bielski. macdonald K orthogonality

J. Math. Anal. Appl. 365 (2010) 195–197 Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa Comment on the orthogonality of the Macdonald functions of imaginary order Radosław Szmytkowski∗, Sebastian Bielski Atomic Physics Division, Department of Atomic Physics and Luminescence, Faculty of Applied Physics and Mathematics, Gda´ nsk University of Technology, Narutowicza 11/12, PL 80-233 Gda´ nsk, Poland article info abstract Article history: Received 24 September 2009Available online 24 October 2009 Submitted by B.C. Berndt Keywords: Macdonald functionsOrthogonal functions Dirac delta distributionRecently, Yakubovich [Opuscula Math. 26 (2006) 161–172] and Passian et al. [J. Math. Anal. Appl. 360 (2009) 380–390] have presented alternative proofs of an orthogonality relationobeyed by the Macdonald functions of imaginary order. In this note, we show that the validity of that relation may be also proven in a simpler way by applying a technique occasionally used in mathematical physics to normalize scattering wave functions to theDirac delta distribution. ©2009 Elsevier Inc. All rights reserved. 1. Introduction In recent papers, Yakubovich [1] and Passian et al. [2] proved the following orthogonality relation for the Macdonald functions of imaginary order: ∞/integraldisplay 0dxKiν(x)Kiν/prime(x) x=π2 2νsinh(πν)δ/parenleftbigν−ν/prime/parenrightbig/parenleftbigν,ν/prime>0/parenrightbig . (1.1) In the first of these works, a proof relied on advanced techniques of the theory of distributions. An approach adopted in the second paper was a two-step one. At first, two heuristic arguments making the relation (1.1) plausible were presented. One of these arguments was based on an integral relation between the Macdonald functions of imaginary order and Mehler’sconical functions, for which a counterpart orthogonality relation had been known for a long time. The second argumentgiven in support of the validity of Eq. (1.1) exploited the fact that the Laplace transform of K iν(x)is a known elementary function. Subsequently, a sophisticated proof of the relation (1.1), different from the one in Ref. [1], was presented. It is the purpose of this note to present still another proof of the relation (1.1). The approach we adopt here is known in mathematical physics, where it is occasionally used for normalization of scattering states to the Dirac delta distribution. Insome sense, it is akin to the standard method used to prove weighted orthogonality relations for eigenfunctions of regularSturm–Liouville problems (cf., e.g., Ref. [3, Section 7.1]). It has several advantages. First, it is elementary compared to the methods used in Refs. [1,2]. Second, it is constructive :o n e derives the orthogonality relation. Finally, it is general and may be used to obtain counterpart orthogonality relations not only for K iν(x), but also for other special functions (e.g., Ref. [4]). *Corresponding author. E-mail addresses: [email protected] (R. Szmytkowski), [email protected] (S. Bielski). 0022-247X/$ – see front matter ©2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2009.10.035 196 R. Szmytkowski, S. Bielski / J. Math. Anal. Appl. 365 (2010) 195–197 2. A summary of relevant properties of the Macdonald functions of imaginary order Before we proceed to the merit, in this short section we shall summarize these properties of the Macdonald functions of imaginary order which will be exploited later in Section 3. The formulas presented below have been excerpted from the collection of Magnus et al. [5]. The function Kiν(x)is a particular solution to the modified Bessel differential equation x2d2F(x) dx2+xdF(x) dx+/parenleftbigν2−x2/parenrightbig F(x)=0. (2.1) Other particular solutions to Eq. (2.1) are the modified Bessel functions of the first kind I±iν(x)=∞/summationdisplay k=01 k!Γ(k+1±iν)/parenleftbiggx 2/parenrightbigg2k±iν . (2.2) The relationship between the three functions is Kiν(x)=π 2iI−iν(x)−Iiν(x) sinh(πν), (2.3) from which it follows immediately that Kiν(x)=K−iν(x). (2.4) For large positive values of x, the function Kiν(x)has the asymptotic representation Kiν(x)x→∞∼/radicalbiggπ 2xe−x/bracketleftbig 1+O/parenleftbig x−1/parenrightbig/bracketrightbig , (2.5) while for x→+ 0 from Eqs. (2.3) and (2.2) and with the aid of the known relationship /vextendsingle/vextendsingleΓ(iν)/vextendsingle/vextendsingle=/radicalbiggπ νsinh(πν)(ν∈R) (2.6) one deduces that Kiν(x)x→+ 0∼/radicalbiggπ νsinh(πν)cos/bracketleftbigg −νlnx 2+argΓ(iν)/bracketrightbigg +O/parenleftbigg x2sin/bracketleftbigg −νlnx 2+argΓ(2+iν)/bracketrightbigg/parenrightbigg (ν∈R). (2.7) 3. Derivation