figuret

# Momentum-Space Scattering Extremizations

Chunchao Wen  College for Advanced Interdisciplinary Studies, National
University of Defense Technology, Changsha 410073, P. R. China.  Nanhu Laser
Laboratory and Hunan Provincial Key Laboratory of Novel Nano-Optoelectronic
Information Materials and Devices, National University of Defense Technology,
Changsha 410073, P. R. China.  Jianfa Zhang<EMAIL_ADDRESS>College for
Advanced Interdisciplinary Studies, National University of Defense Technology,
Changsha 410073, P. R. China.  Nanhu Laser Laboratory and Hunan Provincial Key
Laboratory of Novel Nano-Optoelectronic Information Materials and Devices,
National University of Defense Technology, Changsha 410073, P. R. China.
Shiqiao Qin  College for Advanced Interdisciplinary Studies, National
University of Defense Technology, Changsha 410073, P. R. China.  Nanhu Laser
Laboratory and Hunan Provincial Key Laboratory of Novel Nano-Optoelectronic
Information Materials and Devices, National University of Defense Technology,
Changsha 410073, P. R. China.  Zhihong Zhu  College for Advanced
Interdisciplinary Studies, National University of Defense Technology, Changsha
410073, P. R. China.  Nanhu Laser Laboratory and Hunan Provincial Key
Laboratory of Novel Nano-Optoelectronic Information Materials and Devices,
National University of Defense Technology, Changsha 410073, P. R. China.  Wei
Liu<EMAIL_ADDRESS>College for Advanced Interdisciplinary Studies,
National University of Defense Technology, Changsha 410073, P. R. China.
Nanhu Laser Laboratory and Hunan Provincial Key Laboratory of Novel Nano-
Optoelectronic Information Materials and Devices, National University of
Defense Technology, Changsha 410073, P. R. China.

###### Abstract

Studies into scatterings of photonic structures have been so far
overwhelmingly focused on their dependencies on the spatial and spectral
morphologies of the incident waves. In contrast, the evolution of scattering
properties through another parameter space of incident directions (momentum
space) has attracted comparably little attention, though of profound
importance for various scattering-related applications. Here we investigate,
from the perspective of quasi-normal modes (QNMs), the momentum-space
scattering extremizations with respect to varying incident directions of plane
waves. It is revealed that for effective single-QNM excitations, scatterings
are maximized exactly along those directions where the QNM radiation reaches
its maximum, with matched incident and radiation polarizations. For an
arbitrary direction, when the incident polarization is tuned to be orthogonal
to that of the mode radiation, the QNM cannot be excited and thus the
scatterer becomes invisible with null scatterings. The principles we have
revealed are protected by fundamental laws of reciprocity and energy
conservation (optical theorem), which can be further expanded and applied for
other branches of wave physics.

##  I Introduction

Throughout all disciplines of photonics that involve light-matter
interactions, scattering manipulations (enhancement or suppression e.g.
cloaking) constitute one of the central themes Bohren1983_book ;
YARIV_2006__Photonics ; LIU_2005__Photonic . To control the interactions,
either the incident source or the photonic structure can be engineered to
satisfy different demands of various applications. For source engineering,
previous studies are extensively focused on frequency tuning and/or spatial
phase and polarization morphology structuring
ZUMOFEN_Phys.Rev.Lett._perfect_2008-1 ; GOUESBET_generalized_2011 ;
SHEN_2019_LightSciAppl_Optical ; WAN_2022_Natl.Sci.Rev._Photonic ;
FORBES_Nat.Photonics_Structured . Though it is apparent that light-matter
interactions are largely dependent on incident directions, systematic and
thorough examinations of momentum-space scattering extremizations (maximized
or minimized) have not yet been conducted. All possible incident directions
constitute a closed momentum sphere, and according to the extreme value
theorem RUDIN_1976__Principles  there must be at least one direction along
which the scattering can reach its maximum or minimum.

Here we study the momentum-space scattering extremizations with respect to
varying directions of incident plane waves, from the perspective of QNMs
supported by the scatterers. When only one QNM is effectively excited, it is
discovered that the maximum mode radiation directions correspond exactly to
the momentum-space points where the scattering is maximized, with matched
incident and radiation polarizations. Along an arbitrary direction, when the
incident and mode radiation polarizations are tuned to be orthogonal, all
scatterings are fully eliminated, rendering the scatterer invisible. Our
revelations connecting QNM radiations and momentum-space scattering evolutions
are secured by the fundamental laws of electromagnetic reciprocity and optical
theorem. As a result, the framework we have established can be naturally
extended to apply to other branches of wave physics, incubating applications
in fields such as acoustics, water waves and microscopic matter waves.

Figure 1: (a) A metallic cylinder that supports a linear electric dipolar QNM,
the angular radiation intensity and polarization distributions of which are
shown in (b) and (c), respectively. Three excitation scenarios (i)-(iii) are
also specified in (a), where black and blue arrows denote the propagation and
electric field directions of the incident plane waves, respectively. For (i) &
(ii) the QNM cannot be excited, while for (iii) the excitation efficiency
reaches its maximum.

##  II Theoretical Model Incorporating Reciprocity and Optical Theorem

Throughout this work, we confine our studies to the single-QNM regime: the
spectral regions where only one QNM can be effectively excited, and other QNMs
are either spectrally apart or simply cannot be excited. The incident sources
are plane waves (electric field vector $\mathbf{E}_{\rm{inc}}$, wavevector
$\mathbf{k}_{\rm{inc}}$, angular frequency $\omega$ and vacuum wavelength
$\lambda$) and the QNM excited is characterized by a complex eigenfrequency
$\tilde{\omega}$ LALANNE__LaserPhotonicsRev._Light . In the single-QNM regime,
all scattering properties are decided by the excitation coefficient
$\alpha(\mathbf{E}_{\rm{inc}},\omega,\mathbf{k}_{\rm{inc}})$ of the QNM
CHEN_Phys.Rev.Lett._Extremize :

$\rm
C_{\rm{sca},\rm{ext},\rm{abs}}\propto|\alpha(\mathbf{E}_{\rm{inc}},\omega,\mathbf{k}_{\rm{inc}})|^{2},$
(1)

where $\rm C_{\rm{sca},\rm{ext},\rm{abs}}$ are respectively scattering,
extinction and absorption cross sections;
$|\alpha(\mathbf{E}_{\rm{inc}},\omega,\mathbf{k}_{\rm{inc}})|^{2}$ is the
excitation efficiency, and its spectral curve (dependence on $\omega$)
generally exhibits a typical Lorentzian shape, with the central position and
linewidth decided by the complex $\tilde{\omega}$
LALANNE__LaserPhotonicsRev._Light ; CHEN_Phys.Rev.Lett._Extremize ).

A conventional approach to calculate $\alpha$ and thus the excitation
efficiency relies on cumbersome volume integrations involving detailed near-
field current distributions of the QNM LALANNE__LaserPhotonicsRev._Light . It
is recently shown that, for incident planes waves, $\alpha$ can be directly
calculated in the far field according to the electromagnetic reciprocity
CHEN_Phys.Rev.Lett._Extremize :

$\alpha(\mathbf{E}_{\rm{inc}},\omega,\mathbf{k}_{\rm{inc}})\propto\tilde{\mathbf{E}}_{\rm{rad}}\cdot\mathbf{E}_{\rm{inc}},$
(2)

where $\tilde{\mathbf{E}}_{\rm{rad}}$ is the electric field vector of the QNM
radiation along $\mathbf{k}_{\rm{rad}}=-\mathbf{k}_{\rm{inc}}$. To clarify the
meaning of the dot product in Eq. (2), we employ the normalized Jones vectors
$\mathbf{J_{\rm{inc,rad}}}=(j^{~{}\rm{inc,rad}}_{1},j^{\rm{~{}inc,rad}}_{2})$
to characterize the polarizations of the incident wave and the far-field QNM
radiation, e.g. $\mathbf{J}=\frac{1}{\sqrt{2}}(1,\pm i)$ denoting right- and
left-handed circular polarizations, respectively YARIV_2006__Photonics . Then
Eq. (2) can be expressed as DEHOOP_Appl.sci.Res._reciprocitya ;
CHEN_2021_Phys.Rev.B_Arbitrary :

$\alpha(\mathbf{E}_{\rm{inc}},\omega,\mathbf{k}_{\rm{inc}})\propto\mathbf{J_{\rm{rad}}}\mathbf{J^{\dagger}_{\rm{inc}}}\tilde{\rm{E}}_{\rm{rad}}\rm{{E}}_{\rm{inc}},$
(3)

where $\dagger$ denotes combined operations of complex conjugate and
transpose; $\tilde{\rm{E}}_{\rm{rad}}$ and $\rm{{E}}_{\rm{inc}}$ are scalar
field amplitudes. It is worth mentioning that the Jones vector characterizes
the polarization only and does not contain any information about the light
prorogation direction, e.g. waves of the same polarization while propagating
along opposite directions are represented by the same Jones vector.

Since all cross sections $\rm C_{\rm{sca},\rm{ext},\rm{abs}}$ are defined to
be independent of the incident field strength Bohren1983_book , Eq. (1) can be
then explicitly simplified as:

$\rm
C_{\rm{sca},\rm{ext},\rm{abs}}\propto|\mathbf{J_{\rm{rad}}}\mathbf{J^{\dagger}_{\rm{inc}}}|^{2}|\tilde{\rm{E}}_{\rm{rad}}|^{2}=|\mathbf{J_{\rm{rad}}}\mathbf{J^{\dagger}_{\rm{inc}}}|^{2}{\rm{I}}_{\rm{rad}},$
(4)

where ${\rm{I}}_{\rm{rad}}=|\tilde{\rm{E}}_{\rm{rad}}|^{2}$ is the QNM
radiation intensity along $\mathbf{k}_{\rm{rad}}=-\mathbf{k}_{\rm{inc}}$. For
general reciprocal scatterers, it has been proved, based on the principles of
reciprocity and optical theorem, that for an arbitrary pair of opposite
incident directions ($\pm\mathbf{k}_{\rm{inc}}$) the extinction cross sections
are identical SOUNAS_Opt.Lett.OL_extinction_2014 ;
CHEN_2020_Phys.Rev.Research_Scatteringa . In the single-QNM regime, both cross
sections of scattering and absorption are also identical for
$\pm\mathbf{k}_{\rm{inc}}$ CHEN_Phys.Rev.Lett._Extremize . According to Eq.
(4), it requires that:

$\mathbf{J_{\rm{rad}}}(\mathbf{k}_{\rm{rad}})=\mathbf{J_{\rm{rad}}}(-\mathbf{k}_{\rm{rad}}),~{}~{}{\rm{I}}_{\rm{rad}}(\mathbf{k}_{\rm{rad}})={\rm{I}}_{\rm{rad}}(-\mathbf{k}_{\rm{rad}}).$
(5)

That is to say, in the single-QNM regime, the mode radiations are of the same
polarization and magnitude along two arbitrary opposite directions. A more
intuitive interpretation of this conclusion is as follows: (i) Equation (4)
tells that for an incident plane wave along $\mathbf{k}_{\rm{inc}}$, the
extinction is solely decided by the QNM radiation along the opposite direction
of $\mathbf{k}_{\rm{rad}}=-\mathbf{k}_{\rm{inc}}$; (ii) Optical theorem tells
that the extinction is only related to QNM radiation along
$\mathbf{k}_{\rm{rad}}=\mathbf{k}_{\rm{inc}}$, as only forward radiations
(parallel to the incident direction) can interfere destructively with the
incident wave to extinguish part of its energy to account for all-angle
scatterings and dissipative absorptions Bohren1983_book ; (i) & (ii) are
applicable for arbitrary incident polarizations, which results in Eq. (5).

