`Abstract and Applied AnalysisVolumeย 2008ย (), Article IDย 765920, 12 pageshttp://dx.doi.org/10.1155/2008/765920`
Research Article

## The Analysis of Contour Integrals

1Department of Mathematics, Harran University, Osmanbey Campus, Sanlurfa 63100, Turkey
2Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA

Received 10 November 2007; Accepted 19 January 2008

Academic Editor: Stephen L.ย Clark

Copyright ยฉ 2008 Tanfer Tanriverdi and JohnBryce Mcleod. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For any , the contour integral , is associated with differential equation . Explicit solutions for are obtained. For , eigenvalues, eigenfunctions, spectral function, and eigenfunction expansions are explored. This differential equation which does have solution in terms of the trigonometric functions does not seem to have been explored and it is also one of the purposes of this paper to put it on record.

#### 1. Introduction

When one considers eigenfunction expansions associated with second-order ordinary differential equations, as Titchmarsh does in his book [1], one is concerned with solutions of the equation along with certain boundary conditions, and one tends to say that the only case in which one can solve this equation explicitly in elementary terms for all is the case , when the solutions are of course trigonometric functions.

Now in fact this is not true, and there is in particular one problem which does not seem to have been explored, and it is the purpose of this paper to put it on record. Here is the problem: which can be solved explicitly in elementary terms when is integral. The explicit solution was known to Kamke [2], but Kamke does not anyway explore the consequences for eigenfunction expansions nor does Titchmarsh discuss this problem, although he does discuss problems close to it, for example, on , which leads, when , to an expansion in series involving associated Legendre functions.

It is perhaps worth remarking how our interest in this problem arises. In [3] there is the question of travelling waves and steady solutions for a discrete reaction-diffusion equation of the type where the function is โbistableโ. That is, there exist three numbers , , , , such that with in and in . A prototypical would be where and are positive constants and , so that Such equations arise in a number of different applications, for example, in dislocation theory where is the displacement of the atom in some material, or in neurobiology where is typically the electric potential of the nerve cell, and in both these applications the interest is in monotonic solutions with as , as [3โ6].

The basic question is whether there exist such solutions with the form of a travelling wave, , , or of a steady solution or standing wave, where , and there is an important distinction between these two cases. For a travelling wave, , is clearly a function of the continuous variable , and indeed because of (1.4), a differentiable function of . This leads to the difference-differential equation If, however, , then, as in [3], we have to study the purely difference equation and the solutions may be discontinuous since there is nothing that now connects values of with values of for . It is best therefore to think of the solution of (1.9) as a number of (monotonic) sequences indexed by , each satisfying The simplest case would be that there is just one such sequence (modulo the translation , integral), but it is possible that there may be a finite number, or even a partial or total continuum.

In view of applications, where the distance between atoms or nerve cells is small, it is more natural to think of (1.8) in the form where is small and represents the distance between atoms or nerve cells. A tempting approximation is then and in order to make sense of the scaling, in [3] the authors introduced a factor in front of . This therefore leads to a comparison between the solutions of For the continuous diffusion problem, the answer is both simple and well known [4, 5].

Given a function that is bistable, there is just one possible wave-speed , and this value of is , that is, there is a steady solution if and only if (The proof is a simple phase plane argument, and implies (1.15) follows by multiplying (1.14) by and integrating.)

The solution in the discrete case is however different, as discussed in [6]. There may continue to be steady solutions where (1.15) no longer holds. Consider specifically the case (1.6), so that The case corresponding to (1.15) is , but the authors, in [3], have shown that for sufficiently small, say , there exist precisely two steady solutions of (1.16), and , which of course depends on , can be evaluated for small . Specifically, where the constant is given. For , the solutions move and the equation has travelling wave solutions instead of steady solutions.

