To evaluate the logarithm of a negative quantity we must use

ln ω = ln |ω| + i arg ω, (5.100)

where we will take arg ω to lie in the range ’π < arg ω < π.

Im ω

ω

Re ω

’ω

arg (’ω)

When ω has a small positive imaginary part, arg (’ω) ≈ ’π.

To get an unambiguous answer, we need to give ω an in¬nitesimal imaginary

part ±iµ. Depending on the sign of this imaginary part, we ¬nd that

iπ

I(ω ± iµ) = ± , (5.101)

2ω

so

Π(ω ± iµ) = i·ω. (5.102)

Now the frequency-space version of

¨ ™

Q(t) + · Q + „¦2 Q = Fext (t) (5.103)

is

(’ω 2 ’ i·ω + „¦2 )Q(ω) = Fext (ω), (5.104)

so we must opt for the displacement that gives Π(ω) = ’i·ω. This means

that we must regard ω as having a positive in¬nitesimal imaginary part,

ω ’ ω + iµ. This imaginary part is a good and needful thing: it e¬ects the

replacement of the ill-de¬ned singular integrals

1

∞

?

e’iωt dω,

I= (5.105)

2

ωi ’ ω 2

0

5.5. ANALYTIC PROPERTIES OF GREEN FUNCTIONS 135

which arise as we transform back to real time, with the unambiguous expres-

sions

1

∞

e’iωt dω.

Iµ = (5.106)

2

ωi ’ (ω + iµ)2

0

The latter, we know, give rise to properly causal real-time Green functions.

5.5.2 Plemelj Formul¦

The functions we are meeting can all be cast in the form

1 b ρ(ω )

f (ω) = dω . (5.107)

π a ω ’ω

If ω lies in the integration range [a, b], then we divide by zero as we integrate

over ω = ω. We ought to avoid doing this, but this interval is often exactly

where we desire to evaluate f . As before, we evade the division by zero by

giving ω an in¬ntesimally small imaginary part: ω ’ ω ± iµ. We can then

apply the Plemelj formul¦, which say that

1

f (ω + iµ) ’ f (ω ’ iµ) = iρ(ω),

2

1 1 ρ(ω )

f (ω + iµ) + f (ω ’ iµ) = P dω . (5.108)

“ ω ’ω

2 π

Here, the “P ” in front of the integral stands for principal part. It means that

we are to delete an in¬nitesimal segment of the ω integral lying symmetrically

about the singular point ω = ω.

The Plemelj formula mean that the otherwise smooth and analytic func-

tion f (ω) is discontinuous across the real axis between a and b. If the dis-

continuity ρ(ω) is itself an analytic function then the line joining the points

a and b is a branch cut, and the endpoints of the integral are branch-point

singularities of f (ω).

ω

Re ω

a b

Im ω

The analytic function f (ω) is discontinuous across the real axis between a and b.

136 CHAPTER 5. GREEN FUNCTIONS

The Plemelj formulae may be understood by considering the following

¬gure:

Im g

Re g

ω

ω

ω

ω

Sketch of the real and imaginary parts of g(ω ) = 1/(ω ’ (ω + iµ)).

The singular integrand is a product of ρ(ω ) with

ω’ω

1 iµ

±

= . (5.109)

(ω ’ ω)2 + µ2 (ω ’ ω)2 + µ2

ω ’ (ω ± iµ)

The ¬rst term on the right is a symmetrically cut-o¬ version 1/(ω ’ ω) and

provides the principal part integral. The the second term sharpens and tends

to the delta function ±iπδ(ω ’ ω) as µ ’ 0, and so gives ±iπρ(ω). Because

of this explanation, the Plemelj equations are commonly encoded in physics

papers via the “iµ” cabbala

1 P

± iπδ(ω ’ ω).

= (5.110)

ω ’ (ω ± iµ) ω ’ω

—

If ρ is real, as it often is, then f (ω +i·) = f (ω ’i·) . The discontinuity

across the real axis is then purely imaginary, and

1

f (ω + iµ) + f (ω ’ iµ) (5.111)

2

is purely real. We therefore have

1 Im f (ω )

b

Re f (ω) = P dω . (5.112)

ω ’ω

π a

This is typical of the relations linking the real and imaginary parts of causal

response functions.

5.5. ANALYTIC PROPERTIES OF GREEN FUNCTIONS 137

Example: A practical illustration of such a relation is provided by the com-

plex, frequency-dependent, refractive index , n(ω), of a medium. This is

de¬ned so that a travelling electromagnetic wave takes the form

E(x, t) = E0 ein(ω)kx’iωt . (5.113)