This document discusses outgoing and ingoing Klein-Gordon waves near the event horizons of a black hole. It begins by introducing the Klein-Gordon equation of motion in the background of the Schwarzschild spacetime metric. It then presents the solution to this equation in product form and discusses the radial and time component equations. Finally, it recasts the radial equation using the Regge-Wheeler coordinate and shows that very near the horizon, the equation can be approximated as a simple oscillatory solution. The goal is to obtain outgoing and ingoing wave solutions that have different properties on either side of the black hole's event horizons.

1. Out-Going and In-Going Klein-Gordon Waves Very Near The Blackhole Event
Horizons
Ferdinand Joseph P. Roaa
, Alwielland Q. Bello b
, Engr. Leo Cipriano L. Urbiztondo Jr.c
a
Independent Physics Researcher, 9005 Balingasag, Misamis Oriental
b
Natural Sciences Dept., Bukidnon State University
8700 Malaybalay City, Bukidnon
c
IECEP, Sound Technology Institute of the Philippines
Currently connected as technical consultant/expert for St. Michael College of Caraga (SMCC)
8600 Butuan City, Agusan del Norte
Abstract
In this elementary exercise we consider the Klein-Gordon field in the background of Schwarzschild
space-time metric. Very near the event horizon the radial equation of motion is approximated in form and we
obtain oscillatory solution in the Regge-Wheeler coordinate. The time and radial solutions are then recast in
the outgoing and ingoing coordinates that consequently lead to the outgoing and ingoing waves that have
respectively dissimilar (distinct) analytic properties in the future and past event horizons.
Keywords: Schwarzschild metric, pair production, scalar field, Regge-Wheeler coordinate, event horizons
1. Introduction
This paper is mainly based on our answers
to an exercise presented on page 142 of [1]. The
exercise falls under the topic related to Hawking
radiation although this present document does not
yet tackle the proper details of the cited subject
matter of Hawking radiation. The scope of this
paper only covers the important details in our
solutions to Klein-Gordon field equation against
the background of Schwarzschild space-time
metric [2]. There are basic features of these
solutions that we understand as specifically
relevant to Hawking radiation.
The problem of Hawking radiation was first
explored in the middle of 1970’s in Stephen
Hawking’s paper [3]. By taking quantum
mechanics into account especially in extreme
proximity to a very strong gravitational field of a
blackhole, Hawking realized that blackholes could
emit particles through pair production happening
so asymptotically close to a very strong
gravitational field. In Hawking’s results, this
emission of particles is thermal as if blackholes
were hot bodies whose temperatures are
proportional to blackholes’ surface gravities.
In his pioneering approach, Hawking
illustrated this radiation using a scalar field [3].
We shall no longer present here the lengthy
elaboration in his cited 1975 paper. In Hawking’s
treatment, quantum mechanics was forcefully
implemented in the classical solutions of those
field equations.

2. Immersing a classical field (example, scalar)
in a gravitational field quite complicates
Lagrangians and their resulting equations of
motion because of the presence of non-flat metric
components which represent for gravitational
field. However, a coordinate system can be chosen
so as to recast the equations of motion from which
we can then write their approximate forms
especially so close to event horizons. As earlier
said it is this feature that is so useful in one
approach to the problem of Hawking radiation
2. Klein-Gordon Equation Of Motion In
The Background Of Schwarzschild
Spacetime Metric
We start with the scalar action[4]
𝑆 = ∫ 𝑑4 𝑥 ℒ (1)
along with a Lagrangian given for a scalar field
ℒ = √−𝑔
1
2
( 𝑔 𝜇𝜐(𝜕𝜇 𝜑)( 𝜕𝜐 𝜑) + 2𝑉(𝜑)) (2)
where in the metric signature of positive two (+2)
we take the scalar potential as
𝑉( 𝜑) =
1
2
𝑀2 𝜑2 (3)
To get for the equation of motion for the scalar
field, we vary this action with respect to the
variation of the scalar field. This variation we
write as
𝛿𝑆 = ∫ 𝑑𝜎 𝜇 |
𝛿ℒ
𝛿(𝜕 𝜇 𝜑)
𝛿𝜑|
𝑥 𝐴
𝜇
𝑥 𝐵
𝜇
+ ∫ 𝑑4 𝑥 (
𝛿ℒ
𝛿𝜑
−𝜎 𝜇
𝜕𝜇 (
𝛿ℒ
𝛿(𝜕 𝜇 𝜑)
)) 𝛿𝜑
(4)
This variation is carried out noting that metric
fields 𝑔 𝜇𝜐 are independent of the variation of 𝜑 .
