Finite-Part Integration of the Hilbert Transform Philip Jordan D. Blancas12and Eric A. Galapon1

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Finite-Part Integration of the Hilbert

Transform

Philip Jordan D. Blancas1,2 and Eric A. Galapon1

1Theoretical Physics Group, National Institute of Physics, University of the

Philippines, Diliman, Quezon City, 1101 Philippines

2Research on Optical Science, Engineering, and Systems Laboratory, Department of

Physics, Ateneo de Manila University, Loyola Heights, Quezon City, 1108 Philippines

September 1, 2023

Abstract

The one-sided and full Hilbert transforms are evaluated exactly by

means of the method of ﬁnite-part integration [E.A. Galapon, Proc. Roy.

Soc. A 473, 20160567 (2017)]. In general, the result consists of two

terms—the ﬁrst is an inﬁnite series of ﬁnite-part of divergent integrals,

and the second is a contribution arising from the singularity of the kernel

of transformation. The ﬁrst term is precisely the result obtained when the

kernel of transformation is binomially expanded in positive powers of the

parameter of transformation, followed by term-by-term integration, and

the resulting divergent integrals assigned values equal to their ﬁnite-parts.

In all cases, the ﬁnite-part contribution is present while the presence or

absence of the singular contribution depends on the interval of integra-

tion and on the parity of the function under transformation about the

origin. From the exact evaluation of the Hilbert transform, the dominant

asymptotic behavior for arbitrarily small parameter is obtained.

1 Introduction

A Hilbert integral transform is characterized by a simple pole singularity in

the interior of its contour of integration arising from its kernel of transforma-

tion, and is executed by interpreting the integral as a principal-value integral.

Typically the integration ranges along the entire real line and the transform is

given by

f(ω) = 1

πPV

Z∞

−∞

f(x)

ω−xdx, (1.1)

where PV denotes the principal value. When the function f(x) possesses parity

symmetry, the Hilbert transform (1.1) reduces into either of the forms

πPV

Z∞

xf(x)

ω2−x2dx, 2

πPV

Z∞

ωf(x)

ω2−x2dx, (1.2)

arXiv:2210.14462v2 [math.CV] 31 Aug 2023

depending on whether f(x) is odd or even, respectively. The Hilbert transform

is widely used in signal processing in diverse ﬁelds, such as in acoustics [1],

high-energy physics [2], quantum scattering theory [3], material characterization

[4], optics [5]; moreover, the causality between the dispersion and attenuation

functions in optical systems are related via the Hilbert transform [6].

The Hilbert transform (1.1) and its special reductions (1.2) are routinely

analyzed and evaluated by means of contour integration. In this paper we in-

troduce the method of ﬁnite-part integration [7,8,9,10,11,12,13] in evaluating

exactly the Hilbert transform and its generalizations. Finite-part integration is

a method of evaluating a well-deﬁned integral by means of the ﬁnite-part of

divergent integrals induced from the given integral itself. The method was in-

troduced to solve the problem of missing terms that arise in evaluating the

Stieltjes transform

u(x)

xν(ω+x)dx, 0≤ν < 1, ω > 0,(1.3)

by term-by-term integration that leads to an inﬁnite series of divergent integrals,

with the divergent integrals assigned ﬁnite-values via analytic continuation [8].

It was established in [8,12] that the Stieltjes transform assumes the following

evaluation in terms of ﬁnite-part integrals:

Theorem 1.1. Let u(x)be analytic at x= 0 and let ρ0be the radius of con-

vergence of its Taylor expansion there. If the Stieltjes transform (1.3)exists for

a given positive a≤ ∞ and u(x)is analytic in the interval [0, a], then

u(x)

ω+xdx=

∞

k=0

(−1)kωk\\

u(x)

xk+1 dx−u(−ω) ln(ω),(1.4)

u(x)

xν(ω+x)dx=

∞

k=0

(−ω)k\\

u(x)

xk+ν+1 dx+π

sin(πν)

u(−ω)

ων,0< ν < 1,

(1.5)

for all ω < min(a, ρ0)

In both expressions, the integral \\

0x−k−ν−1u(x) dx(0 ≤ν < 1) denotes the

ﬁnite-part of the divergent integral Ra

0x−k−ν−1u(x) dx. (See Section-2.)

