dirac distribution, the following funda-

mental result follows:

THEOREM 7.27 Let ψ be a function of C

∞

class. Then we have

ψ(x) δ(x) = ψ(0) δ(x) and in particular x δ(x) = 0.

Similarly, we have the relation

ψ(x) X(x) =

n∈Z

ψ(n) δ(x −n).

Proof. If ψ is o f C

∞

class, then by deﬁnition

〈ψδ, ϕ〉 = 〈δ, ψϕ〉 = ψ(0 ) ϕ( 0 ) = ψ(0) 〈δ, ϕ〉 =



ψ(0) δ, ϕ



Since any test function has bounded support, the last formula follows by simple lin-

earity.

It is important to notice that, for instance, the product H δ makes no

sense here!

THEOREM 7.28 The equation x T (x) = 0, with unknown a distribution T , admits

the multiples of the Dirac distribution as solutions, and they are the only solutions:

x T (x) = 0 ⇐⇒



T = αδ, with α ∈ C



Remark 7.29 If ψ is merely continuous, the product ψT cannot be deﬁned for an arbitrary

distribution T , but the product ψδ can in fact still be deﬁned. Similarly, if ψ is of C

class

(in a neighborhood of 0), the product ψδ

′

can be deﬁned (see the next section).

7.3.b De rivative of a distribution

The next goal is to deﬁne derivation in the space of distributions. As before,

consider a locally integrable function f ∈ L

loc

, and assume moreover that it

is differentiable and its derivative is also locally integrable. Then the regular

distribution associated to f

′

is given by



′

, ϕ



′

(x) ϕ(x) dx = −

f (x) ϕ

′

(x) dx = −



f , ϕ

′



The passage to the second integral is done by integrating by par ts; th e bound-

ary terms vanish because ϕ has bounded support (this is one reason for th e

“very restrictive” choice of the space of test functions). This justiﬁes the

following generalization:

192 Distributions I

DEFINITION 7.30 The derivative of a distribution T ∈ DD

DD(RR

RR)

′

is deﬁned by



′

, ϕ



def

= −



T , ϕ

′



for any ϕ ∈ D(R). (7.3)

Similarly, the higher order derivatives are deﬁned by induction by

(m)

, ϕ

def

= (−1)

T , ϕ

(m)

for any ϕ ∈ D and a ny m ∈ N.

Example 7.31 The function x 7→ log |x| is locally integrable and therefore deﬁnes a regular

distribution. However, its derivative (in th e usual sense of functions) x 7→ 1/x is not locally

integrable; therefore it does not deﬁne a (regular) distribution.

However, if the derivation is performed directly in the sense of distributions,



log |x|



′

is a

distribution (it is shown in Exercise 8.4 on page 241 that it is the Cauchy principal value pv

If working on R

, a similar computation justiﬁes the deﬁnition of partial

derivatives of a distribution T ∈ D

′

) by



∂ T

∂ x

, ϕ



def

= −



T ,

∂ϕ

∂ x



for ϕ ∈ D(R

) and 1 ¶ i ¶ n. Partial derivatives of higher order are obtained

by successive applications of these rules: for instance,



∂

∂ x

, ϕ



= (−1)



T ,

∂

∂ x



In particular, on R

, we have



∂ T

∂ x

, ϕ



= −



T ,

∂ϕ

∂ x





∂ T

∂ y

, ϕ



= −



T ,

∂ϕ

∂ y





∂ T

∂ z

, ϕ



= −



T ,

∂ϕ

∂ z



Similarly, the laplacian being a differential operator of order 2, we deﬁne:

DEFINITION 7.32 The laplacian of a distribution on R

is given by

〈△T , ϕ〉

def

= 〈T , △ϕ〉 for any ϕ ∈ D(R

THEOREM 7.33 (Differentiability of distributions) Any distribution on R

ad-

mits partial derivatives of arbitrar y order. Any partial derivative of a distribution is

a distribution.

COROLLARY 7.33 .1 Any locally integrable function on R deﬁning a distribution,

it is inﬁnitely differentiable in the sense of distributions:

∀f ∈ L

loc

(R) ∀k ∈ N f

(k)

∈ D

′

(R).

Dirac and its derivatives 193

At the age of sixteen, Oliver Heaviside (1850—1925), born in a

very poor family, had to leave school. At eighteen, he was hired

by Charles Wheatstone, the inventor of the telegraph. He then

studied t h e differential equations which govern electri c signals

and in particular the “square root of the derivative” (he wrote

d/dt H (t) = 1/

πt), wh ich gave apoplexies to the mathe-

maticians of Cambridge.

One of his papers occasioned the se-

vere disapproval of the Royal Society. He defended vector analysis

(Maxwell was in favor of the use of quaternions instead of vec-

tors), symbolic calculus, and the use of divergent series, and had

faith in methods “whic h work” before being rigorously justiﬁed.

When Laurent Schwartz presented the theory of d istributions, this result

seemed particularly marvelous to the audience: all distributions are inﬁnitely

differentiable and their successive derivatives are themselves distributions.

Thus, if we consider a locally integrable function, however irregular, its as-

sociated distribution is still inﬁnitely differentiable. This is one of the key

advantages of the theory of distributions.

7.4

Dirac and its derivatives

7.4.a The Heaviside distribution

A particularly important e xample is the Heaviside distribution, which we will

use constant ly.