In the classical field theory the varied scalar field
must vanish at the two end points A and B,
𝛿𝜑( 𝐴) = 𝛿𝜑( 𝐵) = 0 and by stationary
condition, the variation of this scalar action must
vanish that is, 𝛿𝑆 = 0. Thus following these
variational (extremal) conditions, we obtain for the
Euler-Lagrange equation for the classical scalar
𝛿ℒ
𝛿𝜑
− 𝜕𝜇 (
𝛿ℒ
𝛿(𝜕𝜇 𝜑)
) = 0
(5)
Upon the substitution of (2) in (5) we get the
equation of motion for the scalar field in curved
spacetime
1
√−𝑔
𝜕𝜇[√−𝑔𝑔 𝜇𝜐(𝜕𝜐 𝜑) ] − 𝑀2 𝜑 = 0
(6)
where we take note of the covariant four-
divergence
1
√−𝑔
𝜕𝜇 [√−𝑔𝑔 𝜇𝜐(𝜕𝜐 𝜑) ] = ∇ 𝜇( 𝑔 𝜇𝜐(𝜕𝜐 𝜑) )
(7)
that is given with metric compatible connections.
In the metric signature of positive two the
fundamental line element in the background of
Schwarzschild spacetime metric is given by
𝑑𝑆2 = −𝜂𝑑𝑡2 + 𝜀𝑑𝑟2 + 𝑟2 𝑑𝜃2 + 𝑟2 𝑠𝑖𝑛2 𝜃𝑑𝜙2
𝜂 = 𝜀−1 = 1 −
2𝐺𝑀 𝑞
𝑟
(8)
We note in these that the square of the speed of
light is unity (𝑐2 = 1, Heaviside units) and that
𝑀 𝑞 is the mass of the gravitational body. In
addition, our spacetime is given with a set of
spacetime coordinates 𝑥 𝜇 = (𝑥0 = 𝑡; 𝑥1 =
𝑟; 𝑥2 = 𝜃; 𝑥3 = 𝜙 )
Our convenient solution to think of is in
product form so as to easily facilitate the
separation of variables. Such product solution is in
the form
𝜑( 𝑥0, 𝑟, 𝜃, 𝜙) = 𝑇( 𝑡) 𝑅(𝑟)Θ(𝜃)𝜓(𝜙)
(9)
This, given the background spacetime of (8), gives
the following component equations of motion
1
𝜓
𝜕 𝜙
2
𝜓 = −𝜇 𝜙
2
(10.1)
1
Θ
1
𝑠𝑖𝑛𝜃
𝜕𝜃[ 𝑠𝑖𝑛𝜃(𝜕𝜃Θ)] −
𝜇 𝜙
2
𝑠𝑖𝑛2 𝜃
= −𝜇 𝜃( 𝜇 𝜃 + 1)
(10.2)
1
𝑇
𝜕0
2
𝑇 = −𝜔2 (10.3)

3. 1
𝑅
1
𝑟2
𝑑
𝑑𝑟
[ 𝜂𝑟2
𝑑𝑅
𝑑𝑟
] −
𝜇 𝜃( 𝜇 𝜃 + 1)
𝑟2 = 𝑀2 −
𝜔2
𝜂
(10.4)
Partial differential equations (Pde’s)
(10.1) and (10.3) can be solved by ordinary
method such as separation of variables. Depending
on the signs of the constants in these equations, the
respective solutions can take oscillatory forms.