Now we proceed to discuss the momentum-space scattering extremizations based
on Eq. (4). The maximization of the scattering requires the simultaneous
maximization of both ${\rm{I}}_{\rm{rad}}$ and
$|\mathbf{J_{\rm{rad}}}\mathbf{J^{\dagger}_{\rm{inc}}}|$: the former
corresponds to that $\mathbf{k}_{\rm{inc}}$ is collinear to the directions
along which the QNM radiation is strongest [according to Eq. (5) there are at
least a pair of such directions which are opposite to each other]; the latter
requires the incident and QNM radiations along $\pm\mathbf{k}_{\rm{inc}}$) are
of the same polarization
($|\mathbf{J_{\rm{rad}}}\mathbf{J^{\dagger}_{\rm{inc}}}|=1$), which is natural
since the time reversal operation does not change the polarization of the
field DEHOOP_Appl.sci.Res._reciprocitya ; CHEN_2021_Phys.Rev.B_Arbitrary .

We have thus so far managed to reveal the recipe to obtain the maximum
scattering in the single-QNM regime: (i) Identify the QNM of complex
eigenfrequency $\tilde{\omega}$; (ii) Calculate its far-field radiations, in
terms of both distributions of radiation intensity ${\rm{I}}_{\rm{rad}}$ and
polarization $\mathbf{J_{\rm{rad}}}$; (iii) Pinpoint the directions (at least
a pair of them) where the scattering reaches its maximum and specify the
polarization; (iv) Shine light along those directions with matched
polarization (with the same polarization as that of the radiation:
$|\mathbf{J_{\rm{rad}}}\mathbf{J^{\dagger}_{\rm{inc}}}|=1$). For a fixed
incident direction, the scattering is maximized when the incident and
radiation polarizations are matched; while for a fixed incident polarization,
the scattering maximization requires simultaneous consideration of
$|\mathbf{J_{\rm{rad}}}\mathbf{J^{\dagger}_{\rm{inc}}}|$ and
${\rm{I}}_{\rm{rad}}$.

Equation (4) also shows that along arbitrary directions, the scattering can be
always fully eliminated $\rm C_{\rm{sca},\rm{ext},\rm{abs}}=0$: along
directions where the QNM radiation is not zero, the elimination is obtained
when the incident and radiation polarizations are orthogonal
($\mathbf{J_{\rm{rad}}}\mathbf{J^{\dagger}_{\rm{inc}}}=0)$; along directions
of vanishing radiations (${\rm{I}}_{\rm{rad}}=0$), there are no scatterings
for incident waves of arbitrary polarizations.

The principles discovered and explicitly described above are schematically
exemplified in Fig. 1: a metallic cylinder [Fig. 1(a)] supports a linear
electric dipolar QNM, the angular radiation intensity and polarization (linear
polarization everywhere along the longitude, except for the poles where the
radiations are singularly zero CHEN_Phys.Rev.Lett._Singularities )
distributions of which are shown respectively in Fig. 1(b) and Fig. 1(c).
Three excitation scenarios are also included in Fig. 1: for (i) & (ii) the
excitation efficiency is zero while for (iii) it is maximized. This can be
easily interpreted from the conventional perspective of near-field
interactions between $\mathbf{E}_{\rm{inc}}$ and the oscillating electrons
along the cylinder axis. Meanwhile, Eq. (4) provides an alternative far-field
interpretation for each scenario: (i) ${\rm{I}}_{\rm{rad}}=0$ and thus
$\alpha=0$; (ii) $\mathbf{J_{\rm{rad}}}\mathbf{J^{\dagger}_{\rm{inc}}}=0$ and
thus $\alpha=0$; (iii) both
$|\mathbf{J_{\rm{rad}}}\mathbf{J^{\dagger}_{\rm{inc}}}|$ and
$\tilde{\mathbf{E}}_{\rm{rad}}$ are maximal, leading to maximized coupling
efficiency and thus also scattering properties. For this elementary example,
the superiority of far-field interpretations based on Eq. (4) is not obvious;
while for more sophisticated structures where direct analysis of charge-field
interactions becomes too complicated, the overwhelming simplicity and strength
of our far-field model would become apparent.

Figure 2: (a) The spherical coordinate system with the azimuthal angle
$\varphi$ and polar angle $\theta$. (b) A gold SRR with all geometric
paramters and its orientation within the coordinate system specified. (c) and
(d) Angular radiation patterns of the two QNMs (of eigenfrequencies
$\tilde{\omega_{1}}$ and $\tilde{\omega_{2}}$) supported by the SRR, where the
selected momentum-space semi-circles are marked by white-dashed lines. For the
higher-order mode in (d), two points of maximum (star) and minimum (cross)
radiation intensity have been marked. (e) and (f) Normalized radiation
intensity or scattering (absorption) cross sections on the momentum-space
semi-circles. (g) and (h) Near-field distributions (in terms of
$\mathbf{E}_{x}$ and $\mathbf{E}_{y}$) for incident waves along the maximum-
radiation direction marked in (d) with matched and orthogonal polarizations,
respectively. (i) Near-field distributions (in terms of $\mathbf{E}_{x}$) for
the incident wave along the minimum-radiation direction marked in (d) with the
matched polarization.

##  III Results and Discussions

To further verify the principles revealed, we employ the seminal structures of
split ring resonators (SRRs), at the same time emphasizing that our theory is
generally applicable to other photonic configurations. Throughout this work,
the SRRs investigated consist of gold with effective bulk permittivity fitted
from data in Ref. Johnson1972_PRB , and numerical results are obtained using
COMSOL Multiphysics. We start with an individual SRR, and all geometric
parameters and its orientation with respect to the coordinate system shown in
Fig. 2(a) are specified in Fig. 2(b). Two QNMs supported by this SRR are
chosen with eigenfrequencies of $\tilde{\omega}_{1}=(2.326\times
10^{14}-1.3258\times 10^{13}\rm{i})$ rad/s (central resonant wavelength
$\lambda_{1}=9.348~{}\mu$m) and $\tilde{\omega}_{2}=(8.055\times
10^{14}-4.8841\times 10^{12}\rm{i})$ rad/s ($\lambda_{2}=2.339~{}\mu$m), and
their corresponding angular radiation patterns (in terms of
${\rm{I}}_{\rm{rad}}$) are shown in Figs. 2(c) and (d), respectively. For each
QNM, we have selected a semi-circle in the momentum-space [denoted by dashed
white lines in Figs. 2(c) & (d)] which covers both directions of maximum and
minimum radiation intensity throughout the momentum space. For calculations of
scattering properties ($\rm C_{\rm{sca}}$ and $\rm C_{\rm{abs}}$) with
incident waves along directions on those momentum-space semi-circles, we fix
the angular frequency of the incident wave ${\omega}=\rm{Re}(\tilde{\omega})$
and make sure the incident polarization is matched to that of the radiation
($|\mathbf{J_{\rm{rad}}}\mathbf{J^{\dagger}_{\rm{inc}}}|=1$), reducing Eq. (4)
to:

$\rm C_{\rm{sca},\rm{ext},\rm{abs}}\propto{\rm{I}}_{\rm{rad}}.$ (6)

The normalized ${\rm{I}}_{\rm{rad}}$ and $\rm C_{\rm{sca,abs}}$ on our
selected momentum-space semi-circles for both QNMs are shown respectively in
Figs. 2(e) and (f). Agreement between them is manifest confirming Eq. (6),
despite some discrepancies along those directions of small
${\rm{I}}_{\rm{rad}}$. This is because small ${\rm{I}}_{\rm{rad}}$ is
synonymous with small excitation efficiency [Eq. (3)] and then contributions
from other QNMs are not fully negligible any more, rendering our single-QNM
approximation less accurate. The smaller is ${\rm{I}}_{\rm{rad}}$, the larger
is the discrepancy, as is also the case for the results shown in Figs. 3(b)
and (c) that will be discussed later.

Figure 3: (a) A pair of coupled parallel SRRs with a displacement of $90$ nm
and a twist angle of $\delta=46.4^{\circ}$ along the y-axis. Each SRR is
identical to that shown in Fig. 2(g). (b) and (c) Angular radiation patterns
for the two selected QNMs (of eigenfrequencies $\tilde{\omega_{3}}$ and
$\tilde{\omega_{4}}$) supported by the SRR pair, where the selected momentum-
space semi-circles are marked by white-dashed lines. (d) and (e) Normalized
radiation intensity or scattering (absorption) cross sections on the selected
momentum-space semi-circles marked.

We have also pinpointed the two directions of maximum and minimum radiation
directions of the higher-order QNM, marked by a star
($\theta=44.3^{\circ},~{}\varphi=0$) and a cross
($\theta=90^{\circ},~{}\varphi=0$) in Fig. 2(d), respectively. When the
incident direction is along the maximum radiation direction, the near-field
distributions for matched
($|\mathbf{J_{\rm{rad}}}\mathbf{J^{\dagger}_{\rm{inc}}}|=1$) and orthogonal
($|\mathbf{J_{\rm{rad}}}\mathbf{J^{\dagger}_{\rm{inc}}}|=0$) incident
polarizations are shown respectively in Fig. 2(g) (largest excitation
efficiency and scattering) and Fig. 2(h) (close to zero excitation efficiency
and scattering). Along the minimum radiation direction
(${\rm{I}}_{\rm{rad}}\approx 0$), it is shown in Fig. 2(i) that, similar to
that shown in Fig. 2(h), scattering is almost absent despite the matched
incident polarization. All those results are consistent with Eq. (4),
confirming that invisibility is obtainable for arbitrary incident directions
with either ${\rm{I}}_{\rm{rad}}=0$ or
$|\mathbf{J_{\rm{rad}}}\mathbf{J^{\dagger}_{\rm{inc}}}|=0$.