DEFINITION 7.34 The Heaviside function is the function H : R → R de-

ﬁned by

H (x) =







0 if x < 0,

1/2 if x = 0,

1 if x > 0.

Mathematicians had been aware for less than a centur y of the fact that a continuous func-

tion could be very far from differentiable (since there exist, for instance, functions continuous

on all of R which are nowhere differentiable, as shown by Peano, Weierstrass, and others).

Enamored of rigor from a recent date only and, therefore, as inﬂexible as any new convert.

The reader interested in the “battle” between Oliver Heaviside and the rigorists of Cambridge

can read with proﬁt the article [4 8 ].

194 Distributions I

DEFINITION 7.35 The Heaviside distribution is the regular distr ibution as-

sociated to the Heaviside function. It is thus deﬁned by

∀ϕ ∈ D(R) 〈H , ϕ〉

def

+∞

ϕ(x) dx.

The derivative, in the sense of distributions, of H is then by deﬁ nit ion



′

, ϕ



= −



H , ϕ

′



= −

+∞

′

(x) dx = −





+∞

= ϕ(0),

from which we deduce

THEOREM 7.36 The derivative of the Heaviside distribution is the Dirac distribution

′

= δ in the sense of distributions.

7.4.b Multidimensional Dirac distrib utions

In many cases, the physicist is working in R

or R

. We will deﬁne here

the notions of point-like, linear, and surface Dirac distribut ions, which occur

frequently in electromagnetism, for instance.

Point-like Dirac distribution

Recall that in three dimensions the Dirac distribution is deﬁned by



(3)

, ϕ



def

= ϕ(0).

The common notation δ

(3)

( r) = δ(x) δ( y) δ(z) will be explained in Sec-

tion 7.6.b. Most authors use δ instead of δ

(3)

since, in general, there is no

possible confusion. Thus a point-like charge in electrostatics is represented by

a distr ibution ρ( r) = q δ( r − a). Note that, when it acts on a space variable,

the three-dimensional Dirac distrib ut ion has the dimension of the inverse of a

volume:

[δ] =

[L]

which implies that ρ( r) has the dimension Q/L

of a volum density of

charge.

Surface Dirac distribution

To describe a surface carrying a uniform charge, we use the surface Dirac

distribution:

Dirac and its derivatives 195

DEFINITION 7.37 Let S be a smooth surface in R

. The (normalized) Dirac

surface distribution on S is given by its action on any test function ϕ ∈

D(R

〈δ

, ϕ〉

def

ϕ d

where d

s is the integration element over the surface S . In electromagnetism,

a uniform surface density σ

on S is described by the distribution σ

This can be extended to nonuniform surface distribution, since we know how

to multiply a distribution by a function of C

∞

class. So a distribution of

charge σ( r) of C

∞

class on a surface S is represented by σ( r) δ

Example 7.38 In R

, for R > 0, we denote by δ



kxk−R



the Di rac surface distribution δ

where S is the sphere S =



x ∈ R

; kxk = R



It should be noted that when working with spacial variables in R

, th e

surface Dirac distribution has the dimension of the inverse of length:

[δ

] =

[L]

which can be recovered by writing (in a heuristic manner, indicated by large

quotes)

〈δ

, ϕ〉 =

“

ZZZ

( r) ϕ( r) d

”

ϕ( r) d

 Exercise 7.1 Let S be a surface in R

and let a ∈ R. Denote by S

′

the surface obtained

from S by homothety with coefﬁcient 1/a. Show that

(a r) =

|a|

′

( r).

◊ Solution: Let ϕ ∈ D(R

). Then we have



(a r), ϕ



def

|a|

, ϕ



E

|a|





r (deﬁnition)

|a|

3−2

′

ϕ( x) d

x =

|a|

〈δ

′

, ϕ〉.

Curvilinear Dirac distribution

By the same method, a density of charge on a curve is deﬁned using the

curvilinear Dirac distribution:

DEFINITION 7.39 Let L be a curve in R

. The unit curvilinear Dirac

distribution on L is deﬁned by its action on any test function ϕ ∈ D(R

)

given by

〈δ

, ϕ〉

def

ϕ d ℓ,

196 Distributions I

+δ/k

−δ/k

k/2

−k/2

Fig. 7.1 — The distribution δ

′

can be seen as two Dirac peaks with opp osite signs inﬁnitely

close to each other — that is, a dipol e.

where dℓ is the integration element over L . In electromagnetism, a density

of charge c( r) supported on the curve L is represented by the distribu-

tion c( r) δ

When working with spacial variables in R

, the curvilinear Dirac distribu-

tion has the dimension of the inverse of a surface:

[δ

] =

[L]

7.4.c The dis tribution δ

′

As seen in the deﬁning formula (7.3), the derivative δ

′

of the Dirac distribu-

tion is deﬁned by



′

, ϕ



def

= −ϕ

′

(0) for any test function ϕ ∈ D.

Note that if ϕ is a test function, then



′

, ϕ



= −ϕ

′

(0) = −lim

k→0





−ϕ



−



= −lim

k→0







x −



−δ



x +



, ϕ



from which we conclude that (note the signs!)

′

(x) = lim

k→0





x +



−δ



x −



This formula is interpreted as follows. The distribution represents a positive

charge 1/k and a negative charge −1/k, situated at a distance equal to k, in

the limit wh ere [k → 0] (see Figure 7.1). Hence the distribution δ

′

represents

a dipole, aligned on the horizont al axis (O x) with dipole moment −1, hence

oriented toward negative values of x.