Pde (10.2) is of Hypergeometric type[5] so as
(10.4), which is complicated by the presence a
non-flat metric component 𝜂.
With the substitution 𝑤 = 𝑐𝑜𝑠𝜃 we
convert (10.2) into a Legendre equation [6].
𝜕 𝑤((1 − 𝑤2) 𝜕 𝑤Θ )
+ ( 𝜇 𝜃( 𝜇 𝜃 + 1) −
𝜇 𝜙
2
1 − 𝑤2
)Θ
= 0
(10.5)
This equation is solved by the associated [7]
Legendre polynomials
Θ 𝜇 𝜃
𝑘 ( 𝑤) = 𝐴 𝜇 𝜃
𝑘 (1 − 𝑤2) 𝑘/2 𝑑 𝑘
𝑑𝑤 𝑘
𝑃𝜇 𝜃
(𝑤) (10.6)
where 𝑘 = | 𝜇 𝜙|and 𝐴 𝜇 𝜃
𝑘 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡. In this, the
Legendre polynomials 𝑃𝜇 𝜃
(𝑤) are derived by
𝑃𝜇 𝜃
( 𝑤) =
1
2 𝜇 𝜃(𝜇 𝜃! )
𝑑 𝜇 𝜃
𝑑𝑤 𝜇 𝜃
( 𝑤2 − 1) 𝜇 𝜃
(10.7.1)
These polynomials solve the Legendre equation
given in the following form
(1 − 𝑤2) 𝜕 𝑤
2 𝑃𝜇 𝜃
− 2𝑤𝜕 𝑤 𝑃𝜇 𝜃
+ 𝜇 𝜃( 𝜇 𝜃 + 1) 𝑃𝜇 𝜃
= 0
(10.7.2)
We can combine solutions (10.6) and
solutions of Pde (10.1) to form spherical
harmonics[8]
𝑌𝜇 𝜃
𝜇 𝜙
= Θ 𝜇 𝜃
𝑘 𝜓 𝜇 𝜙
(10.8)
The component solution
𝜓 𝜇 𝜙
= 𝐵 𝜙 𝑒 𝑖𝜇 𝜙 𝜙
𝐵 𝜙 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (10.9)
satisfies (10.1) for a given integral number 𝜇 𝜙
whose square is greater than zero (𝜇 𝜙
2
> 0). Let us
note here that the absolute values of 𝜇 𝜙 are
restricted up to a given integral value of 𝜇 𝜃. That
is
𝑘 = | 𝜇 𝜙| = 0, 1,2,3, …, 𝜇 𝜃 (10.10)
where
𝜇 𝜃 = 0, 1,2, 3,… , 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 (10.11)
In an asymptotically flat space (or that is
the Minkowski’s spacetime), where the metric
component 𝜂 is unity (𝜂 = 1), the radial equation
of motion Pde(10.4) takes a familiar Bessel form
1
𝑅
1
𝑟2
𝑑
𝑑𝑟
( 𝑟2
𝑑𝑅
𝑑𝑟
) −
𝜇 𝜃( 𝜇 𝜃 + 1)
𝑟2 = −𝛼2
𝛼2 = 𝜔2 − 𝑀2 (10.12)
One convenient form of solution to (10.12) that we
choose is
𝑅 𝑙 = 𝑄𝑙(1/𝑟) 𝑒 𝛼1 𝑟 (10.13)
𝛼1 = ±𝑖𝛼
𝑙 = 𝜇 𝜃 (integral number)
This satisfies the associated Besselradial equation
for a given integral value of 𝑙.
