Analytical Methods 1

7/29/2019 Analytical Methods 1

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Lecture 1: Asymptotic evaluation of integrals

Learning outcomes: By the end of this lecture, the students should be able to

• define an asymptotic expansion,• identify and distinguish Laplace-type and Fourier-type integrals,

• evaluate the asymptotic behaviour of these integrals via the Laplace and stationary-

phase methods,

• generalise and synthesize their knowledge on Laplace-type and Fourier-type integrals

towards complex integrals,

• derive the asymptotic behaviour of complex integrals via the saddle point method.

I. EXAMPLE: CASIMIR–POLDER POTENTIAL

Theoretical physicists often obtain results as integrals which depend on an external pa-

rameter and which are difficult to evaluate analytically. However, it is often possible to

derive approximate results for large values of this parameter. As an example, take the

Casimir–Polder potential of an ground-state atom at distance z from a perfectly conducting

mirror. It is due to the exchange of virtual photons of frequency i ω between the atom and

the mirror. Summing over the contributions from all virtual photons, one finds

U (z ) = −

16π2ε0z 3

∞0

dω α(iω) e−2ωz/c

1 + 2ωz

c+ 2

ω2z 2

c2

,

where α(iω) is the ground-state polarisability of the atom. The distance-dependence of this

potential is not explicit due to the appearance of z inside the integral. However, we may

derive the limiting behaviour of the potential for large distances. For such distances, the

exponential factor effectively restricts the integral to smaller and smaller values ω c/z ,

so that eventually, we may approximate the atomic polarisability by its static counterpart

α(iω)≃α(0). In this limit, the remaining integral can be performed to give (x = ωz/c)

U (z ) ∼ − cα(0)

16π2ε0z 4

∞0

dx e−2x(1 + 2x + 2x2) = − 3 cα(0)

32π2ε0z 4.

II. DEFINITION: ASYMPTOTIC EXPANSION

The large-distance limit of the Casimir–Polder potential is an example of an asymptotic

expansion. In order to generalise beyond this simple case, we need to formally define this

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notion. The asymptotic behaviour of a function f (x) for large arguments x can be described

via the big-O and little-o notations:

Definition. The notation f (x) = O(g(x)) for x → ∞ means that there are finite real con-

stants M and x0 such that |f (x)|≤M |g(x)| ∀x > x0.The notation f (x) = o(g(x)) for x → ∞ means for every constant ǫ > 0 there is a constant

x0 such that |f (x)|≤ǫ|g(x)| ∀x > x0.

In simple terms, the big-O notation says that f (x) grows at most as fast as g(x), while

the little-o notation states that f (x) grows much less than g(x).

Examples: 5+ 6x+7x3 = O(x3),5sin x

x2= O(1/x2), cosh(x) = O(ex),

√ x = o(x).

On this basis, we can now formally define what is meant by an asymptotic expansion:

Definition. An asymptotic scale is a sequence of functions {δ j(x)} ( j = 1, 2, . . .) such

that δ j(x) = o(δ j+1(x)) for x → ∞ ∀ j.

An asymptotic expansion to order δ n(x) of a function f (x) is a series n

j=1 f jδ j(x)

such that

f (x) =n

j=1

f jδ j(x) + O(δ j+1(x)) for x → ∞ ( j = 1, 2, . . . n).

We write

f (x) ∼n

j=1

f jδ j(x) for x → ∞.

Examples: {1/x j+1/2}, { j−x} and (1, log x,x,x2, . . .) are asymptotic scales.

Remarks: In contrast to a Taylor series, an asymptotic expansion need not be convergent

in the limit n→ ∞ for fixed x. However, the remainder rn(x) = f (x)−n j=1 f jδ j(x) does go

to zero for suitable fixed n as x→ ∞. Typically, the best approximation to the function f (x)

for fixed x is given by an asymptotic expansion with some small n. The expansion coefficients

f n are uniquely defined once an asymptotic scale has been chosen. On the contrary, one andthe same asymptotic expansion can represent different functions.

III. LAPLACE-TYPE INTEGRALS: INTEGRATION BY PARTS AND THE

LAPLACE METHOD

We would like to extend our approach used for the Casimir–Polder potential into a general

method for the asymptotic evaluation of integrals. The main feature of the Casimir–Polder

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integral was the appearance of an exponential factor with real exponent. Generalising this

structure, we define

Definition. A Laplace-type integral is an integral of the form

I (λ) =

ba

dx f (x)e−λφ(x)

with a real-valued exponent function φ(x).

Let us first consider the case φ′(x)=0 everywhere, so that the exponent is monotonous.

As in our example, the main contribution to the integral is then due to the region around

that end point where the exponent is smallest. This can be justified more rigorously by

means of integration by parts. Writing

I (λ) =

ba

dx f (x)e−λφ(x) = − ba

dxf (x)

λφ′(x)[−λφ′(x)]e−λφ(x) ,

and integrating by parts, we find

I (λ) = − f (x)

λφ′(x)e−λφ(x)

b

a

+

ba

dx e−λφ(x)d

dx

f (x)

λφ′(x)

.

