Fourier expansions for a logarithmic fundamental solution of the polyharmonic equation
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.
undamental solutions; polyharmonic equation; Fourier series; polynomials; associated
35A08; 31B30; 31C12; 33C05; 42A16
10.1080/10652460YYxxxxxxx \issn1476-8291 \issnp1065-2469 \jvol00 \jnum00 \jyear2009 \jmonthJanuary
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) ), etc. Furthermore, applications of higher-powers of the Laplacian include such varied areas as minimal surfaces , Continuum Mechanics , Mesh deformation , Elasticity , Stokes Flow , Geometric Design , Cubature formulae , mean value theorems (cf. Pizzetti’s formula) , and Hartree-Fock calculations of nuclei .
It is a well-known fact (see for instance Schwartz (1950) (, p. 45), Gelfand & Shilov (1964) (, p. 202) that a fundamental solution of the polyharmonic equation on -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) ), 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)  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) ).
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 -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 , the set , the set of integers is given by and the set represents the rational numbers. For , if and then and , where represents the complex numbers. The set represents the real numbers.
2 Fundamental solution of the polyharmonic equation and the non-logarithmic Fourier series
If satisfies the polyharmonic equation given by
where , for is the Laplacian operator defined by and then is called polyharmonic. If the power of the Laplacian equals two, then (1) is called the biharmonic equation and is called biharmonic. The inhomogeneous polyharmonic equation is given by
where we take to be an integrable function so that a solution to (2) exists. A fundamental solution for the polyharmonic equation on is a function which satisfies the equation
for some , where is the Dirac delta function and . When , we call a fundamental solution of the polyharmonic equation normalized, and denote it by . The Euclidean inner product defined by induces a norm (the Euclidean norm) , on the finite-dimensional vector space , given by In the rest of this paper, we will use the gamma function , which is a natural generalization of the factorial function (see for instance Chapter 5 in Olver et al. (2010) ). A fundamental solution of the polyharmonic equation is given by the following theorem.
Let . Define
where is defined as with being the th harmonic number
then is a normalized fundamental solution for on Euclidean space .
A separable rotationally-invariant coordinate system for the polyharmonic equation (1) on is given by
which is described by an angle and -curvilinear coordinates . A separable rotationally-invariant coordinate system transforms the polyharmonic equation into a set of -uncoupled ordinary differential equations with separation constants and for . For a separable rotationally-invariant coordinate system, this uncoupling is accomplished, in general, by assuming a solution to (1) of the form
where the domains of the functions and , for , and the constants for , depend on the specific rotationally-invariant coordinate system. A rotationally-invariant coordinate system parametrizes points on the -dimensional half-hyperplane given by and using curvilinear coordinates . (For a general description of the theory of separation of variables see Miller (1977) .) The Euclidean distance between two points , expressed in a rotationally-invariant coordinate system, is given by
where the toroidal parameter , is given by
where are defined in (4) for . The hypersurfaces given by equals constant are independent of coordinate system and represent hyper-tori of revolution.
From Theorem 2 we see that, apart from multiplicative constants, the algebraic expression of an unnormalized fundamental solution for the polyharmonic equation in Euclidean space for even, is given by
By expressing in a rotationally-invariant coordinate system (4) we obtain
where . For the polyharmonic equation in even-dimensional Euclidean space with apart from multiplicative constants, the algebraic expression for an unnormalized fundamental solution of the polyharmonic equation is given by
By expressing in a rotationally-invariant coordinate system we obtain
By examining (7) and (8), we see that for computation of Fourier expansions about the azimuthal separation angle of and , all that is required is to compute the Fourier cosine series for the following three functions and defined as
where , and is a fixed parameter.
The Fourier series of is given in Cohl & Dominici (2010)  (cf. (4.4) therein), namely
where the Neumann factor commonly occurs in Fourier series, is the Kronecker delta, and
for and , is the Pochhammer symbol (rising factorial). We have used Whipple’s formula in (9) (see for instance, (8.2.7) in Abramowitz & Stegun (1972) ) to convert the associated Legendre function of the second kind appearing in  to the associated Legendre function of the first kind . The associated Legendre function of the first kind can be defined using the Gauss hypergeometric function, namely (Magnus, Oberhettinger & Soni (1966) , p. 153)
The Gauss hypergeometric function can be defined in terms of the following infinite series
(see for instance Chapter 15 in Olver et al. (2010) ).
where . Since the Fourier series of is computed in Cohl & Dominici (2010) , we understand how to compute Fourier expansions of (8) in separable rotationally-invariant coordinate systems. In order to compute Fourier expansion of (7) in separable rotationally-invariant coordinate systems, all that remains is to determine the Fourier series of . This is the goal of the next two sections.
3 Algebraic approach to the logarithmic Fourier series
Since , one may make the substitution to evaluate the Fourier series of . For instance, it is given in the form of where . For the result is well-known (see for instance Magnus, Oberhettinger & Soni (1966) , p. 259)
which as we will see, should be compared with (10) for , namely
Note that for we may write and therefore as a function of since and therefore Now examine the case for . If we multiply both sides of (11) by and take advantage of the formula
then we have
Collecting the contributions to the Fourier cosine series, we obtain
we notice that the factor appears in both series.
For in , we use (13) and similarly have
If we collect the contributions of the Fourier cosine series, we obtain
This algebraic approach for determining the Fourier series of will now be generalized. By starting with (11) and repeatedly multiplying by factors of , we see that the general Fourier series of can be given in terms of a sequence of polynomials , with and , as
We will refer to as logarithmic polynomials with argument (in our notation and 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 . It then follows that the resulting double sum naturally breaks into two disjoint regions, one triangular
with terms and the other infinite rectangular
By rearranging the order of the and summations in (19), we derive
where are defined as
respectively. We can also write the Fourier series directly in terms of the logarithmic polynomials as follows
Note that by using (9), then we can express as a Fourier series, namely
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, and . By applying the identity
where , to
where (cf. (3.11b) in Cohl & Dominici (2010) ), one can compute the Fourier cosine series of , provided availability of the necessary parameter derivatives.
Note that for and , the associated Legendre function of the first kind vanishes if . This is easily seen using the Rodrigues-type formula (cf. (14.7.11) in Olver et al. (2010) )
and the fact that is a polynomial in of degree . The derivatives are given as follows:
where is the digamma function defined in terms of the derivative of the gamma function
(see for instance (5.2.2) in Olver et al. (2010) ). The degree-derivative of the associated Legendre function of the first kind in (28) is determined using (32) and (33). By collecting terms and using (26), (27), and (28), we obtain
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 . 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 (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 increases, and they have been checked for using Mathematica with the assistance of an algorithm generated using (35) and (36) 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 and we have
for , we derive