Kelly J.J. Graduate Mathematical Physics, With MATHEMATICA Supplements
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106 3 Asymptotic Series
If x is large, the Gaussian factor is sharply peaked in w and we can extend the upper limit
of integration to with little error such that
f x
0
w
x
2
w
2
3
4
4
4
4
4
5
1
k1
2k 1!!
2
k
k!
x
k
6
7
7
7
7
7
8
(3.47)
The integrals can now be evaluated using
M
0
Λ
0
w
Λw
2
1
2
Π
Λ
(3.48)
M
n
Λ
0
w
Λw
2
w
2n
(
(Λ
n
M
0
Π
2
2n 1!!
2
n
Λ
n
1
2
(3.49)
and the series becomes
f x
Π
2x
3
4
4
4
4
4
5
1
k1
2k 1!!
2
k
k!
2
x
2k
6
7
7
7
7
7
8
Π
2x
1
1
4
x
2
9
64
x
4
25
256
x
6
,,,
(3.50)
Unlike the asymptotic expansion for Erfc[x], this series converges for x > 1 because
x > 1
a
k1
a
k
2k 1
2k 1x
2
< 1 (3.51)
and hence is asymptotically equal to f x for large x without need for truncation. Thus, we
obtain an asymptotic series for
I
0
x
2
2
x
2
/ 2
Π x
3
4
4
4
4
4
5
1
k1
2k 1!!
2
k
k!
2
x
2k
6
7
7
7
7
7
8
(3.52)
or
I
0
x
e
x
2Πx
3
4
4
4
4
4
5
1
k1
2k 1!!
2
k
k!
2
2x
k
6
7
7
7
7
7
8
(3.53)
The following function and accompanying plots explore the accuracy of this asymp-
totic series for I
0
z. Figure 3.9 demonstrates that the expansion becomes increasingly
accurate as x increases, but even with just the first term of the series one obtains a few
percent accuracy for x 1. The series converges quite rapidly for x>1. Figure 3.10
shows that the relative error for k
max
5 decreases rapidly as x increases. However, this
series must still be classified as an asymptotic series, even if convergent, because there is
a minimum error for finite x that originates from extending the range of integration.
errorx_, kmax_
1
e
x
2Π x
1
kmax
k1
2k 1!!
2
k
k!
2
2x
k
BesselI0, x
PlotEvaluateTableerrorx, kmax, kmax, 0, 5,
x, 0, 10, PlotRange Automatic, 0.05, 0.05,
Frame True, FrameLabel "x", "relative error",
PlotLabel "Accuracy of Asymptotic Expansion for BesselI0, x"
3.4 Expansion of an Integrand 107
0 2 4 6 8 10
x
0.04
0.02
0
0.02
0.04
Relative error
Figure 3.9. Accuracy of asymptotic expansion for I
0
x as k
max
increases from 0 to 5.
0 20 40 60 80 100
x
0.00001
0.0001
0.001
0.01
Relative error
Figure 3.10. Accuracy of asymptotic expansion for I
0
x with k
max
5.
LogPlotAbserrorx, 5, x, 1, 100,
Frame True, FrameLabel "x", "relative error",
PlotLabel "Accuracy of Asymptotic Expansion for BesselI0, x"
Furthermore, even though our derivation employed real variables, it is clear that the
series is valid more generally for
Π
2
< Argz <
Π
2
. However, as Argz !
Π
2
the accuracy
deteriorates for given r. Figure 3.11 shows the r-dependence of the relative error in I
0
r
Θ
for 0 Θ
3Π
8
in steps of
Π
8
for k
max
5. Although the error for small r increases with Θ,
this expansion still provides reasonable accuracy for modest k
max
over a substantial range
of polar angles.
LogPlot
Evaluate
Table
Abserrorr
Θ
,5, Θ,0,
3Π
8
,
Π
8
,
r, 1, 100, Frame True,
FrameLabel "r", "relative error",
PlotLabel "Accuracy of Asymptotic Expansion for BesselI0, x"
108 3 Asymptotic Series
0 20 40 60 80 100
x
0.00001
0.0001
0.001
0.01
0.1
1
Relative error
Figure 3.11. Accuracy of asymptotic expansion for I
0
r
Θ
with k
max
5and0Θ
3Π
8
in steps
of
Π
8
.