We then proceed to check the more sophisticated configuration of coupled SRRs
schematically shown in Fig. 3(a). Here each SRR is identical to that shown in
Fig. 2(b) and they are parallel-displaced by $90$ nm and twisted by
$\delta=46.4^{\circ}$ with respect to each other along the y-axis. Similarly,
two QNMs supported by the SRR pair are selected with eigenfrequencies of
$\tilde{\omega}_{3}=(1.7825\times 10^{14}-1.1628\times 10^{13}\rm{i})$ rad/s
(central resonant wavelength $\lambda_{3}=10.568~{}\mu$m) and
$\tilde{\omega}_{4}=(7.8753\times 10^{14}-4.1295\times 10^{12}\rm{i})$ rad/s
($\lambda_{4}=2.392~{}\mu$m), and their corresponding radiation patterns are
demonstrated respectively in Figs. 3(b) and (c). The normalized quantities
(${\rm{I}}_{\rm{rad}}$ and $\rm C_{\rm{sca,abs}}$ with matched incident
polarizations) on the selected momentum-space semi-circles [marked by white-
dashed lines in Figs. 3(b) and (c)] are shown in Figs. 3(d) and (e),
respectively. Similar to those already observed in Figs. 2(e) and (f), it is
manifest in Figs. 3(d) and (e) that: along directions with large
${\rm{I}}_{\rm{rad}}$ and thus also large coupling efficiencies, the agreement
between both sets of results are excellent; while for small
${\rm{I}}_{\rm{rad}}$ and thus also low coupling efficiencies, obvious
discrepancies arise, induced by non-negligible contributions from other QNMs.

##  IV Conclusions and Perspectives

We study momentum-space scattering extremizations with respect to varying
incident directions from the perspective the QNMs. When an individual QNM is
effectively excited, we have revealed that: scattering is maximized for waves
incident along the directions of maximum QNM radiations with matched radiation
and incident polarizations; for an arbitrary direction of non-zero radiation,
the scattering is fully eliminated with orthogonal radiation and incident
polarizations; for incident directions along which there are no QNM
radiations, the scatterings are null for arbitrary incident polarizations. We
have effectively revealed the deep connections between QNM radiation patterns
and momentum-space scattering evolutions, which are protected by the
fundamental principles of electromagnetic reciprocity and optical theorem, and
thus can be naturally extended to other disciplines of wave physics.

In this study, we have confined our investigations to fully-polarized incident
waves on the surface of the Poincaré sphere, and similar studies can be
directly extended to the interior of the Poincaré sphere to encompass
partially polarized and unpolarized incident waves. For example, for
unpolarized incident waves, it is obvious that the scattering is maximized and
minimized along directions of maximum and minimum QNM radiations,
respectively.

We have shown in previous studies that the polarization distributions of the
QNM radiations are globally bounded by the Poincaré-Hopf theorem
NEEDHAM__Visuala , which secures the existence of polarization singularities
and a global index sum of Euler characteristic $2$ of the momentum sphere
CHEN_Phys.Rev.Lett._Singularities ; CHEN_2019_ArXiv190409910Math-
PhPhysicsphysics_Linea . In a similar manner, the maximum and minimum QNM
radiation points (together with other critical points) are also globally
bounded by the Morse theory MATSUMOTO_2001__Introduction , and explorations of
those points and their Morse indexes from a photonics perspective can
certainly further expand the richness of the vibrant field of topological
physics.

The presence of discrepancies in Figs. 2(e-f) and Figs. 3(d-e) have already
exposed the limitation of our theoretical model, which is only applicable to
the single-QNM excitation regime. When there are symmetry-protected mode
degeneracies or spectrally-close QNMs that are simultaneously co-excited, the
principles revealed in this paper cannot be directly applied. A more
encompassing model that can systematically solve the problem of the momentum-
space scattering exterminations, for either multi-QNM scatterers or structured
non-plane incident waves, is yet to be established.

## acknowledgement

This research was funded by the National Natural Science Foundation of China
(12274462, 11674396, and 11874426), and the Science and Technology Planning
Project of Hunan Province (2018JJ1033 and 2017RS3039).

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10.1080/10652460YYxxxxxxx 1476-8291 1065-2469 00 00 2009 January

#  Fourier expansions for a logarithmic fundamental solution of the
polyharmonic equation

Howard S. Cohla,b∗
aApplied and Computational Mathematics Division, Information Technology
Laboratory, National Institute of Standards and Technology, Gaithersburg,
Maryland, U.S.A. ;
bDepartment of Mathematics, University of Auckland, Auckland, New Zealand
∗Corresponding author. Email<EMAIL_ADDRESS>
( v3.5 released October 2008)

###### Abstract

In even-dimensional Euclidean space for integer powers of the Laplacian
greater than or equal to the dimension divided by two, a fundamental solution
for the polyharmonic equation has logarithmic behavior. We give two approaches
for developing a Fourier expansion of this logarithmic fundamental solution.
The first approach is algebraic and relies upon the construction of two-
parameter polynomials. We describe some of the properties of these
polynomials, and use them to derive the Fourier expansion for a logarithmic
fundamental solution of the polyharmonic equation. The second approach depends
on the computation of parameter derivatives of Fourier series for a power-law
fundamental solution of the polyharmonic equation. The resulting Fourier
series is given in terms of sums over associated Legendre functions of the
first kind. We conclude by comparing the two approaches and giving the
azimuthal Fourier series for a logarithmic fundamental solution of the
polyharmonic equation in rotationally-invariant coordinate systems.

35A08; 31B30; 31C12; 33C05; 42A16

###### keywords:

fundamental solutions; polyharmonic equation; Fourier series; polynomials;
associated Legendre functions

##  1 Introduction

Solutions of the polyharmonic equation (powers of the Laplacian operator) are
ubiquitous in many areas of computational, pure, applied mathematics, physics
and engineering. We concern ourselves, in this paper, with a fundamental
solution of the polyharmonic equation (Laplace, biharmonic, etc.), which by
convolution yields a solution to the inhomogeneous polyharmonic equation.
Solutions to inhomogeneous polyharmonic equations are useful in many physical
applications including those areas related to Poisson’s equation such as
Newtonian gravity, electrostatics, magnetostatics, quantum direct and exchange
interactions (cf. §1 in Cohl & Dominici (2010) [4]), etc. Furthermore,
applications of higher-powers of the Laplacian include such varied areas as
minimal surfaces [13], Continuum Mechanics [9], Mesh deformation [7],
Elasticity [10], Stokes Flow [8], Geometric Design [20], Cubature formulae
[17], mean value theorems (cf. Pizzetti’s formula) [14], and Hartree-Fock
calculations of nuclei [21].

It is a well-known fact (see for instance Schwartz (1950) ([16], p. 45),
Gel′fand & Shilov (1964) ([6], p. 202) that a fundamental solution of the
polyharmonic equation on $d$-dimensional Euclidean space is given by
combinations of power-law and logarithmic functions of the global distance
between two points. In a recent paper (Cohl & Dominici (2010) [4]), we derived
a complex identity which determined the Fourier coefficients of a power-law
fundamental solution of the polyharmonic equation. The Fourier coefficients
were seen to be given in terms of associated Legendre functions. The present
work is concerned with computing the Fourier coefficients of a logarithmic
fundamental solution of the polyharmonic equation. One obtains a logarithmic
fundamental solution for the polyharmonic equation only in even-dimensional
Euclidean space and only when the power of the Laplacian is greater than or
equal to the dimension divided by two. The most familiar example of a
logarithmic fundamental solution of the polyharmonic equation occurs in two-
dimensions, for a single-power of the Laplacian, i.e., Laplace’s equation.

We present two different approaches for obtaining Fourier series of a
logarithmic fundamental solution for the polyharmonic equation. The first
approach is algebraic and involves the generation of a certain set of
naturally arising two-index polynomials which we refer to as logarithmic
polynomials. The second approach starts with the main result from Cohl &
Dominici (2010) [4] and determines the Fourier series expansion for a
logarithmic fundamental solution of the polyharmonic equation through
parameter differentiation. Series expansions for fundamental solutions of
linear partial differential equations such as the polyharmonic equation are
extremely useful in determining Dirichlet boundary values for solutions on
interior domains (see for example Cohl & Tohline (1999) [5]).

This paper is organized as follows. In §2 we introduce the problem. In §3 we
describe our algebraic approach to computing a Fourier series of a logarithmic
fundamental solution of the polyharmonic equation. In §4 we give our limit
derivative approach for computing the Fourier series of a logarithmic
fundamental solution of the polyharmonic equation. In §5 we give some
comparisons between the two approaches. In §6 we use the results presented in
the previous sections to obtain azimuthal Fourier expansions for a logarithmic
fundamental solution of the polyharmonic equation in rotationally-invariant
coordinate systems which parametrize points in $d$-dimensional Euclidean
space. In Appendix 7 we present some necessary formulae relating to
differentiation of associated Legendre functions of the first kind with
respect to the degree. In Appendix 8 we present some of the properties of the
logarithmic polynomials.

Throughout this paper we rely on the following definitions. The set of natural
numbers is given by ${\mathbf{N}}:=\\{1,2,3,\ldots\\}$, the set
${\mathbf{N}}_{0}:=\\{0,1,2,\ldots\\}={\mathbf{N}}\cup\\{0\\}$, the set of
integers is given by ${\mathbf{Z}}:=\\{0,\pm 1,\pm 2,\ldots\\},$ and the set
${\mathbf{Q}}$ represents the rational numbers. For
$a_{1},a_{2},\ldots\in{\mathbf{C}}$, if $i,j\in{\mathbf{Z}}$ and $j<i$ then
$\sum_{n=i}^{j}a_{n}=0$ and $\prod_{n=i}^{j}a_{n}=1$, where ${\mathbf{C}}$
represents the complex numbers. The set ${\mathbf{R}}$ represents the real
numbers.

