Integral and Series Representations of the Dirac Delta Function
IINTEGRAL AND SERIES REPRESENTATIONSOF THE DIRAC DELTA FUNCTION
Y. T. Li
Department of Mathematics, City University of Hong Kong,Tat Chee Avenue, Kowloon, Hong Kong.
R. Wong
Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong,Tat Chee Avenue, Kowloon, Hong Kong.
Abstract.
Mathematical justifications are given for several integral and seriesrepresentations of the Dirac delta function which appear in the physics liter-ature. These include integrals of products of Airy functions, and of Coulombwave functions; they also include series of products of Laguerre polynomialsand of spherical harmonics. The methods used are essentially based on theasymptotic behavior of these special functions. Introduction
The Dirac delta function δ ( x ) has been used in physics well before the theoryof distributions (generalized functions) was introduced by mathematicians. Themanner in which physicists used this function was to define it by the equations(1.1) δ ( x − a ) = 0 , x (cid:54) = a, and(1.2) (cid:90) ∞−∞ φ ( x ) δ ( x − a ) dx = φ ( a ) , a ∈ R , for any continuous function φ ( x ) on R . However, mathematically, these two equa-tions are inconsistent in the classical sense of a function and an integral, since thevalue of the integral of a function which is zero everywhere except for a finite numberof points should be zero. There are now two mathematically meaningful approachesto help us interpret the delta function given in (1 . − (1 . δ a := δ ( x − a ) as a continuous linear functional acting on a space of smoothfunctions with rapid decay at ±∞ , and the action of δ a on a particular function φ ( x ) is given the value φ ( a ); see [14, p.141] and [15, p.77]. The other approach isto find a sequence of functions δ n ( x − a ) such that(1.3) lim n →∞ (cid:90) ∞−∞ δ n ( x − a ) φ ( x ) dx = φ ( a ) , a ∈ R ;see [7, p.55] and [10, p.17]. Such a sequence is called a delta sequence and we write,symbolically,(1.4) lim n →∞ δ n ( x − a ) = δ ( x − a ) , x ∈ R . Mathematics Subject Classification.
Primary: 46F99; Secondary: 33C10, 33C45.
Key words and phrases.
Dirac delta function, Liouville-Green (WKB) approximation, Airyfunction, Coulomb wave function, Laguerre polynomials, spherical harmonics. a r X i v : . [ m a t h . C A ] M a r Y. T. LI AND R. WONG
It seems that the second approach is more acceptable to physicists and appliedmathematicians.Recently, in the process of preparing some material for the major project “NISTHandbook of Mathematical Functions [13]”, we encountered some very interestingintegral and series representations of the delta function which need mathematicaljustification. For instance, in [3, p.696] the formula(1.5) (cid:90) ∞ xtJ ν ( xt ) J ν ( at ) dt = δ ( x − a ) , Re ν > − , x > , a > , appears, where J ν ( x ) is the Bessel function of the first kind, and in [16, Eq.(122)]one finds the integral representation(1.6) (cid:90) ∞ s ( x, l ; t ) s ( a, l ; t ) dt = δ ( x − a ) , a > , x > , where s ( x, l ; t ) is the Coulomb wave function. A recent reference [17, p.57] on theAiry function Ai( x ) also gives the formula(1.7) (cid:90) ∞−∞ Ai( t − x )Ai( t − a ) dt = δ ( x − a ) . While physicists may find these representations convenient to use in applications,mathematicians would, in general, feel uneasy or even disturbed to see these for-mulas being used since the integrals in (1 . − (1 .
