Logarithmic Sine and Cosine Transforms and Their Applications to Boundary-Value Problems Connected with Sectionally-Harmonic Functions

Let  , r   stand for the polar coordinates in , be a given constant while 2 R 0 a   , u r   satisfies the Laplace equation in the wedge-shaped domain 0 u           0 1 , : 0, , , j j r a        1 n


 
, : denote certain angles such that  , then an explicit expression of   , u r  in terms of eigen-functions can be found through the classical method of separation of variables.But when the boundary condition given on the circular boundary is homogeneous, it is not possible to define a discrete set of eigen-functions.In this paper one shows that if the homogeneous condition in question is of the Dirichlet (or Neumann) type, then the logarithmic sine transform (or logarithmic cosine transform) defined by ) may be effective in solving the problem.The inverses of these transformations are expressed through the same kernels on or 0, a   , a  .Some properties of these transforms are also given in four theorems.An illustrative example, connected with the heat transfer in a two-part wedge domain, shows their effectiveness in getting exact solution.In the example in question the lateral boundaries are assumed to be non-conducting, which are expressed through Neumann type boundary conditions.The application of the method gives also the necessary condition for the solvability of the problem (the already known existence condition!).This kind of problems arise in various domain of applications such as electrostatics, magneto-statics, hydrostatics, heat transfer, mass transfer, acoustics, elasticity, etc.

Introduction
Boundary-value problems connected with sectionallyharmonic functions in wedge-shaped domains , , , where   , r  stand for the polar coordinates in R 2 while a > 0 is a given constant, are important from both pure scientific and engineering points of view.  in the sense of distribution under the boundary conditions As is well-known, when and f r must be different from zero in order to have a non trivial solution .In this case it is not possible to define a set of discrete eigen-functions.To overcome such kind of a difficulty, in the midst of the last century some methods, which are effective when the region consists of D 0 , were proposed.Among them we can mention, for example, the finite Sturm-Liouville transforms introduced by Eringen [1] and Churchill [2], and the finite Mellin transform introduced by Naylor [3,4] (see also [5]).The finite Sturm-Liouville transforms are not appropriate in the case considered here because they are based on the set of eigen-functions which can not be defined in the present case.As to the finite Mellin transforms, they are defined as follows (see, for ex., [5, pp. 462-467]): One can easily check that the first or the second transform is appropriate to reduce the Laplace equation written in the circular polar coordinates to an ordinary differential equation when Dirichlet or Neumann type conditions are prescribed, respectively, on the circular part r = a of the boundary.The inverse transforms consist then of the classical Mellin type integrals.The aim of this note is to show that the transforms of the forms and are effective in getting explicit expressions to the solutions of the problems connected with sectionally-harmonic functions defined in D 0 and D  mentioned above when the boundary condition on the arc r = a is homogeneous and of the Dirichlet or Neumann type.The simplicity of these transforms is that their inverses are also given with the same kernel (see Section 2B below).To clarify the essential properties of the representations ((3a), (b)), in what follows we will consider, without loss of generality, the case where n = 1 (see Figures 2 and 3).
To the best of our knowledge a representation of the form (3b) was first considered by Smythe (see [6, pp. 71-72]) to find the electrostatic potential due to a line source located parallel to a dielectric wedge for which   0, r   .His discussion is based on moot physical arguments and some particular restrictions.As we will show later on, a representation of the form (3b) is not suitable when   0, r   because only the data known for  F  (i.e. the inverse transform).Further- more, whe ) is used to express , which not acom physics is ceptable fr point of view.

Logarithmic Sine and Cosine Transforms
Let a > 0 be a given constant while or for all , a , th the function F  can be determine tely.The theorems th llow concern this inversion problem (for their proof see Appendix).Notice that when The proof of these theorems (except Theorem 1) can easily be achieved by using the already known ones through simple transformations (See for example [5] or [7]).For the sake of fluency of the paper, we prefer to postpone the proofs to the Appendix.In what follows we will denote the inverse transforms given by (5a) and (6a) by Similarly, the inverse transforms given by ( will be denoted by ) and (6b)

Applications to Boundary-Value Problems
W lso

Connected with Sectionally-Harmonic Functions
hen one has a and satisfies a homogeneous boundary condition of the Dirichlet or Neumann type on the circular part of the boundary, namely: and Application of the operator or where for the case of (10b).

An Illustrative
To (17) e following necessary condition is satisfied by the boundary conditions :

 
Since the field Equation ( 14) is equivalent to the equa- This is obtained by first integrating (14) o (18c) then applying the Green's theorem.In what follo we will assume that (16b) is satisfied (it will be used later on!).
In accordance with the definitions of  ws    and Q , one can write The coefficients ) and (18c) as follows:

  
the conditions ( 1), ( Here we put If we first insert (20a)-(20c) into ((19a), (19b)) and then use t he Euler formula to write the cos function through exponential functions, then we get where  is defined with Remark.It is worthwhile to notice here that the convergence of the inverse transform integrals in (22a) and (22b) requires the relation (23).That m tion stated by ( 23) is in fact a condition for the existence of the solution.One can easily check that this is nothing but (16b) (or (16a)).This shows that the application of th log 0. r a Now it is important to observe that the point 0   , to be a which is located on the i double pole of the inte ntegration line, seems grands in (22a) and (22b).But because of the relation (16b) these poles are removable.Indeed, the removability of these dition poles requires the con- which is equivalent to (16b), ( 21) being taken into account.Since the expressions taking place in the brackets in (22a) and (22b) are even functions of , (23) guarantees the removability of the singularity at  = 0. Thus, on the basis of Jordan's lemma, the integral can be computed through the residues at the poles located s in ((22a), (b)) The residue series coming from the poles which occur at the zeros of sinh() and cosh() are connected with the geometry of the wedge in question and, hence, consist of the eigen-functions series.But the terms coming from the poles of metry of th and have no connection with the eigen-functions.
f the geo e wedge eans that the relae transformation in question does not only permit us to obtain an explicit expression of the solution but rather shows also the necessary conditions for the existence of a solution.

A Particular Case. Point Sinks Located on the Lateral Boundaries
Assume more particularly that where   c a  and M stand for two given constant such that (Cf.( 23 In this case the lateral boundaries consist of materials and carry sinks at the points insulating which reduces (22a) to Since in the present case one always has by the Jordan's lemma, the first parts of these integrals can be computed by considering the residues of the poles taking place in the upper half-plane .But, depending on the relative positions of r one can get By straightforward computations we get the following results:  , r a   and assume that an arbitrarily fixed (small) 0 Thus, from (3a) one gets Now, by taking into account the ure A1).
inequalities (see Fig- and sin , 0, x x x   we can choose  so small that From (27b) and (27d) we conclude that for every  0   , however it is small, we can find 0 For this gives the first equati in (4a).To prove the same equality for the case of


Here   1 m   pend on stands for a suitable number which does not de  (For detail see Figure A1).Thus, by choosing  sufficiently small, we guarantee For r = a the latter reduces to the first equality in (4a) meets the following inequality: ) and make the substitutions

Figure 1 
epitomizes a simple case of D 0 which corresponds to n = 2.The sub-regions determined by model the regions filled with different materials having different constitutive parameters.The field function   , u r  satisfies the basic equation

1 .
Figure 1.A composite region D 0 filled with two different materials.

1 ,
on the figure.Here     F r     stands for the density of the exciting sources concentrated on the interface    (if any) while     L u L u and   2 L u are given linear (differential) boundary operations.The boundary conditions in question may also involve certain terms representing the sources localized on the boundary (if any).As to the function     , it has constant values  1 and  2 in the sub-regions in question.Thus on the interface between the sub-regions two transmission conditions of the following forms are satisfied: one can define a set of orthogonal eigen-functions which permit us to obtain an explicit expression of   , u r  in terms of these eigen-functions.The coefficients in the eigen-function series are determined by using the nonhomogeneous boundary condition given on the boundary r = a (i.e. through   f  ) together with the regularity condition to be stated at r = 0.When , at least one of the functions

Figure 2 .
Figure 2. A wedge-shaped region involving only the singular point r = 0.

Figure 3 .
Figure 3.A wedge-shaped region involving only the singular a given function.Then consider the functions through the convergent integral ing p (3a) and (3b).There log stands for the principal branch of the logarithm function.We will refer Values for r  0, r  a and r   As we will see later on (see Section 2B), the expression of the function iece-wise continuous in thes vals.eans that the limit values of the integrals taking place in (3a) and (3b) as r tends to the end point 0 integrals.From applicati the limiting values that are important.Therefore these limit values must be discussed carefully.The two theorems that follow concern this point (for their proof see Appendix). c ), (b)) being taken into account.Here   ˆ, u   orm of stands for the logarithmic sine or cosine transf   , u r  .From (11) one gets show the effectiveness of the repr (b)), in what follows we will give an illustrative example which concerns the heat conduction in a two-part composite region shown in Figure 4.A point source of amount Q exists at the point (b, 0) while the circular part of the boundary is coated by an insulating material.The physical properties of the lateral boundaries (i.e. the boundary conditions on     and    ) will be defined later on.Thus the field function (temperature)   , u r  satisfies the following field equation under the given boundary conditions: j  are two constants which can be chosen appropriately to facilitate the computation.They may also be dependent on .Thus, in the sector

Figure 4 .
Figure 4.A wedge-shaped region with Neumann type boundary conditions.

us
Dirichlet or Neumann boundary conditions on the circular part of the boundary of wedge shaped domains.pli-
r0, consider an arbitrarily given (small)   and choose such that the second part in 0

2
29b)After having fixed A, let us make .By virtue of the well-known Riemann-Lebesgue[5, p. 30], the part in (29a) tends to zero wh )-(29c) one co ludes that for sufficiently small r one has This proves the second equality in (4a).(29a)-(29c) are also valid for r  , which shows the last equality in (4a)2.A Proof for Theorem-The equalities given in (4b) can be shown by repeating Theorem-1.As to the equality given in (4c), owing to the assumption the reasoning made in proving to the first equality in (4a).
Let us insert (5a) into the right-hand side of (3a proves (5a).To prove (5b), one starts from (3b) and repeats (32a)the only exception that (33b) is replaced now ven function Proof of theorem-4 is quite similar to that of Theo-