Weber-Schafheitlin integrals with arbitrary exponent
aa r X i v : . [ m a t h . C A ] A p r Weber-Schafheitlin integrals with arbitrary exponent
Michał WrochnaNovember 9, 2018
Abstract
We present explicit formulae for Weber-Schafheitlin type integrals and give them an interpretation asthe kernel of a physically relevant operator related to the hamiltonian of Aharanov and Bohm. In particular,we derive explicit formulae for Weber-Schafheitlin type integrals with exponent larger or equal 1, whichare distributions on R + . We discuss several special cases. Our aim is to calculate the integral Z ∞ κ ρ J µ ( xκ ) J ν ( κ )d κ, (1.1)for suitable µ, ν ∈ C and x ∈ R + := (0 , ∞ ) , where J µ is the Bessel function of the first kind of order µ . Inthe literature, it is known as the Weber-Schafheitlin discontinuous integral with exponent ρ (or − ρ , dependingon the convention).In the case when Re ρ < (and under some additional assumptions on µ, ν ) it is convergent to a function, ingeneral not continuous in x = 1 . It has been derived in several ways and analysed in many special cases, forwhich we refer in particular to [W] and [DF]. It has been applied in numerous problems, let us mention hereonly two recent works — [HN], [SS].The case Re ρ ≥ is more delicate and requires a distributional approach. Nevertheless, it is quite natural toconsider it; indeed, it appears in some problems where it plays an important role ([KR], [KeR]). In additionto that, we show in this paper that it is the kernel of an operator physically relevant for the Aharanov-Bohmsystem. There have already been successful attempts to derive useful expressions for the distributional caseof (1.1) for special values of parameters, by Kellendonk and Richard [KR] (for ρ = 1 ), by Miroshin [M]( ρ → asymptotic) and by Salamon and Walter [SW] (recurrence formulae and special values of parameterswith ρ ∈ Z ). Motivated by the wish of exhausting all unsolved cases, we provide explicit formulae for (1.1)for arbitrary ρ with positive real part. We discuss also some special cases and compare them with the resultsmentioned earlier.The paper is constructed as follows. Section 2 serves as an additional motivation, linking the hamiltonianof Aharanov and Bohm to the integral we discuss further. In Section 3, we derive formulae for integralsinvolving the modified Bessel function of the second type K µ , as in the approach of Dixon and Ferrar [DF]. Acknowledgements
The author would like to thank especially J. Derezi´nski for suggesting the topic of this paperand proofreading, and V. Georgescu and S. Richard for useful advice. The author is also grateful to the MathematicsDepartment of Cergy-Pontoise University for hospitality and to the Government of France for financial support. A largepart of this paper was written at the Department of Mathematical Physics, University of Warsaw.