7) are all divergent. Thus, itwould seem meaningful and necessary to give a mathematical justification for theserepresentations, and this is exactly the purpose of the present paper.There are also some series representations for the delta function. These includethe following:(1.8) ∞ (cid:88) k =0 (cid:18) k + 12 (cid:19) P k ( x ) P k ( a ) = δ ( x − a ) , (1.9) e − ( x + a ) / √ π ∞ (cid:88) k =0 k k ! H k ( x ) H k ( a ) = δ ( x − a ) , and(1.10) e − ( x + a ) / ∞ (cid:88) k =0 L k ( x ) L k ( a ) = δ ( x − a ) , where P k ( x ), H k ( x ) and L k ( x ) are, respectively, the Legendre, Hermite and La-guerre polynomials. Equations (1 . − (1 .
10) are special cases of an equation inMorse and Feshbach [11, p.729]. Another series representation is given in [3, p.792];that is, ∞ (cid:88) k =0 k (cid:88) l = − k Y kl ( θ , φ ) Y ∗ kl ( θ , φ ) = 1sin θ δ ( θ − θ ) δ ( φ − φ )= δ (cos θ − cos θ ) δ ( φ − φ ) , (1.11)where the functions Y kl ( θ, φ ) are the spherical harmonics (see [3, p.788]) and theasterisk “ ∗ ” denotes complex conjugate.The orthogonal polynomials in (1 . − (1 .
10) and the orthogonal function in (1.11)are the eigenfunctions corresponding to the eigenvalues (discrete spectrum) of somedifferential operators. Likewise, the special functions in (1 . − (1 .
7) can be regarded
HE DIRAC DELTA FUNCTION 3 as the eigenfunctions associated with the continuous spectrum of correspondingdifferential operators. The proofs of the representations in (1 . − (1 .
11) turn outto be much simpler than the proofs of those in (1 . − (1 . . − (1 .
11) all follow from expansion theorems in orthogonalpolynomials, whereas for the representations in (1 . − (1 .
7) we need to providesome new arguments.2.
A generalized Riemann-Lebesgue Lemma
There are already several delta sequences in the literature. For instance, we have δ n ( x − a ) = (cid:114) nπ e − n ( x − a ) , (2.1) δ n ( x − a ) = nπ
11 + n ( x − a ) , (2.2)and(2.3) δ n ( x − a ) = 1 π sin n ( x − a ) x − a ;see [5, pp. 35-38] and [8, pp. 5-13]. To verify whether a given sequence of functionsis a delta sequence, one can apply the criteria given in [5, p.34]. If the function φ ( x )in (1.3) is only piecewise continuous in R , then this equation becomes(2.4) lim n →∞ (cid:90) ∞−∞ δ n ( x − a ) φ ( x ) dx = 12 [ φ ( a + ) + φ ( a − )] , a ∈ R ;see [8, p.16].For convenience in our later argument, we also state and prove the followingresult. Lemma. (A generalized Riemann-Lebesgue lemma) . Let g ( x, R ) be a continuousfunction of x ∈ ( A, B ) and uniformly bounded for R > . If (2.5) lim R →∞ (cid:90) B (cid:48) A (cid:48) g ( x, R ) dx = 0 for any A (cid:48) and B (cid:48) with A < A (cid:48) < B (cid:48) < B , then (2.6) lim R →∞ (cid:90) BA ψ ( x ) g ( x, R ) dx = 0 for any integrable function ψ ( x ) on the finite interval ( A, B ) . If A = 0 and B = + ∞ ,or if A = −∞ and B = + ∞ , then (2.6) holds for any absolutely integrable function ψ ( x ) on the infinite interval ( A, B ) .Proof. First, from (2.5) it is easy to see that (2.6) holds for step functions. Now,let ψ ( x ) be an integrable function on ( A, B ). For any ε >