It can be discerned that the summations in equations (1.4) and (1.5) arise

from the term-by-term integration of the binomial expansion of the kernel (ω+

x)−1about ω= 0 with the resulting divergent integrals assigned the values

equal to their ﬁnite-parts. The second terms in (1.4) and (1.5) are the missing

terms, which are referred to as the singular contribution, when naive term-by-

term integration is performed and the divergent integrals are replaced with their

ﬁnite-parts. The singular contributions were recovered from the fundamental

contour integral representation of the ﬁnite-part integral [7,8]. Here, ﬁnite-part

integration of the Hilbert transform will result in expressions similar to those

given by equations (1.4) and (1.5). But under some circumstances, the singular

contribution of the Hilbert transform integral vanishes.

Our results here do not only provide an exact evaluation of the Hilbert trans-

form but, more importantly, lay the necessary groundwork for the application of

ﬁnite-part integration in the resummation problem of divergent series appear-

ing in perturbation theory in many areas of physics. In [9], Tica and Galapon

devised a prescription to use the result (1.4) for the Stieltjes transform in the

strong asymptotic regime for physical quantities assuming a Stieltjes integral

representation. The prescription yields a more accurate resummation scheme

than the standard scheme by Pade approximants in the non-perturbative regime.

However, the resummation there can only treat alternating divergent series. But

non-alternating series arise also in many contexts, such as in eﬀective action for

the vacuum polarization by a uniform electric ﬁeld [14], in the partition function

for the self-interacting QFT [15], and in QED eﬀective action in time-dependent

electric backgrounds [16], to mention a few. For this case, the Hilbert transform

is expected to play the role of the Stieltjes transform in the alternating case

[9]. A necessary component of the resummation by ﬁnite-parts for the Stieltjes

transform is knowledge of the asymptotic behavior of the Stietljes integral for

arbitrarily small values of the parameter ω. This information dictates the appro-

priate kernel of the Steiltjes transform for the resummation. It is also expected

that the same asymptotic information is necessary in a resummation involv-

ing the Hilbert transform. Therefore, we do not only give an exact evaluation

here but also obtain the explicit dominant behavior of the Hilbert transform for

arbitrarily small values of ω.

In application, we do not expect that the relevant Hilbert transform is re-

stricted to (1.1) and (1.2) with f(x) analytic at the origin as is commonly

assumed. Here, we extend the analysis in the presence of branch point singular-

ity at the origin. In particular, for 0 ≤ν < 1, we evaluate the one-sided Hilbert

transforms

x−νf(x)

ω−xdx, PV

x−νf(x)

ω2−x2dx, PV

x1−νf(x)

ω2−x2dx(1.6)

for any positive ω < a ≤ ∞, and the full transforms

−a

x−νf(x)

ω−xdx, PV

−a

|x|−νf(x)

ω−xdx, PV

−a

|x|−νsgn(x)f(x)

ω−xdx(1.7)

for any real |ω|< a ≤ ∞. These Hilbert transforms can serve as starting points

in evaluating more Hilbert transforms. Using the results in [12], the results

here can be extended to cover cases in the presence of logarithmic singularities

at the origin. The tabulation of Hilbert transforms in terms of ﬁnite-part in-

tegrals is important as they may provide guidance in choosing the appropriate

resummation scheme for non-alternating divergent series.

The rest of the paper is organized as follows. In Section-2, we outline the

method of ﬁnite-part integration. In Sections-3,4, and 5, we present various

theorems involving the one-sided Hilbert transform, full Hilbert transforms, and

their special reductions, respectively. In Section-6, we demonstrate some exam-

ples of using ﬁnite-part integration in evaluating Hilbert transform integrals.

Finally, in Section-7, we show the various methods for evaluating ﬁnite-part

integrals. In Appendix-A, we tabulate a new set of Hilbert transform integrals.

