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453.701 Linear Systems, S.M. Tan, The University of Auckland 1-1

Chapter 1 Dirac delta functions and distributions

1.1 Motivation

In many applications in physics, we would like to be able to handle quantities such as

1. The density (r) of a point mass at r0,

2. The probability density function of a discrete distribution,

3. The force F (t) associated with an instantaneous collision,

4. The derivative of a function with a step discontinuity.

We may regard each of these as the result of a limiting process in which a quantity gets more andmore concentrated at a point in space or time. Consider the sequence of functions

fn(t) =

n for jtj < 12n0 otherwise

: (1.1)

These functions get narrower and taller as n ! 1. The area in any interval enclosing the origintends to one as n becomes large.

These functions do not tend pointwise to any limit as n ! 1 but physically it is useful to think ofpoint sources such as point charges, point masses etc. We would like to invent a limit for thissequence of functions which should have certain properties. It is called the (Dirac) delta functionor the unit impulse and is denoted by (t). We would like (t) to satisfyZ b

a(t) dt =

1 if 0 2 (a; b)0 if 0 =2 [a; b] : (1.2)

Given any function (t) continuous at the origin, we have that

limn

Z

fn(t)(t) dt = (0); (1.3)

and so we would like (t) to have the property thatZ

(t)(t) dt = (0): (1.4)

Many other sequences of functions may be found which also satisfy the limit (1.3) for a suitableclass of functions . These provide alternative representations of the delta function.

1.2 Distributions

We wish to develop a theory which will allow us to deal with objects like delta functions in a correctway. These are not ordinary functions but are rather examples of distributions or generalizedfunctions. The extra conceptual power and elegance of distribution theory will (hopefully) becomeobvious as we proceed. In particular, we shall nd that any distribution can be dierentiated (togive another distribution) any number of times and that distributions justify many operations whichwe often carry out in Physics calculations which are incorrect for ordinary functions.

453.701 Linear Systems, S.M. Tan, The University of Auckland 1-2

1.2.1 Some concepts from mathematical analysis

The following is a rather brief summary of a few denitions and results which will be useful later.They should also serve to emphasize that we do need to be rather careful when manipulatingordinary functions.

Suppose that fxng for n = 1; 2; ::: is a sequence of real or complex numbers. We say that thesequence converges to x as n ! 1 if jxn xj can be made to be as small as we like whenevern is suciently large. Formally, this is written 8 > 0, 9N 2 N, such that n N ) jxn xj < where the quantiers 8 and 9 represent for all and there exist respectively.A function f(t) is said to have limit b as x ! a and we write

limta f(t) = b; (1.5)

provided that jf(t) bj can be made as small as we like whenever t is suciently close (but notequal) to a. Formally, 8 > 0, 9 > 0 such that 0 < jt aj < ) jf(t) bj < .A function f(t) is said to be continuous at a if

limta f(t) = f(a); (1.6)

i.e., 8 > 0, 9 > 0 such that jt aj < ) jf(t) f(a)j < .Note that continuity is a property that a function may have at a point. There are functions whichare continuous only at a single point and are discontinuous everywhere else. Consider for example

f (t) =

t if t is a rational number0 if t is irrational.

(1.7)

You should be able to show (using the denition of continuity) that this function is continuous att = 0 and nowhere else.

It should be familiar result from analysis that a function f(t) which is continuous on a closed andbounded set K is bounded, i.e., 9M 2 R such that jf(t)j M for all t 2 K.We now wish to consider sequences of functions. Suppose ffng is a sequence of functions alldened on some set S. The functions are said to converge pointwise to a function f as n ! 1if for each t 2 S,

f(t) = limn fn(t): (1.8)

Notice that for a given t, the limit is simply the limit of a sequence of numbers.

Examples of sequences of functions

1. Dene

fn(t) =

8 1

:

This sequence of functions (see gure 1.1) converges pointwise to the limit function

f(t) =

0 for t < 11 for t 1 : (1.9)

Note that the limiting function is discontinuous at t = 1; even though every function fn (t)in the sequence is continuous at t = 1:

453.701 Linear Systems, S.M. Tan, The University of Auckland 1-3

0 0.5 1 1.50

0.5

1

1.5

t

f n(t)

n=1n=2n=3n=4

Figure 1.1 An example of a sequence of functions which converges pointwise but not uniformly to alimiting function as n ! 1:

2. The sequence of functions

fn(t) =

n jtj < 12n0 otherwise

; (1.10)

does not converge pointwise to any function since the sequence of numbers ffn(0)g does notconverge.