1e then use them to compute the integrals involving the Hankel functions of the first and second kind — H + µ , H − µ , Z ∞ κ ρ H ± µ ( xκ ) J ν ( κ )d κ, (1.2)closely related to (1.1). Our treatment of the integrals (1.2) is a generalization of the approach adopted in[KR]. We consider separately the cases Re ρ ≤ and Re ρ > . The second one is examinated in Section4 and includes in particular the distributional cases. The main result for the integral (1.1) is contained inProposition 1.1, and follows as a simple consequence of the computations of (1.2). In the following proposition, we give formulae for the Weber-Schaftheitlin integral (1.1) with exponent Re ρ > . The result is a distribution on R + (in general not regular), and its form depends on whether ρ is an integer number (which is not the case if Re ρ ≤ ). We use the notation for the rescaled Gausshypergeometric function F I1 ( a, b ; c ; z ) := 1Γ( c + 1) F ( a, b ; c ; z ) . Proposition 1.1
For any µ, ν ∈ C and Re ρ > satisfying Re ( ρ + ν + 1) > | µ | , and x ∈ R + , the integral R ∞ κ ρ J µ ( xκ ) J ν ( κ )d κ (1.1) equals: • for ρ / ∈ Z , ρ sin πρ (cid:20) (cid:8) ( x − − ρ − sin (cid:0) π − ρ − µ + ν (cid:1) + ( x − − ρ + sin (cid:0) π ρ − µ + ν (cid:1)(cid:9) x − ρ − ν (1 + x ) ρ F I1 (cid:0) − ρ + µ + ν , − ρ − µ + ν ; − ρ + 1; 1 − x − (cid:1) − sin (cid:0) π ρ − µ + ν (cid:1) Γ( ρ + µ + ν )Γ( ρ − µ + ν )Γ( − ρ + µ + ν )Γ( − ρ − µ + ν ) x − − ρ − ν F I1 (cid:0) ρ + µ + ν , ρ − µ + ν ; ρ + 1; 1 − x − (cid:1) (cid:21) • for ρ ∈ Z and − ρ ± µ + ν / ∈ Z , ρ π (cid:20) (cid:26) sin (cid:0) π ρ − µ + ν (cid:1) (cid:2) ( x − − ρ − + ( − ρ ( x − − ρ + (cid:3) + cos (cid:0) π ρ + µ − ν (cid:1) ( − ρ π δ ( ρ − (1 − x )( ρ − (cid:27) × x − ρ − ν (1 + x ) − ρ S µ,ν,ρ (1 − x − ) + ( − ρ Γ( ρ + µ + ν )Γ( ρ − µ + ν )Γ( − ρ + µ + ν )Γ( − ρ − µ + ν ) x − − ρ − ν (cid:26) sin (cid:0) π ρ − µ + ν (cid:1) T µ,ν,ρ (1 − x − ) − (cid:2) sin (cid:0) π ρ − µ + ν (cid:1) log x − ( x + 1) | x − | + cos (cid:0) π ρ − µ + ν (cid:1) πθ ( x − (cid:3) F I1 (cid:0) ρ + µ + ν , ρ − µ + ν ; ρ + 1; 1 − x − (cid:1) (cid:27)(cid:21) . where all the equalities hold in the sense of distributions on R + . In both cases, the RHS is the sum of adistribution multiplied by a smooth function (the first term) and of a locally integrable function (the secondterm). We have denoted θ ( x ) the Heaviside theta function, and the functions S µ,ν,ρ , T µ,ν,ρ are defined for | z | < by S µ,ν,ρ ( z ) := ρ − X k =0 (cid:0) − ρ + µ + ν (cid:1) k (cid:0) − ρ − µ + ν (cid:1) k (1 − ρ ) k k ! z k ,T µ,ν,ρ ( z ) := ∞ X k =0 (cid:0) ρ + µ + ν (cid:1) k (cid:0) ρ − µ + ν (cid:1) k ( ρ + k )! k ! z k × (cid:8) ψ ( k + 1) + ψ ( ρ + k + 1) − ψ ( ρ + µ + ν + k ) − ψ ( ρ − µ + ν + k ) (cid:9) , ith ψ ( y ) := dd y Γ( y ) / Γ( y ) . We use also the definition, for Re λ > − : ( x − λ − := ( | x − | λ , x < , x ≥ , ( x − λ + := ( , x < x − λ , x ≥ , and extend it analytically in the sense of distributions to all values of λ ∈ C (see Remark 4.2). In the following section, we quote some results obtained in [BDG] for the Aharonov-Bohm hamiltonian andconclude from them, that the Weber-Schafheitlin integral describes the integral kernel of a physically relevantoperator. We motivate thus the need for explicit formulae, valid in particular for the distributional Re ρ ≥ case.We consider the Hilbert space L ( R ) and denote its inner product ( ·|· ) . Since it will be convenient to usepolar coordinates r, φ on R , we introduce the unitary transformation