The first term is of order O(1/λ). The second term can be treated by applying the same

procedure again, leading to terms of order O(1/λ2) and higher. So we can conclude:

Method (Integration by parts). A Laplace-type integral with φ′(x) = 0 ( x ∈ [a, b]) has the

leading asymptotic expansion

I (λ) ∼ f (a)

λφ′(a)e−λφ(a) − f (b)

λφ′(b)e−λφ(b) .

When the exponent of a Laplace-type integral has a vanishing derivative at some point,

then the integration by parts methods fails, as can be seen by the appearance of 1 /φ′(x) in

the integrand. However, the main contribution to the integral is still given by the region

where the exponent function φ(x) takes its smallest value. Suppose that the exponent has

a single minimum at a point x0 [φ′(x0) = 0 , φ′′(x0) > 0] which is deeper then the end-point

values [φ(x0) < φ(a), φ(b)]. We restrict the integral to a small region around this minimum

and expand the integrand functions, f (x)≃f (x0), φ(x)≃φ(x0) + 12

φ′′(x0)(x − x0)2, to write

I (λ) ≃ x0+ǫ

x0−ǫ

dx f (x0)e−λφ(x0)−λφ′′(x0)(x−x0)2/2 .

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Introducing the new integration variable y =

λφ′′(x0)/2 (x − x0), we have

I (λ) ≃ f (x0)e−λφ(x0)

2

λφ′′(x0)

ǫ√ λφ′′(x0)/2

−ǫ√

λφ′′(x0)/2

dy e−y2

.

In the limit λ→∞, the last integral takes the value √ π, and we find:

Method (Laplace method). A Laplace-type integral with a single local minimum of its

exponent at x0 [ φ′(x0) = 0, φ′′(x0) > 0 ] with φ(x0) < φ(a), φ(b) has the leading asymptotic

expansion

I (λ) ∼ f (x0)e−λφ(x0)

2π

λφ′′(x0).

A factor 1

2

has to be included if the local minimum occurs at one of the end points, x0 = a

or x0 = b.

Example: The Laplace-type integral I (λ) = ∞0

dx e−λ sinh2(x) has an exponent φ(x) =

sinh2(x) with φ′(x) = 2 sinh(x)cosh(x) and φ′′(x) = 2sinh2(x) + 2 cosh2(x). A local mini-

mum occurs at x0 = 0 where φ′′(x0) = 2. Using the Laplace method, we hence find the

asymptotic expansion I (λ)∼ 12

π/λ.

Remarks: More generally, the asymptotic behaviour of a Laplace-type integral is always

governed by the global minimum of the exponent in the integration range [a, b]. Its asymp-

totic expansion can thus be found by either using integration by parts (if the exponent takes

it lowest value at one of the end points and does not have a local minimum there) or the

Laplace method (applied to the deepest local minimum, which must be deeper than the end

points or coincide with one of them).

IV. FOURIER-TYPE INTEGRALS: STATIONARY-PHASE METHOD

A closely similar class of integrals involves exponential functions with purely imaginary

exponents:

Definition. A Fourier-type integral is an integral of the form

I (λ) =

ba

dx f (x)eiλφ(x)

with a real-valued phase φ(x).

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In the limit λ → ∞, the exponential becomes rapidly oscillating, leading to strong can-

cellations between positive and negative contributions. An exception are regions where the

phase remains relatively constant, which is the case if φ′(x) = 0. Assuming that we have

exactly one such stationary-phase point x0

[φ′(x0) = 0, φ′′(x

0)= 0], the main contribution

to the integral will come from a small region around this point. As before, we expand the

integrand functions around this point, f (x)≃f (x0), φ(x)≃φ(x0) + 12

φ′′(x0)(x − x0)2, to find

I (λ) ≃ x0+ǫ

x0−ǫ

dx f (x0)eiλφ(x0)+iφ′′(x0)(x−x0)2/2 .

Introducing the new integration variable y =

λ|φ′′(x0)|/2 (x − x0), we have

I (λ) ≃ f (x0)e

iλφ(x0) 2

λ|φ′′(x0)| ǫ√

λ|φ′′(x0)|/2

−ǫ√ λ|φ′′(x0)|/2 dy e

±iy2

,

where the two signs represent the cases φ′′(x0)≷0. We will show below that ∞−∞

dy e±iy2

=√

π e±iπ/4 .

In the limit λ→∞, we hence obtain:

Method (Stationary-phase method). A Fourier-type integral with a single stationary-phase

point x0 [ φ′

(x0) = 0, φ′′

(x0)≷0 ] has the leading asymptotic expansion

I (λ) ∼ f (x0)eiλφ(x0)

2π

λ|φ′′(x0)| e±iπ/4 .