𝑑2 𝑄𝑙
𝑑𝑟2 +
2
𝑟
( 𝛼1 𝑟 + 1)
𝑑𝑄𝑙
𝑑𝑟
+ (
2𝛼1
𝑟
−
𝜇 𝜃( 𝜇 𝜃 + 1)
𝑟2
) 𝑄𝑙 = 0
(10.14)
We will only give first two of its solutions.
𝑄𝑙 = 0 =
𝑎0
𝑟
(10.15.1)
𝑄𝑙 = 1 =
𝑎0
𝛼1 𝑟2
( 𝛼1 𝑟 − 1)
(10.15.2)
3. The Time And Radial Equations of
Motion And Their Solutions Given In
Outgoing And Ingoing Coordinates
In the presence of strong gravitational
field we are back with the radial equation of
motion that is given by (10.4), where gravity takes
effect through the metric tensor component, 𝜂. The
authors of this draft have not dwelt on the
solutions to the said equation in terms of the radial
coordinate. We believe that such equation is quite
complicated to obtain for solutions in closed form.
However, this radial equation of motion can be
recast in an alternative radial coordinate so as to
write this equation in a form from which we can

4. obtain for the approximate equation very near the
blackhole horizon.
Using Regge-Wheeler coordinate,
𝑟∗ = 𝑟 + 2𝐺𝑀 𝑞 𝑙𝑛(
𝑟
2𝐺𝑀 𝑞
− 1)
∀𝑟 > 𝑟𝐻 (= 2𝐺𝑀 𝑞 )
𝜕𝑟
𝜕𝑟∗ = (
𝜕𝑟∗
𝜕𝑟
)
−1
= 𝜂
(11.1)
we recast (10.4) into the form
1
𝑅
𝑑2 𝑅
𝑑𝑟∗2 +
1
𝑅
2(𝑟 − 2𝐺𝑀 𝑞)
𝑟2
𝑑𝑅
𝑑𝑟∗ + 𝜔2
= (
𝜇 𝜃( 𝜇 𝜃 + 1)
𝑟2 + 𝑀2) 𝜂
(11.2)
In a region of space so asymptotically close to the
horizon (that is, 𝑟 ≈ 𝑟𝐻), the recast equation of
motion (11.2) can be approximated as
𝑑2 𝑅
𝑑𝑟∗2 + 𝜔2 𝑅 = 0
(11.3)
This is also since very near the horizon, 𝑟 ≈ 𝑟𝐻.
For our present purposes, we admit only
oscillatory solutions
𝑅( 𝑟∗ ) = 𝑅01 𝑒𝑥𝑝(−𝑖𝜔𝑟∗) + 𝑅02 𝑒𝑥𝑝(𝑖𝜔𝑟∗)
(11.4)
For (10.3) we obtain also oscillatory solution
𝑇( 𝑡 ) = 𝑇01 𝑒𝑥𝑝(−𝑖𝜔𝑡) + 𝑇02 𝑒𝑥𝑝(𝑖𝜔𝑡)
(11.5)
where 𝑥0 = 𝑡.
Proceeding from such oscillatory solutions
are approximate wave solutions if we are to define
the Ingoing and Outgoing coordinates respectively
𝑢̃ = 𝑡 + 𝑟∗ (11.6.1)
𝑣̃ = 𝑡 − 𝑟∗ (11.6.2)
These are the Eddington-Finkelstein coordinates
and via these coordinates we can combine the
solutions above into approximate wave solutions.
For the Outgoing wave we have
Φ(𝑟∗ 𝑡)
+
= 𝐴0
+
𝑒𝑥𝑝(−𝑖𝜔𝑣̃ ) (11.7.1)
while for the Ingoing wave
Φ(𝑟∗ 𝑡)
−
= 𝐴0
−
𝑒𝑥𝑝(−𝑖𝜔𝑢̃ ) (11.7.2)
In the contrasting case, we take the limit
as 𝑟 → ∞ as very far from the event horizons and
in this limiting case, from (11.2) we obtain another
approximate radial equation
𝑑2 𝑅
𝑑𝑟∗2
+ (𝜔2 − 𝑀2)𝑅 = 0 (11.8.1)
given for waves very far from the event horizons.
