1

Week 10

Generalised functions, or distributions

1. The Dirac delta function

2. Derivatives of the Dirac delta function

3. Differential equations involving generalised functions

Consider a family of functions δε(x), where x is the variable

and ε is a parameter:

,

.if0

],[if

,if0

)( 12

1

x

x

x

x (1)

1. The Dirac delta function

2

where f(x) is analytic in a certain interval about the point x = 0.

,d)()(

xxfxI

).0(lim0

fI (3)

(2)

Theorem 1:

Consider

Proof:

Represent f(x) by its Taylor series:

...2

)0()0()0()( 2

x

fxffxf (4)

Then

3

hence,

,d...2

)0()0()0(

2

1 2

xx

fxffI

which yields (3) as required. █

).O(032

)0(0)0( 4

2

f

fI

Substitute (1) and (4) into (2):

tends to 0 as ε → 0

4

Consider (2) with f(x) = 1, which yields

At the same time, it follows from (1) that

.1d)(lim0

xx

Q: what kind of function tends to zero at all points except one, but still has non-zero integral (area under the curve)?

.0if0

,0if

,0if0

)(lim0

x

x

x

x

A: It’s called the Dirac delta function, or just delta function. It’s not a usual function though, but a generalised function, or distribution.

5

The delta function [usually denoted by δ(x)] can be defined using infinitely many different families of functions.

,if0

],,0[if)(

],0,[if)(

,if0

)( 2

2

x

xx

xx

x

x

and consider...

Introduce, for example,

(5)

6

,d)()(

xxfxI

),0(lim0

fI

It can be shown that

i.e. the family of functions defined by (5) correspond to the delta function, just like family (1).

where f(x) is analytic in a certain interval about the point x = 0.

7

.1d)(lim0

xx

Then δε(x) corresponds to the delta function, i.e.

)0(d)()(lim0

fxxfx

for any f(x) which is analytic at x = 0.

Let a family of functions δε(x) satisfy

,00)(lim0

xx

Theorem 2:

8

Let

Example 1:

,

.if0

],[if

,if0

)( 2

x

xax

x

x

For which a does this family of functions correspond to δ(x)?

9

Even though generalised functions (GFs) imply an underlying limiting procedure, one often uses a ‘short-hand notation’ treating them as if they were regular functions, e.g.

One should keep in mind, however, that the above equality actually means

).0(d)()( fxxfx

),0(d)()(lim0

fxxfx

where δε(x) is a suitably defined family of functions.

Comment:

Yet, in many cases, the ‘short-hand notation’ can be used to re-arrange expressions involving GFs, and it yields the correct result!

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Example 2:

,d)()(

xxfxI

where δ'(x) is the delta function’s derivative (we haven’t defined it yet, but let’s consider it anyway and see what happens).Treating δ(x) as a regular function and δ'(x) as its derivative, we integrate (6) by parts...

Consider

(6)

11

Assume that

,as0)()( xxfx

Thus,

.d)()()()()()(

xxfxxfxxfxIxx

which kind of agrees with the fact that, in the proper definition of δ(x), the function δε(x) vanishes outside the

interval (–ε, ε).

.d)()(

xxfxI

Now, recall how δ(x) affects test functions.

)0(f

Recalling also definition (6) of I, we obtain...

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Even though this equality was derived without following the proper procedure (families of functions, etc.), we’ll later see that (7) correctly describes how the derivative of the delta function affects a test function.

).0(d)()( fxxfx

(7)

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2. Derivatives of the Dirac delta function

Consider the following family of functions:

.if0

),,0(if

,0if0

),0,(if

,if0

)(2

2

x

x

x

x

x

x

Note that, everywhere except the points x = –ε, 0, +ε, the function δ'ε(x) equals to the derivative of δε(x) defined by

(5). At the ‘exceptional’ points, δε(x) doesn’t have a

derivative, so the values of δ'ε(x) were chosen, more or less, ad hoc.Now, consider...

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).0(lim0

fI

Proof:

Represent f(x) by its Taylor series

,d...6

)0(

2

)0()0()0()( 32

xx

fx

fxffxI

Observe that δ'ε(x) is odd – hence, every other term of the series in [] doesn’t contribute to the integral, and we obtain...

Theorem 3:

where f(x) is analytic in a certain interval about x = 0.

,d)()(

xxfxI

15

,d...6

)0()0(

12

0

32

xx

fxfI

hence,

),O(12

)0()0( 42

ffI as required.

█

tends to 0 as ε → 0

,d...6

)0()0()( 3

xx

fxfxI

hence,

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The minus on the r.-h.s. looks ‘unnatural’, but it actually agrees with the result in Example 2 obtained using the short-hand notation.

The ‘short-hand’ form of Theorem 3 is

).0(d)()( fxxfx

Comment:

The family of functions used in Theorem 3 for representing δ'(x) was obtained by differentiating the family of functions defined by (5) and used to represent δ(x).

In principle, we could’ve used a different family, but there’s still a general rule: if δε(x) represents δ(x), then the

derivative of δε(x) represents δ'(x).

Comment:

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The n-th derivative of δ(x) [denoted by δ(n)(x)] can be defined through any family of functions such that

!.)1(d)(lim )(

0nxxx nnn

,0d)(lim )(

0nmxxx mn

,00)(lim )(

0

xxn

Theorem 4:

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We shall also use δ(x – x0) and δ'(x – x0), such that

),(d)()( 00 xfxxfxx

).(d)()( 00 xfxxfxx

What’s the equivalent of the above equalities for δ"(x – x0)?

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Theorem 5:

Let a function g(x) be smooth and strictly monotonic, i.e.

).()(

1))(( 0

0

xxxg

xg

.0)( 0 xg

Then,

. allfor 0or allfor 0 xgxg

Let also g(x) have a single zero at x = x0, i.e.

.0)( 0 xg

Note that, since g(x) is strictly monotonic,

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.))((

dd

)(

ddd)(d

yxg

yx

xg

yxxxgy

Proof:

Let g'(x0) > 0 and consider

.d)())((

xxfxgI

Let’s change the variable x to y = g(x), so that

(8)

where x(y) is the inverse function to y = g(x).

Observe also that, since y(x0) = 0 and x(y) is the inverse function, it follows that

.)0( 0xx (9)

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Now, (8) becomes

which can be readily evaluated using the definition of δ(x):

,d))((

))(()( y

yxg

yxfyI

Taking into account (9), we obtain

.))0((

))0((

xg

xfI

.)(

)(

0

0

xg

xfI

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Finally, recalling definition (8) of I, we have

which yields the desired result for the case g'(x0) > 0.

The case g'(x0) < 0 is similar. █

,)(

)(d)())((

0

0

xg

xfxxfxg