L ( R ) ∋ f U f ∈ L (0 , ∞ ) ⊗ L ( − π, π ) given by U f ( r, φ ) = √ rf ( r cos φ, r sin φ ) , which allows us to identify L ( R ) with L (0 , ∞ ) ⊗ L ( − π, π ) .The Aharanov-Bohm hamiltonian in polar coordinates is H AB λ := − ∂ r − r ( ∂ φ + i λ ) , understood as the self-adjoint operator associated to the differential expression above (defined on an appro-priate domain). We allow for the moment the parameter λ to be any complex number.Since the self-adjoint operator L := − i ∂ φ has spectrum sp ( L ) = Z , we have the decomposition L ( R ) = ⊕ k ∈ Z H k where H k is the spectral subspace of L for the eigenvalue k . With the help of U we can identify H k with L ( R ) . Since L commutes with H AB λ , we obtain the decomposition U H AB λ U ∗ = ⊕ k ∈ Z H k + λ , where H µ acts as the differential operator − ∂ x + µ − x , when restricted to C c ( R + ) .We now assume µ > − . We gather some results from [BDG] about the operator H µ . We will need firstto define the following symmetric operator, corresponding up to a constant factor to the so-called Hankeltransformation: Definition 2.1 F µ is the operator on L (0 , ∞ ) given by ( F µ f ) ( k ) := Z ∞ J µ ( kx ) √ kxf ( x )d x We have then: 3 heorem 2.2
Let < a < b < ∞ and denote [ a,b ] the characteristic function of the corresponding interval.The integral kernel of [ a,b ] ( H µ ) is [ a,b ] ( H µ )( x, y ) = Z √ b √ a √ xyJ µ ( xκ ) J µ ( yκ ) κ d κ, considered as a quadratic form on C ∞ c ( R + ) , that is, explicitly: (cid:0) f | [ a,b ] ( H µ ) f (cid:1) = Z ∞ [ a,b ] ( κ ) | ( F µ f )( κ ) | d κ for any f ∈ C ∞ c ( R + ) . We may thus identify [ a,b ] ( H µ ) = F µ [ a,b ] ( Q ) F ∗ µ where Q is the self-adjoint position operator, and in consequence, F µ H µ F − µ = Q . Note that in particular, F µ is a unitary involution. For any γ ∈ C , one gets F µ H γµ F − µ = Q γ , and in the sense above, the integral kernel of H γµ is H γµ ( x, y ) = Z ∞ κ γ √ xyJ µ ( xκ ) J µ ( yκ ) κ d κ, which can be expressed in terms of the Weber-Schafheitlin integral with µ = ν .We quote also the following result concerning the wave operators for H µ , assuming now µ ∈ R : Theorem 2.3
For µ, ν > , the Møller wave operators Ω ± µ,ν associated to H µ , H ν exist and Ω ± µ,ν := lim t →±∞ e itH µ e − itH ν = e ± i( µ − ν ) π/ F µ F ν . In particular, the integral kernel of the operator Ω ± µ,ν H γν = H γµ Ω ± µ,ν may be useful in calculations. We haveby the above considerations: Ω ± µ,ν H γν = e ± i( µ − ν ) π/ F µ Q γ F ν and its integral kernel is equal to Ω ± µ,ν H γν ( x, y ) = e ± i( µ − ν ) π/ Z ∞ κ γ √ xyJ µ ( xκ ) J ν ( yκ ) κ d κ = e ± i( µ − ν ) π/ r xy y − γ − Z ∞ κ γ +1 J µ (cid:18) xy κ (cid:19) J ν ( κ ) d κ, where the last integral is of Weber-Schafheitlin type with exponent γ + 1 and argument xy . Re ρ < Proceeding as in [DF], we quote the following classic result [W] for the integral involving the modified Besselfunction of the first and second kind, I µ and K µ : 4 emma 3.1 For Re z > , | z | > , Re ( ν + ρ + 1) > | Re µ | , one has Z ∞ κ ρ K µ ( zκ ) I ν ( κ )d κ = Γ( ρ + µ + ν )Γ( ρ − µ + ν )2 ρ − z − − ρ − ν F I1 (cid:0) ρ + µ + ν , ρ − µ + ν ; ν + 1; z − (cid:1) . Proof.