##  2 Fundamental solution of the polyharmonic equation and the non-
logarithmic Fourier series

If $\Phi$ satisfies the polyharmonic equation given by

$(-\Delta)^{k}\Phi({\bf x})=0,$ (1)

where ${\bf x}\in{\mathbf{R}}^{d},$ $\Delta:C^{p}({\mathbf{R}}^{d})\to
C^{p-2}({\mathbf{R}}^{d})$, for $p\geq 2$ is the Laplacian operator defined by
$\Delta:=\frac{\partial^{2}}{\partial
x_{1}^{2}}+\ldots+\frac{\partial^{2}}{\partial x_{d}^{2}},$ $k\in{\mathbf{N}}$
and $\Phi\in C^{2k}({\mathbf{R}}^{d}),$ then $\Phi$ is called polyharmonic. If
the power $k$ of the Laplacian equals two, then (1) is called the biharmonic
equation and $\Phi$ is called biharmonic. The inhomogeneous polyharmonic
equation is given by

$(-\Delta)^{k}\Phi({\bf x})=\rho({\bf x}),$ (2)

where we take $\rho$ to be an integrable function so that a solution to (2)
exists. A fundamental solution for the polyharmonic equation on
${\mathbf{R}}^{d}$ is a function
${{\mathfrak{g}}}_{k}^{d}:({\mathbf{R}}^{d}\times{\mathbf{R}}^{d})\setminus\\{({\bf
x},{\bf x}):{\bf x}\in{\mathbf{R}}^{d}\\}\to{\mathbf{R}}$ which satisfies the
equation

$(-\Delta)^{k}{{\mathfrak{g}}}_{k}^{d}({\bf x},{\bf x}^{\prime})=c\delta({\bf
x}-{\bf x}^{\prime}),$ (3)

for some $c\in{\mathbf{R}},c\neq 0$, where $\delta$ is the Dirac delta
function and ${{\bf x}^{\prime}}\in{\mathbf{R}}^{d}$. When $c=1$, we call a
fundamental solution of the polyharmonic equation normalized, and denote it by
${\mathcal{G}}_{k}^{d}:({\mathbf{R}}^{d}\times{\mathbf{R}}^{d})\setminus\\{({\bf
x},{\bf x}):{\bf x}\in{\mathbf{R}}^{d}\\}\to{\mathbf{R}}$. The Euclidean inner
product $(\cdot,\cdot):{\mathbf{R}}^{d}\times{\mathbf{R}}^{d}\to{\mathbf{R}}$
defined by $({\bf x},{{\bf
x}^{\prime}}):=x_{1}x_{1}^{\prime}+\ldots+x_{d}x_{d}^{\prime},$ induces a norm
(the Euclidean norm) $\|\cdot\|:{\mathbf{R}}^{d}\to[0,\infty)$, on the finite-
dimensional vector space ${\mathbf{R}}^{d}$, given by $\|{\bf
x}\|:=\sqrt{({\bf x},{\bf x})}.$ In the rest of this paper, we will use the
gamma function $\Gamma:{\mathbf{C}}\setminus-{\mathbf{N}}_{0}\to{\mathbf{C}}$,
which is a natural generalization of the factorial function (see for instance
Chapter 5 in Olver et al. (2010) [15]). A fundamental solution of the
polyharmonic equation is given by the following theorem.

######  Theorem 2.1.

Let $d,k\in{\mathbf{N}}$. Define

${\mathcal{G}}_{k}^{d}({\bf x},{\bf
x}^{\prime})=\left\\{\begin{array}[]{ll}{\displaystyle\frac{(-1)^{k+d/2+1}\
\|{\bf x}-{\bf x}^{\prime}\|^{2k-d}}{(k-1)!\ \left(k-d/2\right)!\
2^{2k-1}\pi^{d/2}}\left(\log\|{\bf x}-{\bf
x}^{\prime}\|-\beta_{k-d/2,d}\right)}\\\\[2.0pt] \hskip
210.55022pt\mathrm{if}\ d\,\,\mathrm{even},\ k\geq d/2,\\\\[10.0pt]
{\displaystyle\frac{\Gamma(d/2-k)\|{\bf x}-{\bf x}^{\prime}\|^{2k-d}}{(k-1)!\
2^{2k}\pi^{d/2}}}\hskip 95.88564pt\mathrm{otherwise},\end{array}\right.$

where $\beta_{p,d}\in{\mathbf{Q}}$ is defined as
$\beta_{p,d}:=\frac{1}{2}\left[H_{p}+H_{d/2+p-1}-H_{d/2-1}\right],$ with
$H_{j}$ being the $j$th harmonic number

$H_{j}:=\sum_{i=1}^{j}\frac{1}{i},$

then ${\mathcal{G}}_{k}^{d}$ is a normalized fundamental solution for
$(-\Delta)^{k}$ on Euclidean space ${\mathbf{R}}^{d}$.

Proof. See Cohl (2010) [3] and Boyling (1996) [2].

A separable rotationally-invariant coordinate system for the polyharmonic
equation (1) on ${\mathbf{R}}^{d}$ is given by

$\left.\begin{array}[]{rcl}x_{1}&=&R(\xi_{1},\ldots,\xi_{d-1})\cos\phi\\\\[2.84544pt]
x_{2}&=&R(\xi_{1},\ldots,\xi_{d-1})\sin\phi\\\\[2.84544pt]
x_{3}&=&x_{3}(\xi_{1},\ldots,\xi_{d-1})\\\\[2.84544pt] &\vdots&\\\\[2.84544pt]
x_{d}&=&x_{d}(\xi_{1},\ldots,\xi_{d-1})\end{array}\quad\right\\},$ (4)

which is described by an angle $\phi\in{\mathbf{R}}$ and $(d-1)$-curvilinear
coordinates $(\xi_{1},\ldots,\xi_{d-1})$. A separable rotationally-invariant
coordinate system transforms the polyharmonic equation into a set of
$d$-uncoupled ordinary differential equations with separation constants
$m\in{\mathbf{Z}}$ and $k_{j}$ for $1\leq j\leq d-2$. For a separable
rotationally-invariant coordinate system, this uncoupling is accomplished, in
general, by assuming a solution to (1) of the form

$\Phi(x)=e^{im\phi}\,{\mathcal{R}}(\xi_{1},\ldots,\xi_{d-1})\prod_{i=1}^{d-1}A_{i}(\xi_{i},m,k_{1},\ldots,k_{d-2}),$

where the domains of the functions ${\mathcal{R}}$ and $A_{i}$, for $1\leq
i\leq d-1$, and the constants $k_{j}$ for $1\leq j\leq d-1$, depend on the
specific rotationally-invariant coordinate system. A rotationally-invariant
coordinate system parametrizes points on the $(d-1)$-dimensional half-
hyperplane given by $\phi=const.$ and $R\geq 0$ using curvilinear coordinates
$(\xi_{1},\ldots,\xi_{d-1})$. (For a general description of the theory of
separation of variables see Miller (1977) [12].) The Euclidean distance
between two points ${\bf x},{{\bf x}^{\prime}}\in{\mathbf{R}}^{d}$, expressed
in a rotationally-invariant coordinate system, is given by

$\displaystyle\|{\bf x}-{{\bf
x}^{\prime}}\|=\sqrt{2RR^{\prime}}\left[\chi-\cos(\phi-\phi^{\prime})\right]^{1/2},$

where the toroidal parameter $\chi>1$, is given by

$\chi:=\frac{R+{R^{\prime}}^{2}+{\displaystyle\sum_{i=3}^{d}(x_{i}-x_{i}^{\prime})^{2}}}{\displaystyle
2RR^{\prime}},$ (5)

where $R,R^{\prime}\in[0,\infty)$ are defined in (4) for ${\bf x},{{\bf
x}^{\prime}}\in{\mathbf{R}}^{d}$. The hypersurfaces given by $\chi>1$ equals
constant are independent of coordinate system and represent hyper-tori of
revolution.

From Theorem 2.1 we see that, apart from multiplicative constants, the
algebraic expression
${\mathfrak{l}}_{k}^{d}:({\mathbf{R}}^{d}\times{\mathbf{R}}^{d})\setminus\\{({\bf
x},{\bf x}):{\bf x}\in{\mathbf{R}}^{d}\\}\to{\mathbf{R}}$ of an unnormalized
fundamental solution for the polyharmonic equation in Euclidean space
${\mathbf{R}}^{d}$ for $d$ even, $k\geq d/2,$ is given by

${\mathfrak{l}}_{k}^{d}({\bf x},{{\bf x}^{\prime}}):=\|{\bf x}-{{\bf
x}^{\prime}}\|^{2k-d}\left(\log\|{\bf x}-{{\bf
x}^{\prime}}\|-\beta_{k-d/2,d}\right).$ (6)

By expressing ${\mathfrak{l}}_{k}^{d}$ in a rotationally-invariant coordinate
system (4) we obtain

$\displaystyle{\mathfrak{l}}_{k}^{d}({\bf x},{{\bf x}^{\prime}})$
$\displaystyle=$
$\displaystyle\left(2RR^{\prime}\right)^{p}\left[\frac{1}{2}\log\left(2RR^{\prime}\right)-\beta_{p,d}\right]\left[\chi-\cos(\phi-\phi^{\prime})\right]^{p}$
(7)
$\displaystyle+\frac{1}{2}\left(2RR^{\prime}\right)^{p}\left[\chi-\cos(\phi-\phi^{\prime})\right]^{p}\log\left[\chi-\cos(\phi-\phi^{\prime})\right],$

where $p=k-d/2\in{\mathbf{N}}_{0}$. For the polyharmonic equation in even-
dimensional Euclidean space ${\mathbf{R}}^{d}$ with $1\leq k\leq d/2-1,$ apart
from multiplicative constants, the algebraic expression for an unnormalized
fundamental solution of the polyharmonic equation
${\mathfrak{h}}_{k}^{d}:({\mathbf{R}}^{d}\times{\mathbf{R}}^{d})\setminus\\{({\bf
x},{\bf x}):{\bf x}\in{\mathbf{R}}^{d}\\}\to{\mathbf{R}}$ is given by

${\mathfrak{h}}_{k}^{d}({\bf x},{{\bf x}^{\prime}}):=\|{\bf x}-{{\bf
x}^{\prime}}\|^{2k-d}.$

By expressing ${\mathfrak{h}}_{k}^{d}$ in a rotationally-invariant coordinate
system we obtain

${\mathfrak{h}}_{k}^{d}({\bf x},{{\bf
x}^{\prime}})=\left(2RR^{\prime}\right)^{-q}\left[\chi-\cos(\phi-\phi^{\prime})\right]^{-q},$
(8)

where $q=2k-d$.

By examining (7) and (8), we see that for computation of Fourier expansions
about the azimuthal separation angle $(\phi-\phi^{\prime})$ of
${\mathfrak{l}}_{k}^{d}$ and ${\mathfrak{h}}_{k}^{d}$, all that is required is
to compute the Fourier cosine series for the following three functions
$f_{\chi},h_{\chi}:{\mathbf{R}}\to(0,\infty)$ and
$g_{\chi}:{\mathbf{R}}\to{\mathbf{R}}$ defined as

$\displaystyle f_{\chi}(\psi):=\left(\chi-\cos\psi\right)^{p},$ $\displaystyle
g_{\chi}(\psi):=\left(\chi-\cos\psi\right)^{p}\log\left(\chi-\cos\psi\right),\qquad\mbox{and}$
$\displaystyle h_{\chi}(\psi):=\left(\chi-\cos\psi\right)^{-q},$

where $p\in{\mathbf{N}}_{0}$, $q\in{\mathbf{N}}$ and $\chi>1$ is a fixed
parameter.