Problems for Chapter 3
1. Landau straggling
If a fast charged particle passes through a thin film of material, it will lose energy by ion-
izing atoms via a stochastic process. Landau used a Laplace transform technique to derive
the probability distribution for this energy-loss process. Without presenting the details,
suffice it to say that the function
FΛ
1
2Π
c
c
p
p
Λp
p (3.54)
plays an important role in the theory. The Laplace parameter c is an arbitrary positive real
number. Use the method of steepest descent to derive an asymptotic approximation for this
function that is useful for ΛI0.
2. stellar reaction rate
A star is powered by nuclear reactions in its core. Suppose that the average rate of nuclear
collisions with center-of-mass energy between E and E E is N
E/T
E E where N is
a constant and T is the core temperature in energy units. Also suppose that the probability
that a collision with energy E results in a nuclear reaction is M
Α/
E
where M and Α
are constants. Use the method of steepest descent to determine the reaction rate assuming
that T IΑ
2
.
3. incomplete gamma function
The incomplete gamma function, Aa, z, can be defined in terms of the contour integral
Aa, z
z
t
t
t
a1
(3.55)
where a and z are complex and where the contour in the complex t-plane starts at z and
approaches t !, 0 smoothly. If necessary, a branch cut is made on the negative real
t-axis. Use the method of steepest descent to determine the leading asymptotic behavior
of the function f zAz 1,z when argz < Π.
Problems for Chapter 3 109
4. Airy function
The Airy functions Aiz and Biz are solutions to the differential equation
w
''
zzwz0 (3.56)
and arise in boundary-value problems in quantum mechanics and electrodynamics. A use-
ful integral representation of Aix for positive real x is given by
Aix
1
2Π
Exp
Ωx
Ω
3
3
Ω (3.57)
Use the method of steepest descent to determine the leading asymptotic behavior of Aix
for large x. (Hint: choose the appropriate saddle point carefully.)
5. Bessel function
The Schläfli integral representation of the Bessel function J
Ν
z takes the form
J
Ν
z
1
2Π
C
t
t
Ν1
Exp
t
1
t
z
2
(3.58)
where the complex t-plane is cut on the negative real axis and where C enters from
just below the cut, circles the origin once in a positive sense, and then exits toward just
above the cut but is otherwise arbitrary. Use the method of steepest descent to obtain the
leading asymptotic behavior, being careful to include both saddle points.
6. exponential integral
Develop an asymptotic series for the exponential integral
E
n
z
1
zt
t
n
t (3.59)
for large z with n>0. You may assume that n is an integer. How many terms can be
included for finite z?
7. asymptotic expansion of Fresnel functions
The Fresnel functions for real variables are defined by
Cx
x
0
Cos
Πt
2
2
t, Sx
x
0
Sin
Πt
2
2
t (3.60)
a) Determine the limiting values C and S. (Hint: use contour integration of an
exponential on a wedge of opening angle Π/ 4.)
b) Obtain asymptotic expansions for the Fresnel functions. (Hint: evaluate corrections to
the asymptotic values by integrating from x to by parts.)
8. method of stationary phase
The method of stationary phase is a somewhat simplified variation of the method of steep-
est descent that is useful for Fourier integrals of the form
f t
gΩ ExpΦΩtΩ (3.61)
where the amplitude gΩ depends relatively slowly upon frequency Ω and the phase ΦΩ
is real. For large times t !, the phase of the exponential factor varies extremely rapidly
110 3 Asymptotic Series
with frequency unless ΦΩ is practically constant. Therefore, the asymptotic value of the
integral tends to be dominated by points of stationary phase.
a) Develop a general formula for the contribution of a point of stationary phase to the
asymptotic behavior of f t. How does one handle several such points?
b) Suppose that Φ
'
Ω 0 anywhere in the interval Ω
1
< Ω < Ω
2
. Use integration by
parts to show that
Ω
2
Ω
1
gΩ ExpΦΩtΩ (3.62)
is of order t
1
and, hence, that intervals lacking a point of stationary phase can be
neglected in the limit t !.
c) Suppose that a wave function is represented by
Ψx, t
1
2Π
˜
Ψk Expkx Ωt k, Ω
k
2
2m
(3.63)
where
˜
Ψk
x
kx
Ψx, 0 (3.64)
is the spectral representation of the initial wave function and varies slowly with k
if Ψx, 0 is well localized. Evaluate the intensity, Ψ
2
,forx !and t !with
constant ratio Ξx/t.