0, we can always find astep function s ( x ) such that(2.7) (cid:90) BA | ψ ( x ) − s ( x ) | dx < ε K , where K = max {| g ( x, R ) | : x ∈ R and R > } . Choose R > (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) BA s ( x ) g ( x, R ) dx (cid:12)(cid:12)(cid:12)(cid:12) < ε R ≥ R . Y. T. LI AND R. WONG
Write (cid:90) BA ψ ( x ) g ( x, R ) dx = (cid:90) BA [ ψ ( x ) − s ( x )] g ( x, R ) dx + (cid:90) BA s ( x ) g ( x, R ) dx. On account of (2.7) and (2.8), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) BA φ ( x ) g ( x, R ) dx (cid:12)(cid:12)(cid:12)(cid:12) < ε for all R ≥ R . Since ε is arbitrary, this proves (2.6) when A and B are finite.If the interval of integration is infinite, and if ψ ( x ) is absolutely integrable there,then we can choose finite numbers A and B such that the integral (cid:82) ψ ( x ) g ( x, R ) dx outside the interval ( A, B ) is small since g ( x, R ) is uniformly bounded. On the finiteinterval ( A, B ), we can apply the result just established. (cid:3) Bessel function
The Bessel function J ν ( xt ) is a solution of the differential equation(3.1) ddt (cid:18) t dydt (cid:19) + (cid:18) x t − ν t (cid:19) y = 0 . With x replaced by a , one obtains a corresponding equation for J ν ( at ). Multiplyingequation (3.1) by J ν ( at ) and the corresponding equation for J ν ( at ) by J ν ( xt ), andsubtracting the two resulting equations, leads to( x − a ) tJ ν ( at ) J ν ( xt ) = ddt { atJ ν ( xt ) J (cid:48) ν ( at ) − xtJ ν ( at ) J (cid:48) ν ( xt ) } . (3.2)Put(3.3) δ R ( x, a ) = x (cid:90) R J ν ( xt ) J ν ( at ) t dt. An integration of (3.2) gives(3.4) δ R ( x, a ) = xx − a (cid:20) at J ν ( xt ) J (cid:48) ν ( at ) − xt J ν ( at ) J (cid:48) ν ( xt ) (cid:21) R . From the ascending power series representation(3.5) J ν ( t ) = (cid:18) t (cid:19) ν ∞ (cid:88) n =0 ( − n Γ( n + ν + 1) n ! (cid:18) t (cid:19) n , it can be shown that the leading terms in the series expansions of at J ν ( xt ) J (cid:48) ν ( at )and xt J ν ( at ) J (cid:48) ν ( xt ) cancel out. Thus, the right-hand side of (3.4) vanishes at thelower limit when ν > −
1, and we obtain(3.6) δ R ( x, a ) = xx − a [ aRJ ν ( xR ) J (cid:48) ν ( aR ) − xRJ ν ( aR ) J (cid:48) ν ( xR )] . Theorem 1.
For a > , ν > − and any piecewise continuously differentiablefunction φ ( x ) on (0 , ∞ ) , we have (3.7) lim R →∞ (cid:90) ∞ φ ( x ) x (cid:18)(cid:90) R J ν ( xt ) J ν ( at ) t dt (cid:19) dx = 12 [ φ ( a + ) + φ ( a − )] , provided that HE DIRAC DELTA FUNCTION 5 (i) (cid:82) ∞ x − | φ ( x ) | dx converges; (ii) (cid:82) x | φ ( x ) | dx converges when ν ≥ − , or (ii (cid:48) ) (cid:82) x ν +1 | φ ( x ) | dx converges when − < ν < − .Proof. In view of the asymptotic formulas(3.8) J ν ( xt ) = (cid:114) πxt (cid:20) cos (cid:18) xt − νπ − π (cid:19) + ε ( x, t ) (cid:21) and(3.9) J (cid:48) ν ( xt ) = − (cid:114) πxt (cid:20) sin (cid:18) xt − νπ − π (cid:19) + ε ( x, t ) (cid:21) where ε j ( x, t ) = O (1 /t ) as t → ∞ uniformly for x ≥ δ > j = 1 ,
2, there areconstants M > M > | J ν ( aR ) | ≤ M R − , | J (cid:48) ν ( aR ) | ≤ M R − and(3.11) | J ν ( xR ) | ≤ M x − R − , | J (cid:48) ν ( xR ) | ≤ M x − R − for x ≥ R ≥