of the orthogonality relation for the Macdonald functions of imaginary order To derive the orthogonality relation for the Macdonald functions of imaginary order, we proceed as follows. If Kiν(x), withν∈R, is substituted for F(x)into Eq. (2.1), this results in the differential identity d dx/parenleftbigg xdKiν(x) dx/parenrightbigg +/parenleftbiggν2 x−x/parenrightbigg Kiν(x)=0. (3.1) The counterpart identity for the function Kiν/prime(x),w i t h ν/prime∈R,i s d dx/parenleftbigg xdKiν/prime(x) dx/parenrightbigg +/parenleftbiggν/prime2 x−x/parenrightbigg Kiν/prime(x)=0. (3.2) Next, we premultiply the first of the above equations by Kiν/prime(x), the second one by Kiν(x), subtract and integrate the result over xfrom x=ξ>0t o x=∞ . After obvious rearrangements, this gives /parenleftbigν2−ν/prime2/parenrightbig∞/integraldisplay ξdxKiν(x)Kiν/prime(x) x=∞/integraldisplay ξdx/bracketleftbigg Kiν(x)d dx/parenleftbigg xdKiν/prime(x) dx/parenrightbigg −Kiν/prime(x)d dx/parenleftbigg xdKiν(x) dx/parenrightbigg/bracketrightbigg . (3.3) The integral on the right-hand side of Eq. (3.3) is easily evaluated by parts; one obtains /parenleftbigν2−ν/prime2/parenrightbig∞/integraldisplay ξdxKiν(x)Kiν/prime(x) x=/bracketleftbigg x/parenleftbigg Kiν(x)dKiν/prime(x) dx−Kiν/prime(x)dKiν(x) dx/parenrightbigg/bracketrightbigg∞ x=ξ. (3.4) By virtue of Eq. (2.5), the expression in the bracket on the right-hand side of the above relation vanishes in the upper limit. Hence, we obtain R. Szmytkowski, S. Bielski / J. Math. Anal. Appl. 365 (2010) 195–197 197 ∞/integraldisplay ξdxKiν(x)Kiν/prime(x) x=−ξKiν(ξ)dKiν/prime(ξ) dξ−Kiν/prime(ξ)dKiν(ξ) dξ ν2−ν/prime2(3.5) and consequently ∞/integraldisplay 0dxKiν(x)Kiν/prime(x) x=− lim ξ→+ 0ξKiν(ξ)dKiν/prime(ξ) dξ−Kiν/prime(ξ)dKiν(ξ) dξ ν2−ν/prime2. (3.6) Using the asymptotic representation (2.7) and elementary trigonometric identities transforms Eq. (3.6) into ∞/integraldisplay 0dxKiν(x)Kiν/prime(x) x=π 2√νν/primesinh(πν)sinh(πν/prime) ×lim ξ→+ 0/braceleftbiggsin[−(ν−ν/prime)lnξ 2+argΓ(iν)−argΓ(iν/prime)] ν−ν/prime +sin[−(ν+ν/prime)lnξ 2+argΓ(iν)+argΓ(iν/prime)] ν+ν/prime/bracerightbigg . (3.7) To evaluate the limit on the right-hand side of Eq. (3.7), we observe that if f(η)is a real analytic function of η∈R,s u c h that f(0)=0 (which implies that lim η→0f(η)/ηis finite), then in the distributional sense it holds that lim a→∞sin[aη+f(η)] πη=1 2πlim a→∞a+f(η)/η/integraldisplay −a−f(η)/ηdαeiαη=1 2πlim a→∞a/integraldisplay −adαeiαη=1 2π∞/integraldisplay −∞dαeiαη. (3.8) In the expression at the extreme right of the above chain of equalities one immediately recognizes the well-known Fourier representation of the Dirac delta distribution δ(η)(cf., e.g., [6, Section 4.5]), so that, provided the function f(η)satisfies the above constraints, one has lim a→∞sin[aη+f(η)] πη=δ(η). (3.9) As for ξ→+ 0 it holds that −ln(ξ/2)→∞ , with the help of the above relationship Eq. (3.7) becomes ∞/integraldisplay 0dxKiν(x)Kiν/prime(x) x=π2 2√νν/primesinh(πν)sinh(πν/prime)/bracketleftbig δ/parenleftbigν−ν/prime/parenrightbig +δ/parenleftbigν+ν/prime/parenrightbig/bracketrightbig . (3.10) Exploiting in Eq. (3.10) the following basic property of the delta distribution [6, Section 4.4]: g/parenleftbigη/prime/parenrightbig δ/parenleftbigη−η/prime/parenrightbig =g(η)δ/parenleftbigη−η/prime/parenrightbig , (3.11) one eventually arrives at the sought orthogonality relation ∞/integraldisplay 0dxKiν(x)Kiν/prime(x) x=π2 2νsinh(πν)/bracketleftbig δ/parenleftbigν−ν/prime/parenrightbig +δ/parenleftbigν+ν/prime/parenrightbig/bracketrightbig . (3.12) If we impose the constraint ν,ν/prime>0, then ν+ν/prime>0 and consequently in the distributional sense we have δ/parenleftbigν+ν/prime/parenrightbig =0/parenleftbigν,ν/prime>0/parenrightbig . (3.13) It is then evident that under the above restriction the orthogonality relation (3.12) turns into the one in Eq. (1.1). References [1] S.B. Yakubovich, A distribution associated with the Kontorovich–Lebedev transform, Opuscula Math. 26 (2006) 161–172. [2] A. Passian, H. Simpson, S. Kouchekian, S.B. Yakubovic, On the orthogonality of the MacDonald’s functions, J. Math. Anal. Appl. 360 (2009) 380–390 . [3] I. Stakgold, Green’s Functions and Boundary Value Problems, 2nd ed., Wiley, New York, 1998.[4] R. Szmytkowski, S. Bielski, An orthogonality relation for the Whittaker functions of the second kind of imaginary order, preprint arXiv:0910.14 92. [5] W. Magnus, F. Oberhettinger, R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed., Springer, Berlin, 1966 . [6] I.N. Sneddon, Fourier Transforms, McGraw–Hill, New York, 1951 [reprinted: Dover, New York, 1995].