And in Appendix-B, we list down the ﬁnite-part integrals used to derive the

Hilbert transform integrals in Appendix-A.

2 Finite-part Integrals

To apply the method of ﬁnite-part integration, we will cast the Hilbert

transform such that the induced divergent integrals are of the form

f(x)

xk+νdxfor k= 1,2,3,..., 0≤ν < 1,0< a ≤ ∞,(2.1)

where the divergence arises from the non-integrable singularity at the origin.

The ﬁnite-part is obtained by replacing the lower limit of integration with some

positive ε<aand the resulting integral decomposed in the form

f(x)

xk+νdx=Cε+Dε,(2.2)

where Cε(Dϵ) constitutes all terms that converge (diverge) in the limit as ε

approaches 0. Then the ﬁnite-part is given by the limit

f(x)

xk+νdx= lim

ε→0Cε.(2.3)

If the upper limit of integration happens to be inﬁnite, the ﬁnite-part is given

Z∞

f(x)

xk+νdx= lim

a→∞ \\

f(x)

xk+νdx, (2.4)

which we assume to exist in this paper. To uniquely deﬁne the ﬁnite part, the

diverging part Dεmust contain only diverging algebraic powers of εand ln ε(see

Section-7.1). By deﬁnition, the ﬁnite-part integral always exists and is unique.

In this paper, we assume that f(x) is a function of the real variable xover

the interval [0, a] (0 < a ≤ ∞) and that it has a complex extension f(z) of

the complex variable zthat is analytic in the interval [0, a]. This means that

f(x) is the restriction of f(z) in the interval [0, a]. By the principle of analytic

continuation, the function f(z) is unique and is completely determined by f(x).

We will refer to a function f(x) over some interval [0, a] having such property as

complex analytic in the given interval. We denote the distance of the singularity

of f(z) closest to the origin by ρ0. If f(z) is entire, then ρ0=∞. Equivalently,

ρ0is the radius of convergence of the Taylor series expansion of f(z) about

z= 0.

Central to the method of ﬁnite-part integration is the fact that the ﬁnite-

part possesses a contour integral representation in the complex plane [8]. The

Figure 1: The contour Cof integration in the contour integral representation

of the ﬁnite-part integrals (2.5) and (2.6). The contour starts at aabove the

branch cut of log zand z−ν, goes around the origin and ends at abelow the cut

without enclosing any singularity of f(z). The same contour is employed in the

contour integral representation of the Hilbert transform.

representation depends on the singularity of the divergent integral at the ori-

gin: a pole singularity (corresponding to ν= 0) or a branch point singularity

(corresponding to ν̸= 0).

Theorem 2.1. Let f(x)be complex analytic in the interval [0, a]with f(0) ̸= 0.

Then

f(x)

xmdx=1

2πi ZC

f(z)

zm[log(z)−iπ] dzfor m= 1,2,3,··· ,(2.5)

f(x)

xm+νdx=1

e−2πiν −1ZC

f(z)

zm+νdzfor 0< ν < 1, m = 1,2,3,··· ,(2.6)

where f(z)is the complex extension of f(x)in the complex plane, log zand zν

take the positive real line as their branch cuts with their values coinciding with

the real-valued functions ln xand xνabove the branch cut, respectively, and the

contour Cis as shown in Figure-1.

3 One-sided Hilbert Transforms

In this section, we execute the ﬁnite-part integration of the one-sided Hilbert

transform of x−νf(x),

f(x)

xν(ω−x)dx, 0≤ν < 1, ω > 0, a ≤ ∞,(3.1)

where f(x) is complex analytic in the interval [0, a].

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Finite-PartIntegrationoftheHilbertTransformPhilipJordanD.Blancas1,2andEricA.Galapon11TheoreticalPhysicsGroup,NationalInstituteofPhysics,UniversityofthePhilippines,Diliman,QuezonCity,1101Philippines2ResearchonOpticalScience,Engineering,andSystemsLaboratory,DepartmentofPhysics,AteneodeManilaUniversity...

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