The Fourier series of $f_{\chi}$ is given in Cohl & Dominici (2010) [4] (cf.
(4.4) therein), namely

$(z-\cos\psi)^{p}=(z^{2}-1)^{p/2}\sum_{n=0}^{p}\epsilon_{n}\cos(n\psi)\frac{(-p)_{n}(p-n)!}{(p+n)!}P_{p}^{n}\left(\frac{z}{\sqrt{z^{2}-1}}\right),$
(9)

where the Neumann factor $\epsilon_{n}=2-\delta_{n,0}$ commonly occurs in
Fourier series, $\delta_{n,0}$ is the Kronecker delta, and

$(z)_{n}:=\prod_{i=1}^{n}(z+i-1),$

for $z\in{\mathbf{C}}$ and $n\in{\mathbf{N}}_{0}$, is the Pochhammer symbol
(rising factorial). We have used Whipple’s formula in (9) (see for instance,
(8.2.7) in Abramowitz & Stegun (1972) [1]) to convert the associated Legendre
function of the second kind $Q_{\nu}^{\mu}:(1,\infty)\to{\mathbf{C}}$
appearing in [4] to the associated Legendre function of the first kind
$P_{\nu}^{\mu}:(1,\infty)\to{\mathbf{R}}$. The associated Legendre function of
the first kind can be defined using the Gauss hypergeometric function, namely
(Magnus, Oberhettinger & Soni (1966) [11], p. 153)

$P_{\nu}^{\mu}(z):=\frac{1}{\Gamma(1-\mu)}\left(\frac{z+1}{z-1}\right)^{\mu/2}{}_{2}F_{1}\left(-\nu,\nu+1;1-\mu;\frac{1-z}{2}\right).$

The Gauss hypergeometric function
${}_{2}F_{1}:{\mathbf{C}}\times{\mathbf{C}}\times({\mathbf{C}}\setminus-{\mathbf{N}}_{0})\times\\{z\in{\mathbf{C}}:|z|<1\\}\to{\mathbf{C}}$
can be defined in terms of the following infinite series

${}_{2}F_{1}(a,b;c;z)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{n!(c)_{n}}z^{n}$

(see for instance Chapter 15 in Olver et al. (2010) [15]).

The Fourier series of $h_{\chi}$ is given in Cohl & Dominici (2010) [4]
(Whipple formula (8.2.7) in Abramowitz & Stegun (1972) [1] and cf. (4.5)
therein), namely

$\frac{1}{(z-\cos\psi)^{q}}=\frac{(z^{2}-1)^{-q/2}}{(q-1)!}\sum_{n=0}^{\infty}\epsilon_{n}\cos(n\psi)(n+q-1)!P_{q-1}^{-n}\left(\frac{z}{\sqrt{z^{2}-1}}\right),$
(10)

where $q\in{\mathbf{N}}$. Since the Fourier series of $h_{\chi}$ is computed
in Cohl & Dominici (2010) [4], we understand how to compute Fourier expansions
of ${\mathfrak{h}}_{k}^{d}$ (8) in separable rotationally-invariant coordinate
systems. In order to compute Fourier expansion of ${\mathfrak{l}}_{k}^{d}$ (7)
in separable rotationally-invariant coordinate systems, all that remains is to
determine the Fourier series of $g_{\chi}$. This is the goal of the next two
sections.

##  3 Algebraic approach to the logarithmic Fourier series

Since $\chi>1$, one may make the substitution $\chi=\cosh\eta$ to evaluate the
Fourier series of $g_{\chi}$. For instance, it is given in the form of
$(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi),$ where
$p\in{\mathbf{N}}_{0}$. For $p=0$ the result is well-known (see for instance
Magnus, Oberhettinger & Soni (1966) [11], p. 259)

$\log(\cosh\eta-\cos\psi)=\eta-\log{2}-2\sum_{n=1}^{\infty}\frac{e^{-n\eta}}{n}\cos(n\psi),$
(11)

which as we will see, should be compared with (10) for $q=1$, namely

$\displaystyle\frac{1}{\cosh\eta-\cos\psi}=\frac{1}{\sinh\eta}\sum_{n=0}^{\infty}\epsilon_{n}\cos(n\psi)e^{-n\eta}.$
(12)

Note that for $\eta>0$ we may write $e^{\eta}$ and therefore $\eta$ as a
function of $\cosh\eta$ since $\sinh\eta=\sqrt{\cosh^{2}\eta-1},$
$e^{\eta}=\cosh\eta+\sqrt{\cosh^{2}\eta-1},$ and therefore
$\eta=\log\left(\cosh\eta+\sqrt{\cosh^{2}\eta-1}\right).$ Now examine the
$p=1$ case for $g_{\chi}$. If we multiply both sides of (11) by
$(\cosh\eta-\cos\psi)$ and take advantage of the formula

$\cos(n\psi)\cos\psi=\frac{1}{2}\Bigl{\\{}\cos[(n+1)\psi]+\cos[(n-1)\psi]\Bigr{\\}},$
(13)

then we have

$\displaystyle(\cosh\eta-\cos\psi)\log(\cosh\eta-\cos\psi)$ $\displaystyle=$
$\displaystyle(\eta-\log 2)\cosh\eta$ (14) $\displaystyle{}-(\eta-\log
2)\cos\psi-2\cosh\eta\sum_{n=1}^{\infty}\frac{e^{-n\eta}}{n}\cos(n\psi)$
$\displaystyle{}+\sum_{n=1}^{\infty}\frac{e^{-n\eta}}{n}\cos[(n+1)\psi]+\sum_{n=1}^{\infty}\frac{e^{-n\eta}}{n}\cos[(n-1)\psi].$

Collecting the contributions to the Fourier cosine series, we obtain

$\displaystyle(\cosh\eta-\cos\psi)\log(\cosh\eta-\cos\psi)$ $\displaystyle=$
$\displaystyle(1+\eta-\log 2)\cosh\eta$ (15)
$\displaystyle{}-\sinh\eta+\cos\psi\left(\log
2-1-\eta-\frac{1}{2}e^{-2\eta}\right)$
$\displaystyle{}+2\sum_{n=2}^{\infty}\frac{e^{-n\eta}\cos
n\psi}{n(n^{2}-1)}(\cosh\eta+n\sinh\eta).$

If we compare (15) with (10) for $q=1$, namely

$\displaystyle\frac{1}{(\cosh\eta-\cos\psi)^{2}}=\frac{1}{\sinh^{3}\eta}\sum_{n=0}^{\infty}\epsilon_{n}\cos(n\psi)e^{-n\eta}(\cosh\eta+n\sinh\eta),$
(16)

we notice that the factor $(\cosh\eta+n\sinh\eta)$ appears in both series.

For $p=2$ in $g_{\chi}$, we use (13) and similarly have

$\displaystyle(\cosh\eta-\cos\psi)^{2}\log(\cosh\eta-\cos\psi)$
$\displaystyle{}=(\eta-\log 2)\cosh^{2}\eta-2(\eta-\log 2)\cosh\eta\cos\psi$
$\displaystyle{}+(\eta-\log
2)\cos^{2}\psi-(2\cosh^{2}\eta+1)\sum_{n=1}^{\infty}\frac{e^{-n\eta}}{n}\cos(n\psi)$
$\displaystyle{}+2\cosh\eta\sum_{n=1}^{\infty}\frac{e^{-n\eta}}{n}\cos[(n+1)\psi]+2\cosh\eta\sum_{n=1}^{\infty}\frac{e^{-n\eta}}{n}\cos[(n-1)\psi]{}$
$\displaystyle{}-\frac{1}{2}\sum_{n=1}^{\infty}\frac{e^{-n\eta}}{n}\cos[(n+2)\psi]-\frac{1}{2}\sum_{n=1}^{\infty}\frac{e^{-n\eta}}{n}\cos[(n-2)\psi].$

If we collect the contributions of the Fourier cosine series, we obtain

$\displaystyle(\cosh\eta-\cos\psi)^{2}\log(\cosh\eta-\cos\psi)$
$\displaystyle=$ $\displaystyle(\eta-\log
2)\left(\cosh^{2}\eta+\frac{1}{2}\right)$ (17)
$\displaystyle{}+2\cosh\eta\,e^{-\eta}-\frac{1}{4}e^{-2\eta}+\Biggl{[}-2(\eta-\log
2)\cosh\eta-\left(2\cosh^{2}\eta+\frac{3}{2}\right)e^{-\eta}$
$\displaystyle{}+\cosh\eta\,e^{-2\eta}-\frac{1}{6}e^{-3\eta}\Biggr{]}\cos\psi+\Biggl{[}\frac{1}{2}(\eta-\log
2)+2\cosh\eta\,e^{-\eta}$
$\displaystyle{}-\frac{1}{2}(2\cosh^{2}\eta+1)e^{-2\eta}+\frac{2}{3}\cosh\eta\,e^{-3\eta}-\frac{1}{8}e^{-4\eta}\Biggr{]}\cos
2\psi$
$\displaystyle{}-4\sum_{n=3}^{\infty}\frac{e^{-n\eta}\cos(n\psi)}{n(n^{2}-1)(n^{2}-4)}\left[(n^{2}-1)\sinh^{2}\eta+3n\sinh\eta\cosh\eta+3\cosh^{2}\eta\right].$

By comparing (17) with (10) for $q=2$, namely

$\displaystyle\displaystyle\frac{1}{(\cosh\eta-\cos\psi)^{3}}$
$\displaystyle=$ $\displaystyle\frac{1}{2\sinh^{5}\eta}$ (18)
$\displaystyle{}\times\sum_{n=0}^{\infty}\epsilon_{n}\cos(n\psi)e^{-n\eta}\left[(n^{2}-1)\sinh^{2}\eta+3n\sinh\eta\cosh\eta+3\cosh^{2}\eta\right],$

then we notice that the factor
$\left((n^{2}-1)\sinh^{2}\eta+3n\sinh\eta\cosh\eta+3\cosh^{2}\eta\right)$
appears in both series. We will demonstrate in §5, why the identification
mentioned in (16) and (18) occurs.

This algebraic approach for determining the Fourier series of $g_{\chi}$ will
now be generalized. By starting with (11) and repeatedly multiplying by
factors of $(\cosh\eta-\cos\psi)$, we see that the general Fourier series of
$g_{\chi}$ can be given in terms of a sequence of polynomials
$R_{p}^{k}:(1,\infty)\to{\mathbf{R}}$, with $p\in{\mathbf{N}}_{0}$ and
$k\in{\mathbf{Z}}$, as

$\displaystyle(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi)$
$\displaystyle=$ $\displaystyle(\eta-\log 2)(\cosh\eta-\cos\psi)^{p}$ (19)
$\displaystyle{}+2\sum_{k=-p}^{p}(-1)^{k+1}R_{p}^{k}(\cosh\eta)\sum_{n=1}^{\infty}\frac{e^{-n\eta}}{n}\cos[(n+k)\psi].$

We will refer to $R_{p}^{k}$ as logarithmic polynomials with argument
$x=\cosh\eta$ (in our notation $p$ and $k$ are both indices) (See Appendix 8
for a description of some of the properties of the logarithmic polynomials).