3. The sequence of functions

fn(t) =

8 0; there is an integer N;such that for all n > N; fn (t) = 0: For all other values of t; the value of fn (t) is zero forevery n.

4. The sequence of functions

fn(t) =1

nsin(n2t) (1.12)

converges pointwise to the zero function.

Example 1 shows that even when all the functions in a sequence are continuous, the pointwiselimit can fail to be continuous (in this case at t = 1).

In example 3, each function fn(t) satisesZ

fn(t) dt = 1: (1.13)

Thus the limiting value of the integral as n ! 1 is one. However, the limiting function is zero andso the integral of the limit function is zero. We see in general that

limn

Z

fn(t) dt 6=Z

f(t) dt even if fn(t) ! f(t) pointwise. (1.14)

453.701 Linear Systems, S.M. Tan, The University of Auckland 1-4

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

t

f n(t)

n=1n=2n=3n=4

Figure 1.2 An example of a sequence of functions which converges pointwise but not uniformly to thefunction which is identically zero.

In example 4, the derivative sequence

f 0n(t) = n cos(n2t) (1.15)

does not converge pointwise to any function, and certainly not to f 0(t), the derivative of the limitingfunction. Thus in general

limn

d

dtfn(t) 6= d

dtf(t) even if fn(t) ! f(t) pointwise. (1.16)

One way of trying to improve matters is to strengthen what we mean by convergence of functions.We thus dene the concept of uniform convergence. A sequence of functions all dened on someset S is said to converge uniformly on S to a function f as n ! 1 if 8 > 0, 9N 2 N such that8t 2 S, jfn(t) f(t)j < whenever n N .It is important to know how this is dierent from the concept of pointwise convergence. Translatingthe denition of pointwise convergence given previously into symbols reads 8t 2 S, 8 > 0, 9N 2 Nsuch that jfn(t) f(t)j < whenever n N . We see that for pointwise convergence, the value ofN depends both on and on t whereas for uniform convergence, the same N must work for allt 2 S. Notice that if a sequence of functions converges uniformly to some limit, the sequence mustalso converge pointwise to that limit, but the converse does not necessarily hold.

Exercise: Show that the convergence in examples 1 and 3 is not uniform on R but that convergenceis uniform in example 4.

Another way of checking whether a pointwise-convergent sequence is uniformly convergent is tolook at the dierence sequence dn(t) = fn(t)f(t); which is another sequence of functions. Theconvergence of fn(t) to f(t) is uniform on S provided that for all t 2 S; jdn(t)j may be boundedabove by a sequence which converges to zero. i.e., if we can nd a sequence Mn such that Mn ! 0as n ! 1 and jdn(t)j Mn for all t 2 S.If fn converges to f uniformly on S, then

1. If each fn is continuous at a 2 S, f is also continuous at a.

453.701 Linear Systems, S.M. Tan, The University of Auckland 1-5

2. Provided that fn and f are integrable, the limit of the integral is equal to the integral of thelimit, i.e.,

limn

Z xa

fn(t) dt =

Z xa

f(t) dt; (1.17)

where [a; x] is contained in S. The convergence of the integral is uniform.

Exercise: Prove the above assertions using the denitions given.

However, example 4 shows that we still cannot interchange the limiting process and a derivative,even given uniform convergence. This whole situation is rather unsatisfactory, especially sinceuniform convergence is quite a strong condition which may be dicult to establish in practicalsituations. By contrast, by using the theory of distributions, we shall see that it is always possibleto dierentiate distributions and to interchange the limiting process with derivatives of any order.

1.2.2 Test functions, functionals and distributions

In equation (1.4) we found it convenient to think about the delta function in terms of its actionon another function which is called a test function. The delta function may be regarded as amachine which operates on the test function to produce the number (0) as the result. We shallconsider such machines (Figure 1.3) which take test functions as inputs and produce numbers asoutputs in more detail.

Figure 1.3 A Dirac delta function acts on a test function to return its value at zero. More generally,a functional is a machine which acts on a test function to produce a number.

We must rst dene the class of test functions on which our machines work. This class is denotedby D, which we shall require to be a vector space. For technical reasons, we shall initially