The conditions for µ, ν, ρ , for the convergence of the integral, are established using asymptotic seriesfor the Bessel functions of the corresponding type.For the derivation of the integral, we first rescale the variable κ by the factor z − , expand I ν ( κ/z ) as a powerseries and then use, assuming | z | > , Z ∞ K µ ( κ ) κ β − = Z ∞ Z ∞ e − uκ ( u − µ − κ β − d u d κ = 2 β − Γ (cid:16) β − µ (cid:17) Γ (cid:16) β + µ (cid:17) , where we have substituted for K µ the corresponding integral representation, and computed the obtained ex-pression, integrating first with respect to κ . It remains to compare the obtained series with the hypergeometric F function on the RHS. ✷ We recall the relation between the Bessel function of the first kind J µ and I ν : I ν ( z ) = i − ν J ν (i z ) . Using Z ∞ κ ρ K µ ( zκ ) I ν ( κ )d κ = z − ρ − Z ∞ κ ρ K µ ( κ ) I ν ( κ/z )d κ, Z ∞ κ ρ K µ ( zκ ) J ν ( κ )d κ = z − ρ − Z ∞ κ ρ K µ ( κ ) J ν ( κ/z )d κ, we get as a straightforward corollary of Lemma 3.1: Corollary 3.2
For Re z > , Re ( ν + ρ + 1) > | Re µ | , Z ∞ κ ρ K µ ( zκ ) J ν ( κ )d κ = Γ( ρ + µ + ν )Γ( ρ − µ + ν )2 ρ − z − − ρ − ν F I1 (cid:0) ρ + µ + ν , ρ − µ + ν ; ν + 1; − z − (cid:1) . (3.1)Note that the superfluous assumption | z | > has been eliminated by analytic continuation with respect to z ,using the analyticity of the Gauss hypergeometric function F on C \ [ 1 , ∞ [ .We denote the Hankel function of the first and second kind, respectively — H + , H − , and recall that they arerelated to K µ as follows, for any y ∈ C : H + µ ( y ) = 2i π e − i πµ K µ ( − i y ) ,H − µ ( y ) = − π e i πµ K µ (i y ) . It follows that the integrals Z ∞ κ ρ H ± µ ( xκ ) J ν ( κ )d κ x ∈ R + are both the limiting case of R ∞ κ ρ K µ ( zκ ) J ν ( κ )d κ for purely imaginary z . We use the resultsobtained in Corollary 3.2 for Re z > , setting first z = ∓ i( x ± i ε ) , which gives: Z ∞ κ ρ H ± µ ( xκ ) J ν ( κ )d κ = lim ε ց Z ∞ κ ρ H ± µ ( x ± i εκ ) J ν ( κ )d κ = 2i π e ∓ i πµ lim ε ց Z ∞ κ ρ K µ ( ∓ i( x ± i ε ) κ ) J ν ( κ )d κ = ± ρ i π e ± i π ρ − µ + ν Γ( ρ + µ + ν )Γ( ρ − µ + ν ) lim ε ց f ( x ± i ε ) , where f ( x ± i ε ) := ( x ± i ε ) − − ρ − ν F I1 (cid:0) ρ + µ + ν , ρ − µ + ν ; ν + 1; ( x ± i ε ) − (cid:1) . Since we have also the relation J µ = (cid:0) H + µ + H − µ (cid:1) [W], it follows that Z ∞ κ ρ J µ ( xκ ) J ν ( κ )d κ = 2 ρ − i π Γ( ρ + µ + ν )Γ( ρ − µ + ν ) (cid:20) e i π ρ − µ + ν lim ε ց f ( x + i ε ) − e − i π ρ − µ + ν lim ε ց f ( x − i ε ) (cid:21) . (3.2)Therefore, in order to derive the Weber-Schafheitlin integral, as well as the integrals involving H ± µ , it isenough to examine the limit lim ε ց f ( x ± i ε ) . Note that f ( z ) depends on the parameters µ, ν, ρ and it willfollow that it is convenient to treat the cases Re ρ < and Re ρ > separately. Proposition 3.3
For any µ, ν ∈ C and Re ρ < satisfying Re ( ρ + ν + 1) > | µ | , and x ∈ R + , the integral R ∞ κ ρ J µ ( xκ ) J ν ( κ )d κ is equal to: ρ Γ( ρ + µ + ν )Γ( − ρ − µ + ν ) x µ F I1 (cid:0) ρ + µ + ν , ρ + µ − ν ; µ + 1; x − (cid:1) for x < , and ρ Γ( ρ + µ + ν )Γ( − ρ + µ − ν ) x − − ρ − ν F I1 (cid:0) ρ + µ + ν , ρ − µ + ν ; ν + 1; x − (cid:1) for x > . Proof.