1. From (3.6), it follows that | δ R ( x, a ) | ≤ M M x | x − a | (cid:18) ax / + 1 x / (cid:19) . Hence, for b > max { a, } , we have (cid:90) ∞ b | δ R ( x, a ) φ ( x ) | dx ≤ M (cid:90) ∞ b x − | φ ( x ) | dx, where M = M M ( a + 1) b / ( b − a ). Since the last integral is convergent bycondition (i), for any ε > c > b such that(3.12) (cid:90) ∞ c | δ R ( x, a ) φ ( x ) | dx < ε . Let 0 < ρ < min { a, } . To estimate the integral of δ R ( x, a ) φ ( x ) on the interval(0 , ρ ), we divide our discussion into two cases: (i) ν ≥ − , and (ii) − < ν < − .In the first case, we have from (3.5) a positive constant M (cid:48) such that(3.13a) | J ν ( xR ) | ≤ M (cid:48) ( xR ) ν ≤ M (cid:48) ( xR ) − and(3.13b) | J (cid:48) ν ( xR ) | ≤ M (cid:48) ( xR ) ν − ≤ M (cid:48) x − R − ≤ M (cid:48) x − R − for 0 < xR ≤ R ≥
1. From (3.8) and (3.9), we also have(3.14) | J ν ( xR ) | ≤ M (cid:48)(cid:48) x − R − , | J (cid:48) ν ( xR ) | ≤ M (cid:48)(cid:48) x − R − for xR ≥
1. Coupling (3.13) and (3.14), we obtain | J ν ( xR ) | ≤ M x − R − , | J (cid:48) ν ( xR ) | ≤ M x − R − for 0 < x ≤ R ≥
1, with M = max { M (cid:48) , M (cid:48)(cid:48) } . Thus, | δ R ( x, a ) | ≤ M M ( a + 1) x a − ρ , < x ≤ , Y. T. LI AND R. WONG and(3.15) (cid:90) ρ | δ R ( x, a ) φ ( x ) | dx ≤ M (cid:90) ρ x | φ ( x ) | dx, where M = M M ( a + 1) / ( a − ρ ).In the second case, there are constants M (cid:48) > M (cid:48)(cid:48) > | J ν ( xR ) | ≤ M (cid:48) ( xR ) ν ≤ M (cid:48) x ν R − , | J (cid:48) ν ( xR ) | ≤ M (cid:48) ( xR ) ν − ≤ M (cid:48) x ν − R − for 0 < xR ≤ R ≥
1, and | J ν ( xR ) | ≤ M (cid:48)(cid:48) x − R − ≤ M (cid:48)(cid:48) x ν R − , | J (cid:48) ν ( xR ) | ≤ M (cid:48)(cid:48) x − R − ≤ M (cid:48)(cid:48) x ν − R − for xR ≥ R ≥
1. With M = max { M (cid:48) , M (cid:48)(cid:48) } , it follows that | J ν ( xR ) | ≤ M x ν R − and | J (cid:48) ν ( xR ) | ≤ M x ν − R − . Therefore, | δ R ( x, a ) | ≤ M M a − ρ ( a + 1) x ν +1 and(3.16) (cid:90) ρ | δ R ( x, a ) φ ( x ) | dx ≤ M (cid:90) ρ x ν +1 | φ ( x ) | dx, where M = M M ( a + 1) / ( a − ρ ). On account of conditions (ii) and (ii (cid:48) ), for any ε > < d < ρ such that(3.17) (cid:90) d | δ R ( x, a ) φ ( x ) | dx < ε . For d < x < c , a combination of (3.6), (3.8) and (3.9) gives δ R ( x, a ) = 2 π xx − a (cid:26)(cid:114) xa sin ζ ( x, R ) cos ζ ( a, R ) − (cid:114) ax cos ζ ( x, R ) sin ζ ( a, R ) + ε ( x, a ; R ) (cid:27) , (3.18)where c and d are given in (3.12) and (3.17), respectively, ζ ( x, R ) = xR − νπ − π and ε ( x, a ; R ) = O (1 /R ) as R → ∞ uniformly for x ∈ ( d, c ). Note that ε ( x, a ; R )is continuously differentiable, ε ( x, a ; R ) / ( x − a ) is continuous in x and uniformlybounded in x and R , and φ ( x ) is piecewise continuous in ( d, c ). Hence,(3.19) lim R →∞ (cid:90) cd ε ( x, a ; R ) x − a xφ ( x ) dx = 0 . By the Riemann-Lebesgue lemma, we also have(3.20) lim R →∞ (cid:90) cd (cid:18)(cid:114) xa − (cid:19) sin ζ ( x, R ) x − a xφ ( x ) dx = 0and(3.21) lim R →∞ (cid:90) cd (cid:18)(cid:114) ax − (cid:19) cos ζ ( x, R ) x − a xφ ( x ) dx = 0 . HE DIRAC DELTA FUNCTION 7
A combination of the results in (3 . − (3 .