If we choose to have the scalar field as massless
𝑀 = 0, then (11.8.1) can take identical form as
that of (11.3).
𝑑2 𝑅
𝑑𝑟∗2
+ 𝜔2 𝑅 = 0 (11.8.2)
Let us note that in the previous case (for waves
very near the event horizons), it does not matter
whether we have a massless scalar or not since the
mass term in (11.2) drops off because of the
vanishing metric component 𝜂 very near the event
horizons. That is, very near the event horizons,
effectively we have a massless scalar that
corresponds to a massless scalar field very far
from the said horizons.
4. Analytic Properties Of The Wave
Solutions
As we have noted in the previous section,
very near the event horizon the scalar field is
effectively massless and very far from the said
horizon, there corresponds the same radial
equation of motion for a massless scalar field. For
our present discussion purposes we would only
have to take the crude approximation that we have
the same out-going solution (11.7.1) for the two
cases of waves very near the horizon and waves
very far from the horizon. That is, we have to
assume that we have the same out-going waves
propagating very near the horizon that have
reached very far from the horizon.
Because the given waves are massless, we
will assume that the out-going waves travel along
the out-going null path 𝛾+, while the ingoing
waves along the infalling null path 𝛾−.
Respectively, these paths are given by
𝛾+ : 𝜒 − 𝜂 = −𝑎+ (12.1.1)
𝛾−: 𝜒 + 𝜂 = 𝑎− (12.1.2)

5. Since our radial coordinate is within the interval
𝑟𝐻 < 𝑟 < ∞, the waves under concern here
belong to region I of the Carter-Penrose diagram.
(Fig.1)
In Figure 1, we have only drawn region I
as bounded by future event horizon 𝐻+, future null
infinity ℑ+, past null infinity ℑ− and past event
horizon 𝐻−. We approximate these boundaries as
straight lines
𝐻+: 𝜒 − 𝜂 = −𝜋 (12.1.3)
𝐻−: 𝜒 + 𝜂 = −𝜋 (12.1.4)
ℑ+: 𝜒 + 𝜂 = 𝜋 (12.1.5)
ℑ−: 𝜒 − 𝜂 = 𝜋 (12.1.6)
These are obtained from the given changes of
coordinates
𝜒 + 𝜂 = 2𝑢̃′, tan 𝑢̃′ = 𝑢̃ (12.2.1)
and
𝜒 − 𝜂 = −2𝑣̃ ′ , tan 𝑣̃ ′ = 𝑣̃ (12.2.2)
(Remark: Let us not confuse 𝜂 in here with the 0th
component of the metric tensor in (8). In this
section this eta is just a vertical coordinate on the
Carter-Penrose graph.)
Let us take the infalling wave along an
infalling null path given for a constant (of non-
infinite values) infalling coordinate so we may
write 𝑡 = 𝑐𝑜𝑛𝑠𝑡 − 𝑟∗. As we have noted very
near the horizon, the Regge-Wheeler coordinate is
pushed off to negative infinity (𝑟∗ → −∞) and as
a consequence, the coordinate time t approaches
(positive) infinity ( 𝑡 → ∞). The future event
horizon is where all 𝑟∗ → −∞ and 𝑡 → ∞. So,
along 𝑢̃ = 𝑐𝑜𝑛𝑠𝑡, very near the horizon (𝑟 ≈ 𝑟𝐻),
the infalling wave hits 𝐻+ as it moves with
positive infinite values of coordinate time t – into
an infinite coordinate future.
In the limit as 𝑟∗ → −∞ and 𝑡 → ∞, the
out-going coordinate takes on positive infinite
values, 𝑣̃ = ∞ so that the outgoing wave is not
defined on the future event horizon.