9. surface waves on deep water
One-dimensional surfaces waves on deep water in a narrow channel take the form
Ψx, t
˙
x y, tΨy, 0y
x y,t
˙
Ψy, 0y (3.65)
where
˙
f x, t(f x, t/ (t and
x, t
SinΩt
Ω
kx
k
2Π
(3.66)
with Ω
gk (g is the gravitational acceleration). Here Ψx, 0 and
˙
Ψx, 0 represent the
initial displacement and vertical velocity profiles and we evaluate Ψx,t for later times, t 2
0. (See A. L. Fetter and J. D. Walecka, Theoretical Mechanics of Particles and Continua,
McGraw-Hill, 1980.) Use the method of stationary phase to evaluate x, t for t !
and x !simultaneously in the constant ratio Ξx/t. Sketch and interpret this function.
4 Generalized Functions
Abstract. The analysis of physical systems is often simplified using idealizations
that represent an impulse being delivered instantaneously or a charge distribution
being confined to a mathematical point without volume. Here we develop the Dirac
delta function in terms of limiting properties of nascent delta functions. Similar
methods are used for related generalized functions (or distributions).
4.1 Motivation
Often it is useful to idealize an external force as instantaneous or a charge distribution
as two-dimensional. For example, if the response of a system is much slower than the
duration of an applied force, it becomes useful to employ impulse, defined as the time
integral of the force
t
0
Τ/ 2
t
0
Τ/ 2
Ftt $p (4.1)
in a limiting process where the duration Τ!0 approaches zero while the peak force
Ft
0
!such that the product ΤFt
0
$p remains constant. Thus, we imagine that the
entire change of momentum, $p, is delivered instantaneously at time t
0
. If the response
of the system is much slower than the duration of the force, it becomes insensitive to the
detailed shape of Ft and responds only to the net impulse $p. There is then considerable
latitude in choosing a convenient representation for Ft.
Consider a Gaussian function
∆x, Σ
Exp
x
2
2Σ
2
2ΠΣ
2
(4.2)
defined with unit integral
∆x, Σx 1 (4.3)
As the width Σ decreases, the height increases to maintain a constant integral. Thus, we
could represent an impulsive force by
Ftlim
Τ!0
∆t,Τ (4.4)
such that the integral
Ε
Ε
Ftt (4.5)
Graduate Mathematical Physics. James J. Kelly
Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-40637-9
112 4 Generalized Functions
x
f #x', G #x,V '
f #x'
G #x,V '
x
f #x', G #x,V '
Figure 4.1. Integration of a broad function, f x, against a narrow function, ∆x, Σ.
over any small but finite interval, t < Ε with Ε!0, provides the same impulse . There-
fore, one of the representations of the Dirac delta function ∆x is offered by the limit
∆xlim
Σ!0
∆x, Σ lim
Σ!0
Exp
x
2
2Σ
2
2ΠΣ
2
(4.6)
However, ∆x is clearly not an ordinary function – this limit vanishes for any x 0
and does not exist for x 0.
x 0 lim
Σ!0
Exp
x
2
2Σ
2
2ΠΣ
2
0 (4.7)
x ! 0
Exp
x
2
2Σ
2
2ΠΣ
2
!
1
Σ
2Π
! (4.8)
Despite the fact that the limit of the integrand vanishes everywhere but the one point where
it diverges, the integral remains finite, in this representation constant, and converges to
unity. We describe ∆x, Σ as a nascent delta function, a delta function in the making.
Consider an ordinary function f x which is continuous at x 0. Despite the peculiar
divergence of ∆x, the integral
f x∆xx f0 (4.9)
converges simply to f0. Figure 4.1 illustrates the evaluation of this integral. We assume
that f x is continuous in some interval around x 0 and choose Σ small enough to
ensure that f x is well behaved throughout the domain where ∆x, Σ has appreciable
strength. As Σ is reduced further, the product ∆x, Σfx is approximated increasingly
well by f 0∆x, Σ because the variation of f x within the width of ∆x, Σ can be made
arbitrarily small. Consequently, we can extract the constant f 0 from the integral
2
f x∆x, Σ x * f 0
∆x, Σx f0 (4.10)
4.2 Properties of the Dirac Delta Function 113
to obtain
f x∆xx lim
Σ!0
f x∆x, Σ x f 0 (4.11)
without reference to the detailed shape of ∆x, Σ. Notice that the limits of integration need
not be infinite, just much larger than Σ.