Consider the factor F I1 (cid:0) ρ + µ + ν , ρ − µ + ν ; ν + 1; ( x ± i ε ) − (cid:1) , appearing in the definition of f ( x +i ε ) . Denoting a := ρ + µ + ν , b := ρ − µ + ν and c := ν +1 , we check that Re ( c − a − b ) = − Re ρ < ,which ensures the limit ε ց exists.For x > , the argument of the F function has real part smaller than , thus by analyticity it follows thatthe limits with +i ε and − i ε coincide. Obtaining the desired expression is then just a matter of rewriting thephase factors in terms of Γ functions, using (cid:16) e i π ρ − µ + ν − e − i π ρ − µ + ν (cid:17) = sin (cid:0) π ρ − µ + ν (cid:1) = π Γ (cid:0) ρ + ν − µ (cid:1) Γ (cid:0) − ρ − ν + µ (cid:1) . For x < , we can use the result above with µ and ν interchanged, thanks to Z ∞ κ ρ J µ ( xκ ) J ν ( κ )d κ = x − ρ − Z ∞ κ ρ J µ ( κ ) J ν ( κ/x )d κ. ✷ The case Re ρ > Assuming Re ρ > , we examine the limit lim ε ց f ( x ± i ε ) = lim ε ց ( x ± i ε ) − − ρ − ν F I1 (cid:0) ρ + µ + ν , ρ − µ + ν ; ν + 1; ( x ± i ε ) − (cid:1) . Denoting a := ρ + µ + ν , b := ρ − µ + ν and c := ν + 1 , we check that Re ( c − a − b ) = − Re ρ > , whichimplies a singular behaviour of the F I1 factor at x = 1 in the limit ε → . We therefore use the followingsymmetry of the F I1 function: F I1 (cid:0) a, b ; c ; z − (cid:1) = (1 − z − ) c − a − b F I1 (cid:0) c − a, c − b ; c ; z − (cid:1) = (1 − z − ) − ρ F I1 (cid:0) − ρ + µ + ν , − ρ − µ + ν ; ν + 1; z − (cid:1) , in order to isolate the whole singular factor (1 − z − ) − ρ , the second term on the RHS being well defined forall z ∈ C . Consequently, f ( x ± i ε ) = ( x ± i ε ) − − ρ − ν (1 − ( x ± i ε ) − ) − ρ F I1 (cid:0) − ρ + µ + ν , − ρ − µ + ν ; ν + 1; ( x ± i ε ) − (cid:1) = ( x ± i ε ) − ρ − ν ( x + 1 ± i ε ) − ρ ( x − ± i ε ) − ρ F I1 (cid:0) − ρ + µ + ν , − ρ − µ + ν ; ν + 1; ( x ± i ε ) − (cid:1) . Because of the singular part (i.e. ( x − ± i ε ) − ρ ), the limit ε ց requires a distributional approach. Let usconcentrate now on the first factors of the above expression. We will need the following: Definition 4.1
We define ( x − ± i0) λ := lim ε ց ( x − ± i ε ) λ , as distributions on R + . Remark 4.2
The distributions ( x − ± i0) λ are well defined and can be represented in a more explicit wayas follows (see [H] for properties of the analogously defined distributions ( x ± i0) λ on R ): • for Re λ > − , we have the equality between locally integrable functions ( x − ± i0) λ = e ± i λπ ( x − λ − + ( x − λ + , where: ( x − λ − := ( | x − | λ , x < , x ≥ , ( x − λ + := ( , x < x − λ , x ≥ • for Re λ > − k and λ / ∈ Z , where k ∈ N , we have the equality ( x − ± i0) λ = e ± i λπ ( x − λ − + ( x − λ + , where ( x − λ ± are now distributions defined by their action on an arbitrary test function φ ∈ C ∞ c ( R + ) (or equivalently, as the analytic continuation of the distributions defined in the preceding case): (cid:10) ( x − λ − , φ (cid:11) := Z (1 − x ) λ + k φ ( k ) ( x ) / (( λ + 1) . . . ( λ + k )) d x, (cid:10) ( x − λ + , φ (cid:11) := ( − k Z ∞ ( x − λ + k φ ( k ) ( x ) / (( λ + 1) . . . ( λ + k )) d x. for λ = − k , where k ∈ N , we have ( x − ± i0) − k := ( x − − k − + ( − k ( x − − k + ± ( − k i π δ ( k − ( x − k − , where (cid:10) ( x − − k − , φ (cid:11) := ( − k − Z log(1 − x ) φ ( k ) ( x )( k − x + ( − k − φ ( k − (1) (cid:16)P k − j =1 j − (cid:17) ( k − , (cid:10) ( x − − k + , φ (cid:11) := − Z ∞ log( x − φ ( k ) ( x )( k − x + φ ( k − (1) (cid:16)P k − j =1 j − (cid:17) ( k − . Lemma 4.3