In the case for the out-going wave let us
take it along an out-going null path with constant
out-going coordinate. So we may write the
coordinate time t as 𝑡 = 𝑐𝑜𝑛𝑠𝑡+ 𝑟∗. As this wave
propagates very near the event horizon where the
Regge-Wheeler coordinate is pushed off to
negative infinity consequently, coordinate t
approaches to negative infinity. Let us note then
that past event horizon is where all space-time
points have 𝑟∗ → −∞ and 𝑡 → − ∞. That is to
say then, along 𝑣̃ = 𝑐𝑜𝑛𝑠𝑡, very near the horizon,
the out-going wave hits the past event horizon as it
moves with negative infinite values of coordinate
time t – into an infinite coordinate past.
We also have to note that as 𝑟∗ → −∞
and 𝑡 → − ∞, the infalling coordinate takes on
negative infinite values so that the infalling wave
is not defined on the past event horizon.
5. Fourier Components OfThe Out-going
Waves
We shall skip the detailed consideration
that leads to the appropriate parametrization of the
wave solutions as discussed in [1]. Such
consideration applies parallel transport to arrive at
the required outgoing wave solution ultimately
parametrized in terms of the infalling coordinate.
In our convenience we have deviated a
little from [1] and take a parametrization in the
given form
𝑎+ = 2𝑣̃ ′(𝑢̃) (13.1.1)

6. This differs from what is arrived at [on pp. 126 of
[1]] in the numerical factor 2GM as given in this
draft. So in this draft we write
𝑣̃( 𝑢̃) = −2𝐺𝑀𝑙𝑛(−𝑢̃) (13.1.2)
−∞ < 𝑢̃ < 0
In the limit as 𝑢̃ → −∞, we have 𝑢̃ ′ =
−𝜋/2, 𝑎− = −𝜋, 𝛾− → 𝐻−, while 𝑣̃ = −∞,
𝑣̃ ′ = −𝜋/2 . Consequently, 𝑎+ = −𝜋 and 𝛾+ →
ℑ−. As the outgoing null path comes so
asymptotically close to the line of the past null
infinity all outgoing waves along this path can
only be viewed in the infinite past since 𝑣̃ = 𝑡 −
𝑟∗ = −∞, 𝑡 → − ∞, 𝑟∗ → ∞, while as the
infalling null path comes so infinitely close to the
line of past event horizon, all infalling waves
along the infalling null path can only be viewed in
the infinite past since 𝑢̃ = 𝑡 + 𝑟∗ = −∞, 𝑡 →
− ∞, 𝑟∗ → −∞, 𝑟 ≈ 𝑟𝐻.
For the case as 𝑢̃ → 0 we have 𝑢̃′ = 0,
𝑎− = 0 and 𝛾− is now along the line 𝜒 + 𝜂 = 0.
In this limiting case, 𝑣̃ = ∞ so that 𝑣̃ ′ = 𝜋/2 and
consequently, 𝑎+ = 𝜋 and 𝛾+ → 𝐻+. What this
means is that waves along the outgoing null path
as this path comes so very close to the line of
future event horizon can only be viewed in the
infinite future given 𝑣̃ = 𝑡 − 𝑟∗ = ∞, 𝑡 → ∞
and 𝑟∗ → −∞.
Following from the parametrization of the
out-going coordinate in terms of the infalling
coordinate, the outgoing wave becomes a
parametrized function of the infalling coordinate.
Such parametrized function must satisfy the
following boundary conditions
Φ( 𝑟∗ 𝑡)
+
[𝑣̃( 𝑢̃)] = 𝐴0
+
𝑒𝑥𝑝(𝑖2𝜔𝐺𝑀𝑙𝑛(−𝑢̃) )
−∞ < 𝑢̃ < 0
Φ( 𝑟∗ 𝑡)
+
[𝑣̃( 𝑢̃)] = 0, 𝑢̃ > 0
(13.2.1)
In here, we shall consider only the Fourier
integration involving positive frequency modes,
𝜔′ > 0.