V. COMPLEX INTEGRALS: SADDLE-POINT METHOD

So far, we have treated integrals with exponentials whose exponents were either purely

real or purely imaginary. More generally, we finally consider an integrals of the type

I (λ) =

C

dz f (z )eλφ(z)

where the exponent φ(z ) is an arbitrary complex-valued function. Such integrals are best

evaluated by means of contour integration, which is why we have now allowed for a complex

integration path C . For both Laplace- and Fourier-type integrals, we have found that the

main contribution is due to regions where the exponent has an extremum. Generalising this

idea, we now consider saddle points:

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Definition. A simple saddle point of a complex-valued function φ(z ) is a point z 0 where

φ′(z 0) = 0, but φ′′(z 0)= 0.

The notion is justified by the behaviour of φ(z ) near z 0: Expanding φ(z ) ≃ φ(z 0) +

12φ′′(z 0)(z −z 0)2, and writing φ′′(z 0)≡a+ib, z −z 0≡∆x+i∆y, we find

Re φ(z ) ≃ Re φ(z 0) + 12

Re[(a + ib)(∆x + i∆y)2 = Re φ(z 0) + 12

[a(∆x2 − ∆y2) − 2b∆x∆y]

= Re φ(z 0) +1

2(∆x, ∆y)·M·

∆x

∆y

with M =

a −b

−b −a

.

The eigenvalues of this quadratic form have opposite signs,

det(M− λI) = λ2 − (a2 + b2) = 0 ⇒ λ = ±√

a2 + b2 ,

meaning that the real part of φ(z ) indeed has a saddle point at z 0 (the same can be shown

for the imaginary part). A trajectory going through the saddle point can thus exhibit a

maximum or a minimum, depending on the angle from which the saddle point is approached.

If we choose the right angle, we can make Re φ(z ) exhibit a steep maximum while Im φ(z )

remains constant. This is best seen in polar coordinates. Writing φ′′(z 0) = |φ′′(z 0)|eiθ and

z −z 0 = ρeiϕ, we have

12

φ′′(z 0)(z

−z 0)2 = 1

2

|φ′′(z 0)

|ρ2ei(θ+2ϕ) .

The real part will obviously reach its steepest maximum at z 0 if ei(θ+2φ) = −1, implying an

angle ϕ =−θ/2±π/2.

The original complex integral can now be evaluated at follows: We make use to Cauchy’s

theorem to transform the original integration contour C to a new, equivalent contour C ′

that passes through the saddle point z 0 at an angle ϕ. The main contribution to the integral

is due to the maximum of Re φ(z ) near this point and can be evaluated by means of the

Laplace method [d(z

−z 0) = eiϕdρ]

I (λ) =

C ′

dz f (z )eλφ(z) ≃ f (z 0)eλφ(z0)eiϕ ∞−∞

dρ e−λ|φ′′(z0)|ρ2/2 .

Evaluating the integral result in:

Method (Saddle point method). A complex integral with a single simple saddle point z 0

[ φ′(z 0) = 0, φ′′(z 0)= 0 ] has the leading asymptotic expansion

I (λ) ∼ f (z 0)eλφ(z0) eiϕ

2π

λ

|φ′′(z 0)

|

.

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Here, ϕ =−θ/2±π/2 with φ′′(z 0) = |φ′′(z 0)|eiθ.

There are two possible angles ϕ such that we cross the saddle point in a direction of

steepest descent, since we can follow the respective trajectory forwards or backwards. The

correct choice can be made by looking at the global behaviour of Re φ(z ). We have to choose

an integration path C ′ such that Re φ(z ) remains as small as possible. Often, this is the

most direct path between the end points that crosses z 0 at one of the angles ϕ.

Example: Using the saddle point method, we can evaluate the integral ∞−∞

dz eiz2

as

needed for the stationary phase method. The exponent φ(z ) = iz 2 exhibits a saddle point

at z = = 0 where φ′′(z 0) = 2i = 2eiπ/2 and hence θ = π/2. A look at the global behaviour of

Re φ(z ) =−2xy reveals that ϕ =−θ/2+π/2 = π/4 is the correct choice for the angle, so that ∞−∞ dz eiz

2

≃√ πeiπ/4.

Remarks: Note that we have neglected contributions from the end points of the integration

contour as well as those from critical points where the integrand is not analytic. Furthermore,

the integral may have several critical points, in which case its asymptotic expansion involves

a sum over the contributions from each critical point and the relative phases need to carefully

be considered.

FURTHER READING

• Complex Variables: Introduction and Applications , M. J. Ablowitz and A. S. Fokas,

2nd Ed., Chap. 6, pp. 411-513 (Cambridge University Press, Cambridge, 2003).

• Waves and Fields in Inhomogeneous Media , W. C. Chew, Chap. 2.5, pp. 79–92 (IEEE

Press, New York, 1995).

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