The double sum in (19) is simplified by making the replacement $n+k\mapsto n$.
It then follows that the resulting double sum naturally breaks into two
disjoint regions, one triangular

${\mathcal{A}}:=\\{(k,n):-p\leq k\leq p-1,\ k+1\leq n\leq p\\},$

with $p(2p+1)$ terms and the other infinite rectangular

${\mathcal{B}}:=\\{(k,n):-p\leq k\leq p,\ p+1<n<\infty\\}.$

By rearranging the order of the $k$ and $n$ summations in (19), we derive

$\displaystyle\displaystyle(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi)$
$\displaystyle=$ $\displaystyle(\eta-\log 2)(\cosh\eta-\cos\psi)^{p}$ (20)
$\displaystyle\displaystyle+\sum_{n=0}^{p}\cos(n\psi)e^{-n\eta}\,\mathfrak{r}_{n,p}^{-p,n-1}(\cosh\eta)+\sum_{n=1}^{p-1}\cos(n\psi)e^{n\eta}\,\mathfrak{r}_{-n,p}^{-p,-n-1}(\cosh\eta)$
$\displaystyle\displaystyle+\sum_{n=p+1}^{\infty}\frac{\cos(n\psi)e^{-n\eta}}{n(n^{2}-1)\cdots(n^{2}-p^{2})}\,\Re_{n,p}(\cosh\eta),$

where $\mathfrak{r}_{n,p}^{k_{1},k_{2}},\Re_{n,p}:(1,\infty)\to{\mathbf{R}}$
are defined as

$\mathfrak{r}_{n,p}^{k_{1},k_{2}}(\cosh\eta):=2\sum_{k=k_{1}}^{k_{2}}\frac{(-1)^{k+1}e^{k\eta}R_{p}^{k}(\cosh\eta)}{n-k},$

and

$\Re_{n,p}(\cosh\eta):=\frac{(n+p)!}{(n-p-1)!}\mathfrak{r}_{n,p}^{-p,p}(\cosh\eta),\quad(n\geq
p+1),$

respectively. We can also write the Fourier series directly in terms of the
logarithmic polynomials $R_{p}^{k}$ as follows

$\displaystyle(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi)$
$\displaystyle=$ $\displaystyle(\eta-\log 2)(\cosh\eta-\cos\psi)^{p}$ (21)
$\displaystyle+2\sum_{n=0}^{p}\cos(n\psi)e^{-n\eta}\sum_{k=-p}^{n-1}\frac{(-1)^{k+1}e^{k\eta}R_{p}^{k}(\cosh\eta)}{n-k}$
$\displaystyle-2\sum_{n=1}^{p-1}\cos(n\psi)e^{n\eta}\sum_{k=-p}^{-n-1}\frac{(-1)^{k+1}e^{k\eta}R_{p}^{k}(\cosh\eta)}{n+k}$
$\displaystyle+2\sum_{n=p+1}^{\infty}\cos(n\psi)e^{-n\eta}\sum_{k=-p}^{p}\frac{(-1)^{k+1}e^{k\eta}R_{p}^{k}(\cosh\eta)}{n-k}.$

Note that by using (9), then we can express $(\cosh\eta-\cos\psi)^{p}$ as a
Fourier series, namely

$(\cosh\eta-\cos\psi)^{p}=\sinh^{p}\eta\sum_{n=0}^{p}\epsilon_{n}\cos(n\psi)\frac{(-p)_{n}(p-n)!}{(p+n)!}P_{p}^{n}(\coth\eta),$
(22)

where $p\in{\mathbf{N}}_{0}$.

If we define the function $\mathfrak{P}_{n,p}:(1,\infty)\to{\mathbf{R}}$ such
that

$\displaystyle\mathfrak{P}_{n,p}(\cosh\eta):=\left\\{\begin{array}[]{ll}\displaystyle\mathfrak{r}_{0,p}^{-p,-1}(\cosh\eta)\hskip
130.25679pt\mathrm{if}\ n=0,\\\\[5.69046pt] \displaystyle
D_{n,p}(\cosh\eta)+E_{n,p}(\cosh\eta)\hskip 62.3116pt\mathrm{if}\ 1\leq n\leq
p-1,\\\\[5.69046pt] \displaystyle
e^{-p\eta}\mathfrak{r}_{p,p}^{-p,p-1}(\cosh\eta)\hskip 104.70593pt\mathrm{if}\
1\leq n=p,\\\\[5.69046pt]
\displaystyle\frac{e^{-n\eta}}{(n^{2}-p^{2})\cdots(n^{2}-1)n}\,\Re_{n,p}(\cosh\eta)\hskip
21.33955pt\mathrm{if}\ n\geq p+1,\end{array}\right.$ (24)

where $D_{n,p},E_{n,p}:(1,\infty)\to{\mathbf{R}}$ are defined as

$D_{n,p}(\cosh\eta)=\left\\{\begin{array}[]{ll}\displaystyle
e^{n\eta}\mathfrak{r}_{-n,p}^{-p,-n-1}(\cosh\eta)&\qquad\mathrm{if}\ p\geq
2,\\\\[5.69046pt] \displaystyle 0&\qquad\mathrm{if}\
p\in\\{0,1\\},\end{array}\right.$

and

$E_{n,p}(\cosh\eta)=\left\\{\begin{array}[]{ll}\displaystyle
e^{-n\eta}\mathfrak{r}_{n,p}^{-p,n-1}(\cosh\eta)&\qquad\mathrm{if}\ p\geq
1,\\\\[5.69046pt] \displaystyle 0&\qquad\mathrm{if}\ p=0,\end{array}\right.$

respectively, then we can write (20) as

$\displaystyle\displaystyle(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi)$
$\displaystyle=$ $\displaystyle(\eta-\log 2)(\cosh\eta-\cos\psi)^{p}$
$\displaystyle+\sum_{n=0}^{\infty}\cos(n\psi)\mathfrak{P}_{n,p}(\cosh\eta).$

In fact, if we use (22), then we can express $g_{\chi}$ as

$\displaystyle\hskip
5.69046pt\displaystyle(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi)$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\cos(n\psi)\mathfrak{P}_{n,p}(\cosh\eta)$
$\displaystyle{}+(\eta-\log
2)\sinh^{p}\eta\sum_{n=0}^{p}\epsilon_{n}\cos(n\psi)\frac{(-p)_{n}(p-n)!}{(p+n)!}P_{p}^{n}(\coth\eta).$

Furthermore, if we define $\mathfrak{Q}_{n,p}:(1,\infty)\to{\mathbf{R}}$ as

$\mathfrak{Q}_{n,p}(\cosh\eta):=\mathfrak{P}_{n,p}(\cosh\eta)+\frac{\epsilon_{n}(-p)_{n}(p-n)!}{(p+n)!}(\eta-\log
2)\sinh^{p}\eta P_{p}^{n}(\coth\eta),$

then we can express $g_{\chi}$ as

$\displaystyle(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi)=\sum_{n=0}^{\infty}\cos(n\psi)\mathfrak{Q}_{n,p}(\cosh\eta).$
(25)

##  4 Limit derivative approach to the logarithmic Fourier series

We now use a second approach to compute the Fourier series for a logarithmic
fundamental solution of the polyharmonic equation (6). We would like to match
our results to the computations in §3, which clearly demonstrate different
behaviors for the two regimes, $0\leq n\leq p$ and $n\geq p+1$. By applying
the identity

$(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi)=\lim_{\nu\rightarrow
0}\frac{\partial}{\partial\nu}(\cosh\eta-\cos\psi)^{\nu+p},$ (26)

where $p\in{\mathbf{N}}_{0}$, to

$\displaystyle(\cosh\eta-\cos\psi)^{\nu}$ $\displaystyle=$
$\displaystyle\sinh^{\nu}\eta\sum_{n=0}^{\infty}\frac{(-1)^{n}\epsilon_{n}\cos(n\psi)}{(\nu+1)_{n}}P_{\nu}^{n}(\coth\eta),$
(27)

where $\nu\in{\mathbf{C}}\setminus-{\mathbf{N}}$ (cf. (3.11b) in Cohl &
Dominici (2010) [4]), one can compute the Fourier cosine series of
$(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi)$, provided availability of
the necessary parameter derivatives.

Applying (26) to (27), we obtain

$\displaystyle(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi)$
$\displaystyle=\left[\lim_{\nu\to
0}\frac{\partial}{\partial\nu}\sinh^{\nu+p}\eta\right]\sum_{n=0}^{\infty}\frac{(-1)^{n}\epsilon_{n}\cos(n\psi)}{(p+1)_{n}}P_{p}^{n}(\coth\eta)$
$\displaystyle+\sinh^{p}\eta\sum_{n=0}^{\infty}(-1)^{n}\epsilon_{n}\cos(n\psi)\left[\lim_{\nu\to
0}\frac{\partial}{\partial\nu}\frac{1}{(\nu+p+1)_{n}}\right]P_{p}^{n}(\coth\eta)$
$\displaystyle+\sinh^{p}\eta\sum_{n=0}^{\infty}\frac{(-1)^{n}\epsilon_{n}\cos(n\psi)}{(p+1)_{n}}\left[\lim_{\nu\to
0}\frac{\partial}{\partial\nu}P_{\nu+p}^{n}(\coth\eta)\right].$

Note that for $p\in{\mathbf{Z}}$ and $n\in{\mathbf{N}}_{0}$, the associated
Legendre function of the first kind $P_{p}^{n}$ vanishes if $n\geq p+1$. This
is easily seen using the Rodrigues-type formula (cf. (14.7.11) in Olver et al.
(2010) [15])

$P_{p}^{n}(z)=(z^{2}-1)^{n/2}\frac{d^{n}P_{p}(z)}{dz^{n}},$

and the fact that $P_{p}(z)$ is a polynomial in $z$ of degree $p$. The
derivatives are given as follows:

$\lim_{\nu\to
0}\frac{\partial}{\partial\nu}\sinh^{\nu+p}\eta=\sinh^{p}\eta\log\sinh\eta,$
(28) $\lim_{\nu\to
0}\frac{\partial}{\partial\nu}\frac{1}{(\nu+p+1)_{n}}=\frac{p!}{(p+n)!}\left[\psi(p+1)-\psi(p+1+n)\right],$
(29)

and

$\lim_{\nu\to
0}\frac{\partial}{\partial\nu}P_{\nu+p}^{n}(\coth\eta)=\left[\frac{\partial}{\partial\nu}P_{\nu}^{n}(\coth\eta)\right]_{\nu=p},$
(30)

where $\psi:{\mathbf{C}}\setminus-{\mathbf{N}}_{0}\to{\mathbf{C}}$ is the
digamma function defined in terms of the derivative of the gamma function

$\frac{d}{dz}\Gamma(z)=:\psi(z)\Gamma(z)$

(see for instance (5.2.2) in Olver et al. (2010) [15]). The degree-derivative
of the associated Legendre function of the first kind in (30) is determined
using (34) and (35). By collecting terms and using (28), (29), and (30), we
obtain