In order to prove results such as (1.17), one has to regard (1.16) as a singular perturbation of the steady continuous-diffusion equation for which the solution (satisfying , ) is . If we linearize (1.18) about , we obtain But multiplying (1.18) by and integrating lead to , so that since , we have . The linearization (1.19) thus becomes , which is of course (1.2) with and . Thus the selfadjoint operator given (in ) by has an eigenvalue at , with eigenfunction (differentiation of (1.18) shows that satisfies (1.19)). This fact, together with the additional fact that the spectrum of is continuous above (since ), is highly relevant to the work in [3] and led to our interest more generally in the spectral problem (1.2).

The explicit solution for any using contour integrals different from what Kamke did is known to [7]. For more information on this problem one can see [7, 8].

#### 2. Preliminaries

We want to know expansion of an arbitrary function in terms of eigenfunctions. So one needs to know the following. Let and be the solutions of (1.1) such that where is real. . The general solution of (1.1) is of the form The spectrum is defined by means of the function which exists for all real and is a nondecreasing function. The expansion of a function in terms of the spectral function depends on the following lemmas taken from [1].

Lemma 2.1. Without detailing, let the interval be : If has poles, then

Lemma 2.2. Without detailing, let the interval be . If is an even function, then . So the expansion formula is where

#### 3. Main Results

We are now dealing with (1.2) in the case where is integral. Without loss of generality, we may suppose , but since reduces (1.2) to the simple trigonometric case, we are in fact interested only in . We first prove that a solution is given by where the contour is taken round the point and no other zero of . This is slight variant of a form which Titchmarsh uses in his discussion of (1.3). The proof below will show (3.1), being continuous at least formally, to be a solution of (1.2) where is not an integer, but the difficulty then is to choose a suitable contour, since the integrand has a branch point at .

Remark 3.1. We also remark that it is obvious that we can express the solution (3.1) equivalently ignoring some multiplicative constants as

Theorem 3.2. The contour integral (3.1) satisfies the differential equation (1.2).

Proof. We see that Integrating (3.1) by parts, so that Comparing (3.4) and (3.6), we see that , so that satisfies (1.2), as required.

Remark 3.3. We now point out that the factor played little part in the argument. Certainly, the argument would have washed equally well if we had replaced by :

Theorem 3.4. The contour integral (3.7) satisfies the differential equation (1.2).

Proof. Proof is the same as the above theorem. So we omit it.

Remark 3.5. Furthermore, once the integrands have poles at , the solution can be evaluated by calculating the relevant residues. For example, in the trivial case , when we should recover the trigonometric functions, the residues of are so that the solution (3.7) becomes multiples of (similarly ), as we expect.
We can generalize Theorems (3.2) and (3.4) by defining the following operator:where is a differentiable function as long as one can pick up residue.

Corollary 3.6. If then we obtain Theorems (3.2) and (3.4). The operator is also linear.

#### 4. The Explicit Solution Given by Residues for ๐=1

We now require the residues of Since we see that the residue at is so that one solution is By examining the residue of the second equation of (4.1), we see that a second solution is

Remark 4.1. The solution can also be obtained from (3.2). For we have already seen, from our brief discussion of the case , that the integral in (3.2) is just a multiple of (or of if we replace by ) hence in the first case, (3.2) gives a multiple of in accordance with (4.4).

Remark 4.2. Wronskian . We now have two linearly independent solutions.

Remark 4.3. The general solution is .

Lemma 4.4. is an odd function but is an even function.

Proof.

#### 5. Eigenvalues and Eigenfunctions for ๐=1 When ๐ฆ(0)=๐ฆ(๐)=0

Theorem 5.1. Eigenvalues with associated boundary conditions are the zeros of furthermore, one and only one eigenvalue lies in the interval for every integral .

Proof. One can see from Remark 4.3 that if then . So that implies One can see immediately that the eigenvalues belong to the interval (5.2).
To prove the second part we use the following strategy. Multiply (5.1) by and set , then denote . So Notice that is an even and does not intersect -axis, where . If , then and if , then . So the monotonicity of implies that has only one zero belonging to the interval (5.2) for every integral .

Remark 5.2. So is the eigenfunction in the form of (4.4):

Corollary 5.3. One can orthonormalize the eigenfunctions.