Notice that the order of integration and limit operations are not interchangeable – one
must evaluate the integral before the limit because the limit of the nascent delta func-
tion, ∆x, Σ, is not a respectable function. Therefore, ∆x is only defined within the con-
text of integration against a continuous function. The Dirac delta function is an example of
a generalized function defined in terms of its behavior under an integral. Mathematicians
prefer to call generalized functions distributions instead, but to this author the term distri-
bution hardly seems applicable to an entity that vanishes at all but one point. Other useful
distributions include the Heaviside step function, the square function, and derivatives of
the delta function. Familiarity with the properties of generalized functions can greatly sim-
ply the evaluation of integrals or the solutions of differential equations that involve large
disparities between the spatial or temporal profile of one component and the response of
another. Of course, one could employ only well-behaved narrow functions and evaluate
the appropriate limits explicitly, thereby recreating a generalized function, but who has the
time?
4.2 Properties of the Dirac Delta Function
If f x is continuous at x
0
, then
f x∆x x
0
x f x
0
(4.12)
selects the value at x
0
. This property is easily verified using the change of variables
y x x
0
f x∆x x
0
x
f y x
0
∆yy (4.13)
and the definition
gy∆yy g0 (4.14)
where
gy f y x
0
g0 f x
0
(4.15)
Similarly, one can prove that
a 0
f x∆axx
f 0
a
(4.16)
114 4 Generalized Functions
using
a>0
f x∆axx
1
a
f
y
a
∆yy
f 0
a
f 0
a
(4.17)
a<0
f x∆axx
1
a
f
y
a
∆yy
f 0
a
f 0
a
(4.18)
Thus, we write
a 0 ∆ax
∆x
a
(4.19)
as an equality even though neither side represents a function because the replacement of
one side with the other is operationally correct within the context of an appropriate integral.
Notice that this result suggests that ∆x is even; hence, it is useful to require that property
of nascent delta functions.
How should one interpret an expression of the form ∆x
2
a
2
? The interpretation is
based upon the behavior of that generalized function under an integral. We assume that
a is real and, without loss of generality, that a>0. Recognizing that x
2
a
2
has roots
at x a, we can separate the integral into two contributions
f x∆x
2
a
2
x
0
f x∆x ax a x
0
f x∆x ax a x
(4.20)
where in the first integral
x !a x a !2a
0
f x∆x ax a x
f a
2a
(4.21)
while in the second
x ! a x a ! 2a
0
f x∆x ax a x
f a
2a
(4.22)
such that
a 0
f x∆x
2
a
2
x
f a f a
2a
(4.23)
provided only that f x is continuous near both x a and that a 0. Therefore, from an
operational point of view, we may write
a 0 ∆x
2
a
2
∆x a∆x a
2a
(4.24)
because both sides produce the same result when integrated against any suitable function.
(Note, however, that ∆x
2
is uninterpretable.)
Finally, we can generalize this procedure to evaluate ∆
gx
where gx is a function
with an arbitrary number of isolated roots.
4.3 Other Useful Generalized Functions 115
Theorem 19. Suppose that gx has isolated simple roots at x
n
,n 1, 2,..., such that
x * x
n
gx*x x
n
g
'
x
n
with g
'
x
n
0. Then
∆
gx
n
∆x x
n
g
'
x
n
(4.25)
If the roots of gx are isolated, we may evaluate the integral
f x∆
gx
x
n
x
n
Ε
x
n
Ε
f x∆
x x
n
g
'
x
n
x (4.26)
as a sum over contributions from each root of gx where Ε is a positive number smaller
than the spacing between roots. Then using the scaling property ∆ax∆x/ a, we obtain
f x∆
gx
x
n
f x
n
g
'
x
n
(4.27)
Therefore, we obtain an operational definition
gx
n
0,g
'
x
n
0∆
gx
n
∆x x
n
g
'
x
n
(4.28)
that incorporates all preceding results from this section.
4.3 Other Useful Generalized Functions
4.3.1 Heaviside Step Function
A useful version of the unit step function, known as the Heaviside function, can be con-
structed by integration over a delta function, whereby
<x
x
∆yy (4.29)
serves as an operational definition. A more controlled definition of a nascent step function
is based upon a suitable nascent delta function, such that
<x, Σ
x
∆y, Σy (4.30)
<xlim
Σ!0
<x, Σ (4.31)
If we choose the Gaussian representation, then we obtain
∆x, Σ
Exp
x
2
2Σ
2
2ΠΣ
2
<x, Σ
1
2
1 Erf
x
2Σ
(4.32)
where Erfx is the error function. Figure 4.2 demonstrates that the transition between
values of 0 for x I 0or1forx D 1 becomes very sharp as Σ becomes small. Furthermore,