$\displaystyle(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi)$
$\displaystyle=$ $\displaystyle(\eta-\log 2)(\cosh\eta-\cos\psi)^{p}$ (31)
$\displaystyle+p!\sinh^{p}\eta\sum_{n=0}^{p}\frac{(-1)^{n}\epsilon_{n}\cos(n\psi)}{(p+n)!}$
$\displaystyle\times\left[2\psi(2p+1)-\psi(p+1+n)-\psi(p+1-n)\right]P_{p}^{n}(\coth\eta)$
$\displaystyle+(-1)^{p}p!\sinh^{p}\eta\sum_{n=0}^{p-1}\frac{\epsilon_{n}\cos(n\psi)}{(p+n)!}$
$\displaystyle\times\sum_{k=0}^{p-n-1}\frac{(-1)^{k}(2n+2k+1)\left[1+\frac{k!(p+n)!}{(2n+k)!(p-n)!}\right]}{(p-n-k)(p+n+k+1)}P_{n+k}^{n}(\coth\eta)$
$\displaystyle+2(-1)^{p}p!\sinh^{p}\eta\sum_{n=1}^{p}\frac{(-1)^{n}\cos(n\psi)}{(p-n)!}$
$\displaystyle\times\sum_{k=0}^{n-1}\frac{(-1)^{k}(2k+1)}{(p-k)(p+k+1)}P_{k}^{-n}(\coth\eta)$
$\displaystyle+2(-1)^{p+1}p!\sinh^{p}\eta\sum_{n=p+1}^{\infty}\cos(n\psi)(n-p-1)!P_{p}^{-n}(\coth\eta).$

##  5 Comparison of the two approaches

The limit derivative approach presented in §4 might be considered, of the two
methods, preferred for computing the azimuthal Fourier series for a
logarithmic fundamental solution of the polyharmonic equation. This is because
it produces azimuthal Fourier coefficients in terms of the well-known special
functions, associated Legendre functions. On the other hand, the algebraic
approach presented in §3 produces results in terms of the two-parameter
logarithmic polynomials $R_{p}^{k}(x)$. As far as the author is aware, these
polynomials are previously unencountered in the literature. By comparison of
the two approaches we see how the logarithmic polynomials $R_{p}^{k}(x)$
(potentially a new type of special function) are intimately related to the
associated Legendre functions. In this section we make this comparison
concrete. We should also mention that the following comparison equations
resolve to become quite complicated as $p$ increases, and they have been
checked for $0\leq p\leq 10$ using Mathematica with the assistance of an
algorithm generated using (37) and (38) from Appendix 8.

By equating the Fourier coefficients using the two approaches we can obtain
summation formulae which are satisfied by the logarithmic polynomials. For
$n=0$ and $p\geq 1$ we have

$\displaystyle\sum_{k=1}^{p}\frac{(-1)^{k+1}e^{-k\eta}R_{p}^{k}(\cosh\eta)}{k}$
$\displaystyle{}=\sinh^{p}\eta\left[\psi(2p+1)-\psi(p+1)\right]P_{p}(\coth\eta)$
$\displaystyle{}+(-1)^{p}\sinh^{p}\eta\sum_{k=0}^{p-1}\frac{(-1)^{k}(2k+1)}{(p-k)(p+k+1)}P_{k}(\coth\eta),$

for $1\leq n\leq p-1$, $p\geq 2$ we derive

$\displaystyle\sum_{k=-p}^{n-1}\frac{(-1)^{k+1}e^{k\eta}R_{p}^{k}(\cosh\eta)}{n-k}+e^{2n\eta}\sum_{k=-p}^{-n-1}\frac{(-1)^{k}e^{k\eta}R_{p}^{k}(\cosh\eta)}{n+k}$
$\displaystyle=$ $\displaystyle p!e^{n\eta}\sinh^{p}\eta$
$\displaystyle{}\times\Biggl{\\{}\frac{(-1)^{n}}{(p+n)!}\left[2\psi(2p+1)-\psi(p+1+n)-\psi(p+1-n)\right]P_{p}^{n}(\coth\eta)$
$\displaystyle{}+\frac{(-1)^{p}}{(p+n)!}\sum_{k=0}^{p-n-1}\frac{(-1)^{k}(2n+2k+1)\left[1+\frac{k!(p+n)!}{(2n+k)!(p-n)!}\right]}{(p-n-k)(p+n+k+1)}P_{n+k}^{n}(\coth\eta)$
$\displaystyle{}+\frac{(-1)^{p+n}}{(p-n)!}\sum_{k=0}^{n-1}\frac{(-1)^{k}(2k+1)}{(p-k)(p+k+1)}P_{k}^{-n}(\coth\eta)\Biggr{\\}},$

for $n=p$, $p\geq 1$ we obtain

$\displaystyle\sum_{k=-p}^{p-1}\frac{(-1)^{k+1}e^{k\eta}R_{p}^{k}(\cosh\eta)}{p-k}$
$\displaystyle=$ $\displaystyle p!e^{p\eta}\sinh^{p}\eta$
$\displaystyle{}\times\Biggl{\\{}\frac{(-1)^{p}}{(2p)!}\left[\psi(2p+1)-\psi(1)\right]P_{p}^{p}(\coth\eta)$
$\displaystyle{}+\sum_{k=0}^{p-1}\frac{(-1)^{k}(2k+1)}{(p-k)(p+k+1)}P_{k}^{-p}(\coth\eta)\Biggr{\\}},$

and for $n\geq p+1$, $p\geq 0,$ we see that

$\displaystyle\sum_{k=-p}^{p}\frac{(-1)^{k+1}e^{k\eta}R_{p}^{k}(\cosh\eta)}{n-k}$
(32) $\displaystyle=(-1)^{p+1}p!(n-p-1)!e^{n\eta}\sinh^{p}\eta
P_{p}^{-n}(\coth\eta).$

We now have closed-form expressions for the finite terms given by (20), (25)
and (20) in terms of associated Legendre functions of the first kind. We also
have a proof of the correspondence for the “ending” function $\Re_{n,q-1}$
mentioned in §3. Through (32), the function $\Re_{n,p}$ (cf. (15), (16), (17),
(18), and (20)) is directly related to the associated Legendre function of the
first kind, namely

$\Re_{n,p}(\cosh\eta)=2(-1)^{p+1}p!(p+n)!e^{n\eta}\sinh^{p}\eta
P_{p}^{-n}(\coth\eta).$

Therefore through (10) we have

$\frac{1}{(\cosh\eta-\cos\psi)^{q}}=\frac{(-1)^{q}}{2[(q-1)!]^{2}\sinh^{2q-1}\eta}\sum_{n=0}^{\infty}\epsilon_{n}\cos(n\psi)e^{-n\eta}\Re_{n,q-1}(\cosh\eta),$

which demonstrates the correspondences which was mentioned near (12), (16),
and (18) for (20) and (31).

##  6 Fourier expansion for a logarithmic fundamental solution of the
polyharmonic equation

Now that we have computed the Fourier series for $g_{\chi}$, namely (20) (cf.
(25)) and (31), we are in a position to compute the azimuthal Fourier series
for a logarithmic fundamental solution of the polyharmonic equation (6).

For instance, using (7), (22) and (31), we have

$\displaystyle\hskip 2.84544pt{\mathfrak{l}}_{k}^{d}({\bf x},{{\bf
x}^{\prime}})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(2RR^{\prime}\right)^{p}\left[\log\left(RR^{\prime}\right)+\eta-\beta_{p,d}\right](\chi^{2}-1)^{p/2}$
$\displaystyle\times\sum_{n=0}^{p}\epsilon_{n}\cos[n(\phi-\phi^{\prime})]\frac{(-p)_{n}(p-n)!}{(p+n)!}P_{p}^{n}\Biggl{(}\frac{\chi}{\sqrt{\chi^{2}-1}}\Biggr{)}$
$\displaystyle+\frac{1}{2}(2RR^{\prime})^{p}p!(\chi^{2}-1)^{p/2}\sum_{n=0}^{p}\frac{(-1)^{n}\epsilon_{n}\cos[n(\phi-\phi^{\prime})]}{(p+n)!}$
$\displaystyle\times\left[2\psi(2p+1)-\psi(p+1+n)-\psi(p+1-n)\right]P_{p}^{n}\Biggl{(}\frac{\chi}{\sqrt{\chi^{2}-1}}\Biggr{)}$
$\displaystyle+\frac{1}{2}(2RR^{\prime})^{p}p!(\chi^{2}-1)^{p/2}\sum_{n=0}^{p-1}\frac{\epsilon_{n}\cos[n(\phi-\phi^{\prime})]}{(p+n)!}$
$\displaystyle\times\sum_{k=0}^{p-n-1}\frac{(-1)^{k}(2n+2k+1)\left[1+\frac{k!(p+n)!}{(2n+k)!(p-n)!}\right]}{(p-n-k)(p+n+k+1)}P_{n+k}^{n}\Biggl{(}\frac{\chi}{\sqrt{\chi^{2}-1}}\Biggr{)}$
$\displaystyle+(2RR^{\prime})^{p}(-1)^{p}p!(\chi^{2}-1)^{p/2}\sum_{n=1}^{p}\frac{(-1)^{n}\cos[n(\phi-\phi^{\prime})]}{(p-n)!}$
$\displaystyle\times\sum_{k=0}^{n-1}\frac{(-1)^{k}(2k+1)}{(p-k)(p+k+1)}P_{k}^{-n}\Biggl{(}\frac{\chi}{\sqrt{\chi^{2}-1}}\Biggr{)}$
$\displaystyle+(2RR^{\prime})^{p}(-1)^{p+1}p!(\chi^{2}-1)^{p/2}$
$\displaystyle\times\sum_{n=p+1}^{\infty}\cos[n(\phi-\phi^{\prime})](n-p-1)!P_{p}^{-n}\Biggl{(}\frac{\chi}{\sqrt{\chi^{2}-1}}\Biggr{)}.$

Alternatively, using (7), (9), and (21) we obtain

$\displaystyle\hskip 14.22636pt{\mathfrak{l}}_{k}^{d}({\bf x},{{\bf
x}^{\prime}})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(2RR^{\prime}\right)^{p}\left[\log\left(RR^{\prime}\right)+\eta-\beta_{p,d}\right](\chi^{2}-1)^{p/2}$
$\displaystyle\hskip
56.9055pt\times\sum_{n=0}^{p}\epsilon_{n}\cos[n(\phi-\phi^{\prime}]\frac{(-p)_{n}(p-n)!}{(p+n)!}P_{p}^{n}\left(\frac{\chi}{\sqrt{\chi^{2}-1}}\right),$
$\displaystyle+(2RR^{\prime})^{p}\sum_{n=0}^{p}\cos[n(\phi-\phi^{\prime})]\left(\chi-\sqrt{\chi^{2}-1}\right)^{n}$
$\displaystyle\hskip
56.9055pt\times\sum_{k=-p}^{n-1}\frac{(-1)^{k+1}\left(\chi+\sqrt{\chi^{2}-1}\right)^{k}R_{p}^{k}(\chi)}{n-k}$
$\displaystyle-(2RR^{\prime})^{p}\sum_{n=1}^{p-1}\cos[n(\phi-\phi^{\prime})]\left(\chi+\sqrt{\chi^{2}-1}\right)^{n}$
$\displaystyle\hskip
56.9055pt\times\sum_{k=-p}^{-n-1}\frac{(-1)^{k+1}\left(\chi+\sqrt{\chi^{2}-1}\right)^{k}R_{p}^{k}(\chi)}{n+k}$
$\displaystyle+(2RR^{\prime})^{p}\sum_{n=p+1}^{\infty}\cos[n(\phi-\phi^{\prime})]\left(\chi-\sqrt{\chi^{2}-1}\right)^{n}$
$\displaystyle\hskip
56.9055pt\times\sum_{k=-p}^{p}\frac{(-1)^{k+1}\left(\chi+\sqrt{\chi^{2}-1}\right)^{k}R_{p}^{k}(\chi)}{n-k}.$