Proof. The orthonormalized eigenfunctions denoted by , .

Remark 5.4. An arbitrary function in terms of eigenfunctions follows:

#### 6. Eigenvalues and Eigenfunctions for ๐=1 When ๐ฆ๎(0)=๐ฆ๎(๐)=0

Theorem 6.1. The eigenvalues with are the zeros of (6.1); furthermore, there exists one and only one eigenvalue lying in the interval (5.2) for every integral .

Proof. One can see from Remark 4.3 that if then either or . If then the associated eigenfunction is zero. So this is useless. Hence, . implies So it is obvious that the zeros (eigenvalues) belong to the interval (5.2).
Set . (). Equation (6.1) is denoted by : It is enough to show that is monotonic: We see that everywhere. We therefore conclude that is monotonic.

Remark 6.2. The associated eigenfunctions are

Corollary 6.3. One can orthonormalize the eigenfunctions.

Proof. The orthonormalized eigenfunctions denoted by , .

Remark 6.4. Therefore, an arbitrary in terms of orthonormalized eigenfunctions is

#### 7. Spectral Function ๐(๐) over (0,โ) and Expansion

Now let and be the solutions of which satisfy (2.1); so that Now we need to find . This suggests that To get demanded result, one needs to find the asymptotics of and as and : Finally, we must arrange the linear combination so that the terms cancel. That is, . Hence, We see that the spectrum is continuous for . But we have a point spectrum for . So that an arbitrary is a linear combination of integrand and series. The spectral function is calculated from (2.3). Hence, In particular, if , then So that in this case there is no eigenvalue for . From Lemma 2.1, one can see the expansion of Similarly, if , then Hence, there exits only one eigenvalue at and the corresponding eigenfunction where is (7.2). So the spectrum calculated by (2.3) is Therefore, from Lemma 2.1, an expansion of function in terms of eigenfunctions and spectral function follows: where is a constant. So we have proved the following theorem regarding the nature of .

Theorem 7.1. If there is no eigenvalue where . If , then there exists only one eigenvalue at and the corresponding eigenfunction is .

Finally, one can ask what is the range of in the case of and .

Theorem 7.2. If , then there are precisely two eigenvalues except at . Hence there are two eigenfunctions, namely, and .

Proof. We check the zeros of both the numerator and denominator of (7.6). After working out the algebra, we see that the zeros of the numerator and the denominator are If has poles, then they are the eigenvalues. If so, the eigenfunctions are and . Finally, there is one thing to be proved in this case . To do this, we expand the zeros around and at the end : Similarly, That completes the proof.

#### 8. Expansion over (โโ,โ)

Now consider the interval instead of . From (2.7) the following can be calculated: Now one can use (7.2), (7.3), and (8.1) gets the following expansion from Lemma (2.2): where is a constant.

#### References

1. E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I, Oxford University Press, Oxford, UK, 2nd edition, 1962.
2. E. Kamke, Differentialgleichhungen, Akademische Verlagsgesellschaft, Leipzig, Germany, 1943.
3. S. P. Hastings and J. B. McLeod, โTravelling waves and steady solutions for a discrete reaction-diffusion equation,โ Preprin.
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5. P. C. Fife and J. B. McLeod, โA phase plane discussion of convergence to travelling fronts for nonlinear diffusion,โ Archive for Rational Mechanics and Analysis, vol. 75, no. 4, pp. 281โ314, 1981.
6. A. Carpio, S. J. Chapman, S. P. Hastings, and J. B. McLeod, โWave solutions for a discrete reaction-diffusion equation,โ European Journal of Applied Mathematics, vol. 11, no. 4, pp. 399โ412, 2000.
7. T. Tanriverdi, Boundary-value problems in ODE, Ph.D. thesis, University of Pittsburgh, Pittsburgh, Pa, USA, 2001.
8. T. Tanriverdi and J. B. Mcleod, โGeneralization of the eigenvalues by contour integrals,โ Applied Mathematics and Computation, vol. 189, no. 2, pp. 1765โ1773, 2007.