Let λ ∈ C and let g ( z ) be a function holomorphic in the neighborhood of the halfline R + . Then lim ε ց (cid:2) g ( x + i ε )( x − ± i ε ) λ (cid:3) = g ( x )( x − ± i0) λ , in the sense of distributions on R + . Proof.
It is clear that lim ε ց (cid:2) g ( x + i ε )( x − ± i ε ) λ (cid:3) = g ( x )( x − ± i0) λ , (4.1)for Re λ > − , since the RHS is then a regular distribution and the convergence of each of the factors asfunctions is uniform on every compact subset of R + . Assuming λ = 0 , we differentiate (4.1) and obtain lim ε ց (cid:2) g ′ ( x + i ε )( x − ± i ε ) λ + λg ( x + i ε )( x − ± i ε ) λ − (cid:3) = g ′ ( x )( x − ± i0) λ + λg ( x )( x − ± i0) λ − , where we have used the fact that dd x ( x − ± i0) λ = λ ( x − ± i0) λ − [H]. We then use (4.1) again tosubstract the first term of both sides. We have thus proved (4.1) for Re λ > − , λ = − .We consider the case λ = − separately, and differentiate instead the equality between the locally integrablefunctions: lim ε ց [ g ( x + i ε ) log( x − ± i ε )] = g ( x ) log( x − ± i0) , where log( x − ± i0) := lim ε ց log( x − ± i ε ) = log | x − | ± i πθ ( x − , and its distributional derivativeis ( x − ± i0) − (this can be seen by setting first g ( x ) ≡ in the above equality and differentiating).By induction, we prove (4.1) for the remaining values of λ ∈ C . ✷ Recalling our expression for f ( x + i ε ) , we have f ( x + i ε ) = ( x + i ε ) − ρ − ν (1 + x + i ε ) − ρ ( x − ε ) − ρ q ( x + i ε ) , where q ( x + i ε ) := F I1 (cid:0) − ρ + µ + ν , − ρ − µ + ν ; ν + 1; ( x + i ε ) − (cid:1) , which converges uniformly on every compact subset of R + to q ( x + i0) := F I1 (cid:0) − ρ + µ + ν , − ρ − µ + ν ; ν + 1; x − (cid:1) , considered as a single-valued function. Note that the Gauss hypergeometric function F has a branch cut [ 1 , ∞ [ . Here, the limit is taken by approaching the halfline from below ( Im ( x + i ε ) − < ), and wehave denoted F ( a, b ; c ; x ) = lim ǫ ց F ( a, b ; c ; x − i ǫ ) on the branch cut. On the other hand, the limit lim ε ց f ( x − i ε ) corresponds to approaching the halfline from above in the argument of the F factor,therefore q ( x − i0) := lim ε ց q ( x − i ε ) is not equal to q ( x + i0) . 8 emark 4.4 The function x q ( x ± i0) is not differentiable, and therefore the meaning of the product ( x − ± i0) − ρ q ( x ± i0) is unclear. To prove that it exists despite this apparent problem, we show that q ( x ± i0) can be written as q ( x ± i0) = h ( x ) + ( x − ± i0) ρ h ± ( x ) , (4.2) where h ( x ) is smooth and both h +2 ( x ) , h − ( x ) belong to L loc1 ( R + ) . Then, the equality ( x − ± i0) − ρ q ( x ± i0) := ( x − ± i0) − ρ h ( x ) + h ± ( x ) defines the desired product well, being the sum of a distribution multiplied by a smooth function and of alocally integrable function. Proof.