Φ̃( 𝑟∗ 𝑡)
+ ( 𝜔, 𝜔′) = ∫ 𝑑𝑢̃
∞
−∞
Φ( 𝑟∗ 𝑡)
+
(𝑢̃, 𝜔)𝑒−𝑖𝜔′ 𝑢̃
= 𝐴0
+
∫ 𝑑𝑢̃ 𝑒𝑥𝑝(𝑖2𝜔𝐺𝑀𝑙𝑛(−𝑢̃) ) 𝑒−𝑖𝜔′ 𝑢̃0
−∞
(13.2.2)
This integration is facilitated with the aid of close
contour integration on a complex z-plane, where
for a point P(x, y) in a rectangular coordinate
system there corresponds a complex number, 𝑧 =
𝑥 + 𝑖𝑦. The said close contour is given by
∮ 𝑑𝑧 𝑓( 𝑧) 𝑒−𝑖𝜔′𝑧 = ∫ 𝑑𝑧 𝑓( 𝑧) 𝑒−𝑖𝜔′𝑧
𝑂𝐴Γ
+ ∫ 𝑑𝑧 𝑓( 𝑧) 𝑒−𝑖𝜔′𝑧
𝐴𝐵
+ ∫ 𝑑𝑧 𝑓( 𝑧) 𝑒−𝑖𝜔′𝑧
𝐵𝑂
(13.2.3)
We may choose this close integral in the
left upper quadrant to involve the points O(0, 0) at
the origin of the rectangular coordinate system
with a corresponding complex number z = 0 +
iO = 0, A(-R, 0) with z = −R + i0 = −R and
B(0, R) for the complex number z = 0 + iR =
iR. There are three integral paths connecting these
points: straight line along the (real) x axis from O
to A, a quarter circle from A to B and straight line
along (imaginary) y axis from B back to point of
origin.
Briefly, we may examine for the
singularities of the function 𝑓( 𝑧) =
exp(𝑖2𝜔𝐺𝑀𝑙𝑛(−𝑧)) to get for the basic reason
why we can choose ∀𝑥 > 0 to be the branch cut
and why we are having the close integral in the
upper left quadrant.
𝑓( 𝑧) = exp(𝑖2𝜔𝐺𝑀𝑙𝑛(−𝑧)) (13.2.4)
𝑧 = 𝑥 + 𝑖𝑦
At y = 0, where z = x, we have
𝑓( 𝑧 = 𝑥) = exp(𝑖2𝜔𝐺𝑀𝑙𝑛(−𝑥))
(13.2.5.1)
Note in this case that 𝑓( 𝑧 = 𝑥) is singular that is,
not defined on ∀𝑥 > 0 since
𝑙𝑛(−𝑥) (13.2.5.2)
is undefined on ∀𝑥 > 0. So in this case we shall
think of ∀𝑥 > 0 as the branch cut region.

7. On the other hand at x = 0, where z = iy,
we get
𝑓( 𝑧 = 𝑖𝑦) = exp(𝜋𝜔𝐺𝑀) exp(𝑖2𝜔𝐺𝑀𝑙𝑛𝑦)
(13.2.6)
This is singular on ∀𝑦 < 0 because clearly
𝑙𝑛𝑦 (13.2.7)
is not defined for all y less than zero. So in this
case, ∀𝑦 < 0 is also a branch cut here.
Proceeding with (13.2.3) let us take the
path that brings us from O to A. This path is along
the real x axis, where y = 0.
∫ 𝑑𝑧 𝑓( 𝑧) 𝑒−𝑖𝜔′𝑧
𝑂𝐴
= − ∫ 𝑑𝑥 exp(𝑖2𝜔𝐺𝑀𝑙𝑛(−𝑥))𝑒−𝑖𝜔′𝑥
0
−𝑅
(13.2.8)
The right-hand side is negative since the
integration goes against the direction of the path
OA.