Using these formulae, we can, for instance, obtain the axisymmetric component
of a logarithmic fundamental solution of the polyharmonic equation, namely

$\displaystyle\left.{\mathfrak{l}}_{k}^{d}({\bf x},{{\bf
x}^{\prime}})\right|_{n=0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(2RR^{\prime})^{p}(\chi^{2}-1)^{p/2}$
$\displaystyle{}\times\left[\log(RR^{\prime})+\eta-\beta_{p,d}+2\psi(2p+1)-2\psi(p+1)\right]P_{p}\Biggl{(}\frac{\chi}{\sqrt{\chi^{2}-1}}\Biggr{)}$
$\displaystyle{}+(2RR^{\prime})^{p}(\chi^{2}-1)^{p/2}\sum_{k=0}^{p-1}\frac{(-1)^{k}(2k+1)}{(p-k)(p+k+1)}P_{k}\Biggl{(}\frac{\chi}{\sqrt{\chi^{2}-1}}\Biggr{)},$

or

$\displaystyle\left.{\mathfrak{l}}_{k}^{d}({\bf x},{{\bf
x}^{\prime}})\right|_{n=0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(2RR^{\prime})^{p}(\chi^{2}-1)^{p/2}\left[\log(RR^{\prime})+\eta-\beta_{p,d}\right]P_{p}\Biggl{(}\frac{\chi}{\sqrt{\chi^{2}-1}}\Biggr{)}$
$\displaystyle{}\hskip
51.21504pt+(2RR^{\prime})^{p}\sum_{k=1}^{p}\frac{(-1)^{k+1}(\chi-\sqrt{\chi^{2}-1})^{k}R_{p}^{k}(\chi)}{k}.$

The above expressions for the axisymmetric component of a fundamental solution
of the polyharmonic equation is one type of expression sought after in Tsai,
Chen & Hsu (2009) [19].

##  7 Derivatives with respect to the degree of certain integer-order
associated Legendre functions of the first kind

The derivative with respect to its degree for the associated Legendre function
of the first kind evaluated at the zero degree is given in §4.4.3 of Magnus,
Oberhettinger & Soni (1966) [11] as

$\left[\frac{\partial}{\partial\nu}P_{\nu}(z)\right]_{\nu=0}=\frac{z-1}{2}{}_{2}F_{1}\left(1,1;2;\frac{1-z}{2}\right).$
(33)

An important generalization of this formula has recently been derived in
Szmytkowski (2011) [18]. The degree-derivative of the associated Legendre
function of the first kind for $p,m\in{\mathbf{N}}_{0}$ and $0\leq m\leq p$
(cf. (5.12) in Szmytkowski (2011) [18]) is given by

$\displaystyle\left[\frac{\partial}{\partial\nu}P_{\nu}^{m}(z)\right]_{\nu=p}$
$\displaystyle=$ $\displaystyle P_{p}^{m}(z)\log\frac{z+1}{2}$ (34)
$\displaystyle{}+\left[2\psi(2p+1)-\psi(p+1)-\psi(p-m+1)\right]P_{p}^{m}(z)$
$\displaystyle{}+(-1)^{p+m}\sum_{k=0}^{p-m-1}(-1)^{k}\frac{(2k+2m+1)\left[1+\frac{k!(p+m)!}{(k+2m)!(p-m)!}\right]}{(p-m-k)(p+m+k+1)}P_{k+m}^{m}(z)$
$\displaystyle{}+(-1)^{p}\frac{(p+m)!}{(p-m)!}\sum_{k=0}^{m-1}(-1)^{k}\frac{2k+1}{(p-k)(p+k+1)}P_{k}^{-m}(z),$

and for $m\geq p+1$ (cf. (5.16) in Szmytkowski (2011) [18]) there is

$\left[\frac{\partial}{\partial\nu}P_{\nu}^{m}(z)\right]_{\nu=p}=(-1)^{p+n+1}(p+n)!(n-p-1)!P_{p}^{-n}(z).$
(35)

Some special cases of (34) include for $m=0$

$\displaystyle\left[\frac{\partial}{\partial\nu}P_{\nu}(z)\right]_{\nu=p}$
$\displaystyle=$ $\displaystyle
P_{p}(z)\log\frac{z+1}{2}+2\left[\psi(2p+1)-\psi(p+1)\right]P_{p}(z)$
$\displaystyle{}+2(-1)^{p}\sum_{k=0}^{p-1}(-1)^{k}\frac{2k+1}{(p-k)(p+k+1)}P_{k}(z),$

for $m=p$

$\displaystyle\left[\frac{\partial}{\partial\nu}P_{\nu}^{p}(z)\right]_{\nu=p}$
$\displaystyle=$ $\displaystyle
P_{p}^{p}(z)\log\frac{z+1}{2}+\left[2\psi(2p+1)-\psi(p+1)+\gamma\right]P_{p}^{p}(z)$
$\displaystyle{}+(-1)^{p}(2p)!\sum_{k=0}^{p-1}(-1)^{k}\frac{2k+1}{(p-k)(p+k+1)}P_{k}^{-p}(z),$

where $\gamma=-\psi(1)$ is Euler’s constant $\gamma\approx 0.57721566490$. Of
course we also have for $m=p=0$

$\left[\frac{\partial}{\partial\nu}P_{\nu}(z)\right]_{\nu=0}=\log\frac{z+1}{2},$

which exactly matches (33).

##  8 The logarithmic polynomials

The logarithmic polynomials $R_{p}^{k}$ are nonvanishing only for $-p\leq
k\leq p$ and by construction, they are satisfied by the following recurrence
relation

$R_{p}^{k}(x)=\frac{1}{2}R_{p-1}^{k-1}(x)+xR_{p-1}^{k}(x)+\frac{1}{2}R_{p-1}^{k+1}(x).$
(36)

From (11) we have that $R_{0}^{0}(x)=1$. This gives us the starting point for
the recursion. We conjecture that the derivative of the logarithmic
polynomials $R_{p}^{k}$ is given by

$\frac{d}{dx}R_{p}^{\pm k}(x)=pR_{p-1}^{\pm k}(x).$

It is evident by construction that these polynomials are even in the index
$k$, i.e.,

$R_{p}^{k}(x)=R_{p}^{-k}(x).$

Some of the first few logarithmic polynomials are given by

$\displaystyle R_{0}^{0}(x)=1,$ $\displaystyle R_{1}^{0}(x)=x,\ R_{1}^{\pm
1}(x)=\frac{1}{2}$ $\displaystyle R_{2}^{0}(x)=\frac{1}{2}+x^{2},\ R_{2}^{\pm
1}(x)=x,\ R_{2}^{\pm 2}(x)=\frac{1}{4},$ $\displaystyle
R_{3}^{0}(x)=\frac{3}{2}x+x^{3},\ R_{3}^{\pm
1}(x)=\frac{3}{8}+\frac{3}{2}x^{2},\ R_{3}^{\pm 2}(x)=\frac{3}{4}x,\
R_{3}^{\pm 3}(x)=\frac{1}{8}.$

We can find the generating function for the logarithmic polynomials as
follows. Let

$F(x,y,z)=\sum_{p=0}^{\infty}\sum_{k=-\infty}^{\infty}R_{p}^{k}(x)y^{k}z^{p}$

be the generating function for the logarithmic polynomials $R_{p}^{k}$. If we
define the function

$S_{p}(x,y)=\sum_{k=-\infty}^{\infty}R_{p}^{k}(x)y^{k},$

then using the recurrence relation for $R_{p}^{k}$ (36) we can show

$S_{p}(x,y)={\displaystyle\left(x+\frac{1}{2}\left(y+\frac{1}{y}\right)\right)}S_{p-1}(x,y).$

Combining this result along with the fact that $R_{0}^{0}(x)=1$, we have

$S_{p}(x,y)={\displaystyle\left(x+\frac{1}{2}\left(y+\frac{1}{y}\right)\right)^{p}},$

so therefore the generating function for the logarithmic polynomials
$R_{p}^{k}$ is given by

$F(x,y,z)=\frac{1}{\displaystyle
1-z\left(x+\frac{1}{2}\left(y+\frac{1}{y}\right)\right)}.$

An algorithm for generating the logarithmic polynomials can be obtained by
solving the following set of difference equations

$\left.\begin{array}[]{rcl}a_{0}(p)&=&\frac{1}{2}a_{0}(p-1)\\\\[8.5359pt]
a_{1}(p)&=&\frac{1}{2}a_{1}(p-1)+xa_{0}(p-1)\\\\[8.5359pt]
a_{2}(p)&=&\frac{1}{2}a_{2}(p-1)+xa_{1}(p-1)+\frac{1}{2}a_{0}(p-1)\\\\[8.5359pt]
&\vdots&\\\\[8.5359pt]
a_{n}(p)&=&\frac{1}{2}a_{n}(p-1)+xa_{n-1}(p-1)+\frac{1}{2}a_{n-2}(p-1)\end{array}\quad\right\\},$
(37)

subject to the boundary conditions

$\left.\begin{array}[]{rcl}a_{0}(0)&=&1\\\\[5.69046pt]
a_{1}(1)&=&xa_{0}(0)\\\\[5.69046pt]
a_{2}(2)&=&xa_{1}(1)+a_{0}(1)\\\\[5.69046pt] &\vdots&\\\\[5.69046pt]
a_{n}(n)&=&xa_{n-1}(n-1)+a_{n-2}(n-1)\end{array}\quad\right\\},$ (38)

where $R_{p}^{k}=a_{p-|k|}(p)$ is given along diagonals for a fixed $p-|k|$.
For instance, one can obtain $R_{p}^{\pm p}(x)=\frac{1}{2^{p}},$ for $p\geq
0$, $R_{p}^{\pm(p-1)}(x)=\frac{p}{2^{p-1}}x,$ for $p\geq 1$ and
$R_{p}^{\pm(p-2)}(x)=\frac{p}{2^{p}}+\frac{p(p-1)}{2^{p-1}}x^{2},$ for $p\geq
2,$ etc.

## Acknowledgements

I would like to thank Tom ter Elst, Matthew Auger, and Radosław Szmytkowski
for valuable discussions. I would also like to thank Shenghui Yang at Wolfram
Research for valuable assistance in generating an algorithm to symbolically
compute the logarithmic polynomials. I acknowledge funding for time to write
this paper from the Dean of the Faculty of Science at the University of
Auckland in the form of a three month stipend to enhance University of
Auckland 2012 PBRF Performance. Part of this work was conducted while H. S.
Cohl was a National Research Council Research Postdoctoral Associate in the
Information Technology Laboratory at the National Institute of Standards and
Technology, Gaithersburg, Maryland, U.S.A.

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