For ρ / ∈ Z , the decomposition (4.2) is possible due to the following formula, holding for z ∈ C ([BS],eq. (B.9)): F I1 (cid:0) − ρ + µ + ν , − ρ − µ + ν ; ν + 1; z (cid:1) = π sin πρ (cid:26) ρ + µ + ν )Γ( ρ − µ + ν ) F I1 (cid:0) − ρ + µ + ν , − ρ − µ + ν ; − ρ + 1; 1 − z (cid:1) − − ρ + µ + ν )Γ( − ρ − µ + ν ) (1 − z ) ρ F I1 (cid:0) ρ + µ + ν , ρ − µ + ν ; ρ + 1; 1 − z (cid:1) (cid:27) . (4.3)We take z = ( x ± i ε ) − and pass to the limit ε ց with both the expressions. Note that the only differencebetween the limit with +i ε and − i ε appears in the factor lim ε ց (1 − ( x ± i ε ) − ) ρ = x − ρ (1+ x ) ρ ( x − ± i0) ρ ,since the F factors on the RHS are analytic on the required domain. It is clear that (4.2) holds, with h +2 ( x ) = h − ( x ) in particular.For ρ ∈ Z , we have instead ([BS], eq. (B.10)): F I1 (cid:0) − ρ + µ + ν , − ρ − µ + ν ; ν + 1; z (cid:1) =1Γ( ρ + µ + ν )Γ( ρ − µ + ν ) ρ − X k =0 ( − k ( ρ − k − (cid:0) − ρ + µ + ν (cid:1) k (cid:0) − ρ − µ + ν (cid:1) k k ! (1 − z ) k +( − ρ Γ( − ρ + µ + ν )Γ( − ρ − µ + ν ) (1 − z ) ρ ∞ X k =0 (cid:0) ρ + µ + ν (cid:1) k (cid:0) ρ − µ + ν (cid:1) k ( ρ + k )! k ! { ψ ( k + 1) + ψ ( ρ + k + 1) − ψ ( ρ + µ + ν + k ) − ψ ( ρ − µ + ν + k ) − log(1 − z ) (cid:9) (1 − z ) k , for − ρ ± µ + ν / ∈ Z , where ψ ( y ) = Γ ′ ( y ) / Γ( y ) .As in the preceding case, we take z = ( x ± i ε ) − and pass to the limit. We obtain a similar decomposition.Note, however, that the limits log(1 − x ∓ i0) differ, and thus h +2 ( x ) = h − ( x ) The cases where at least one of the parameters − ρ ± µ + ν is an integer are treated the same way, only thefunctions h ( x ) h ± ( x ) being then different ([BS], eq. (B.11), (B.12)). ✷ Proposition 4.5
In the sense of distributions on R + , lim ε ց f ( x ± i ε ) = x − ρ − ν ( x + 1) − ρ (cid:2) ( x − ± i0) − ρ h ( x ) + h ± ( x ) (cid:3) , where the functions h ( x ) and h ± ( x ) are defined in Remark 4.4. Proof.