The path that goes from A to B is along
the quarter circle 𝐶 𝑅 ∶ | 𝑧| = 𝑅.
∫ 𝑑𝑧 𝑓( 𝑧) 𝑒−𝑖𝜔′𝑧
𝐴𝐵 = ∫ 𝑑𝑧 𝑓( 𝑧) 𝑒−𝑖𝜔′𝑧
𝐶 𝑅
(13.2.9)
Then the path from B back to the origin O.
This is at x = 0, along the imaginary y axis.
∫ 𝑑𝑧 𝑓( 𝑧) 𝑒−𝑖𝜔′𝑧
𝐵𝑂
= −𝑖 ∫ 𝑑𝑦 exp(𝑖2𝜔𝐺𝑀𝑙𝑛(𝑒−𝑖𝜋/2 𝑦))𝑒 𝜔′𝑦
𝑅
0
(13.2.10)
Note that we have taken real integral limit R, the
radius of the circle. As earlier said we choose our
close contour so as to avoid the singularities of the
function f(z) and since this contour does not
contain the said singularities consequently, the
close contour integration adds up to zero. That is,
∮ 𝑑𝑧 𝑓( 𝑧) 𝑒−𝑖𝜔′𝑧 = 0
Γ
(13.2.11)
Then we take limit as 𝑅 → ∞ and take
note that (13.2.9) vanishes upon this limit.
𝑙𝑖𝑚 𝑅 → ∞ ∫ 𝑑𝑧 𝑓( 𝑧) 𝑒−𝑖𝜔′𝑧
𝐶 𝑅
= 0
(13.2.12)
Thus, from (13.2.3) we get the result
∫ 𝑑𝑥 exp(𝑖2𝜔𝐺𝑀𝑙𝑛(−𝑥))𝑒−𝑖𝜔′𝑥
0
−∞
= −𝑖𝑒 𝜋𝜔𝐺𝑀 ∫ 𝑑𝑦 exp(𝑖2𝜔𝐺𝑀𝑙𝑛(𝑦))𝑒 𝜔′𝑦
∞
0
(13.2.13)
To arrive at the desired end result we do
some tricks on this latest equation.
Say we let 𝑥 = 𝑖𝑦 and write −𝑥 =
𝑒
𝑖𝜋
2 (−𝑦), while retaining the real infinite integral
limits. So from (13.2.13) we are led to
∫ 𝑑𝑦 exp(𝑖2𝜔𝐺𝑀𝑙𝑛(−𝑦))𝑒 𝜔′𝑦0
−∞ =
− 𝑒2𝜋𝜔𝐺𝑀 ∫ 𝑑𝑦 exp(𝑖2𝜔𝐺𝑀𝑙𝑛(𝑦))𝑒 𝜔′𝑦∞
0
(13.2.14)
Then we are to note from this to substitute back in
(13.2.13) to have
∫ 𝑑𝑥 exp(𝑖2𝜔𝐺𝑀𝑙𝑛(−𝑥))𝑒−𝑖𝜔′𝑥
0
−∞
= 𝑖𝑒− 𝜋𝜔𝐺𝑀 ∫ 𝑑𝑦 exp(𝑖2𝜔𝐺𝑀𝑙𝑛(−𝑦))𝑒 𝜔′𝑦
0
−∞
(13.2.15)
and put back −𝑦 = 𝑒
𝑖𝜋
2 (−𝑥) in (13.2.15), while
retaining the real infinite integral limits, thereby
obtaining
∫ 𝑑𝑥 exp(𝑖2𝜔𝐺𝑀𝑙𝑛(−𝑥))𝑒−𝑖𝜔′𝑥
0
−∞
= − 𝑒−2𝜋𝜔𝐺𝑀 ∫ 𝑑𝑥 exp(𝑖2𝜔𝐺𝑀𝑙𝑛(−𝑥))𝑒 𝑖𝜔′𝑥
0
−∞
(13.2.16)
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