We have to prove that as ε ց , ( x − ± i ε ) − ρ ( x ± i ε ) − ρ − ν ( x + 1 ± i ε ) − ρ h ( x ± i ε ) −→ ( x − ± i0) − ρ x − ρ − ν ( x + 1) − ρ h ( x ) ( x ± i ε ) − ρ − ν ( x + 1 ± i ε ) − ρ ( x − ± i ε ) − ρ ( x − ± i ε ) ρ h ( x ± i ε ) −→ x − ρ − ν (1 + x ) − ρ h ± ( x ) . The first limit is a consequence of Lemma 4.3. The second one is clearly true, since the convergence of thecorresponding functions is uniform on each compact subset of R + . ✷ Remark 4.6
The case when ρ ∈ Z and at least one of the numbers − ρ ± µ + ν is an integer are treatedsimilarly. One can deduce from the expansions given in [BS] (eq. (B.11), (B.12)), the explicit expressions forthe functions h ( x ) and h ± ( x ) , following step by step the proof of Remark 4.4 in the degenerate case. Corollary 4.7
Using Equation 3.2 and Proposition 4.5, we get the results gathered in Proposition 1.1. Theformulae in Proposition 1.1 are written in terms of the distributions ( x − − ρ ± rather than ( x − ± i0) − ρ ,using the relations listed in Remark 4.2. We end up commenting on some special cases, involving much simplier expressions.
Remark 4.8
An explicit formula in the special case ρ = 1 has been derived in [KR]. It can be recoveredfrom our general expression, by substituting the (well-known) equality ( x − ± i0) − = Pv (cid:18) x − (cid:19) ∓ i πδ ( x − , where Pv denotes the Cauchy principal value. Furthermore, S µ,ν, ( x ) ≡ by definition, and it remains touse (4.3) back again to get F functions with argument x ± instead of − x ± . Remark 4.9
Much simplier formulae can be derived in the special case − ρ + µ ± ν = 0 , since the Gausshypergeometric function with a parameter set to zero is trivial, i.e. F (0 , . . . ; · ) ≡ . Note that the above remarks hold for the integral involving H ± µ instead of J µ as well. References [BDG] Bruneau L., Derezi´nski J., Georgescu V.:
Homogeneous operators on halfline , arXiv:0911.5569v1,2009[BS] Becken W., Schmelcher P.:
The analytic continuation of the Gaussian hypergeometric function forarbitrary parameters , Journal of Computational and Applied Mathematics 126, p. 449-478, 2000[DF] Dixon A. L., Ferrar W. L.:
Infinite integrals in the theory of Bessel Functions , Quarterly Journal ofMathematics 126, p. 122-145, 1930[H] H¨ormander I.:
The analysis of linear partial differential operators , volume 1 , 2nd edition, Springer,1990[HN] Hongo K., Naqvi Q.A.:
Diffraction of electromagnetic wave by disk and circular hole in a perfectlyconducting plane , Progress in Electromagnetics Research, 113150, 2007[KeR] Keating J.P.1; Robbins J.M:
Force and impulse from an Aharonov-Bohm flux line , Journal of PhysicsA: Mathematical and General, vol. 34 no. 4, 200110KR] Kellendonk J., Richard S.:
Weber-Schafheitlin type integrals with exponent 1 , Integral Transforms andSpecial Functions 20 no. 2, 2009[KR] Kellendonk J., Richard S.:
New formulae for the Aharonov-Bohm wave operators , arXiv:0811.3963,2008[M] Miroshin, R.N.:
An asymptotic series for the Weber-Schafheitlin integral , Math. Notes 70, no. 5-6 p.682687, 2001[SS] Suzuki H., Sato H.-T.:
On Bogoliubov transformation of scalar wave functions in de Sitter space ,Mod. Phys. Lett. A9 3673-3684, 1994[SW] Salamon N.J., Walter G.G.:
Limits of Lipschitz-Hankel Integrals , J. Inst. Maths Applics 24, 237-254,1979[W] Watson G.N.:
A treatise on the theory of Bessel functions , 2nd edition, Cambridge University Press,1966 M ICHAŁ W ROCHNA , Research Training Group “Mathematical Structures in Modern Quantum Physics”,Mathematisches Institut, Universit¨at G¨ottingen, Bunsenstr. 3-5, D - 37073 G¨ottingen, Germanye-mail: [email protected]@uni-math.gwdg.de