21) yieldslim R →∞ (cid:90) cd δ R ( x, a ) φ ( x ) dx = lim R →∞ π (cid:90) cd sin { ζ ( x, R ) − ζ ( a, R ) } x − a xφ ( x ) dx = lim R →∞ (cid:90) cd sin R ( x − a ) π ( x − a ) · xx + a φ ( x ) dx On account of Jordan’s theorem on the Dirichlet kernel [2, p.473], we conclude(3.22) lim R →∞ (cid:90) cd δ R ( x, a ) φ ( x ) dx = 12 [ φ ( a − ) + φ ( a + )] . Since the number ε in (3.12) and (3.17) is arbitrary, it follows from (3.22) thatlim R →∞ (cid:90) ∞ δ R ( x, a ) φ ( x ) dx = 12 [ φ ( a − ) + φ ( a + )] , which is equivalent to (3.7). (cid:3) Coulomb wave function
The Coulomb wave function s ( x, l ; r ) is a solution of the Coulomb wave equation(4.1) d ydr + (cid:26) x + (cid:18) r − l ( l + 1) r (cid:19)(cid:27) y = 0 , that satisfies the initial conditions(4.2) s ( x, l ; 0) = s (cid:48) ( x, l ; 0) = 0and has the asymptotic behavior s ( x, l ; r ) = 1 √ π x − [sin ζ ( x, l ; r ) + ε ( x, l ; r )] , (4.3) s (cid:48) ( x, l ; r ) = 1 √ π x [cos ζ ( x, l ; r ) + ε ( x, l ; r )] , (4.4)where ε j ( x, l ; r ) = O (1 /r ) as r → ∞ uniformly for x ≥ δ > , j = 1 ,
2, and(4.5) ζ ( x, l ; r ) = kr + 1 k ln (2 kr ) − lπ (cid:18) l + 1 − ik (cid:19) with k = √ x ; see [16, p.236]. For a > x >
0, we define(4.6) δ R ( x, a ) = (cid:90) R s ( x, l ; r ) s ( a, l ; r ) dr. From (4.1) and (4.2), one can show as in Sec. 3 that(4.7) δ R ( x, a ) = s ( x, l ; R ) s (cid:48) ( a, l ; R ) − s (cid:48) ( x, l ; R ) s ( a, l ; R ) x − a . Theorem 2.
For any a > and any piecewise continuously differentiable function φ ( x ) on (0 , ∞ ) , we have (4.8) lim R →∞ (cid:90) ∞ φ ( x ) (cid:90) R s ( x, l ; r ) s ( a, l ; r ) dr dx = 12 [ φ ( a + ) + φ ( a − )] , provided that the integrals (4.9) (cid:90) x − | φ ( x ) | dx and (cid:90) ∞ x − | φ ( x ) | dx are convergent. Y. T. LI AND R. WONG
Proof.
From formulas (4.3) and (4.4), for any b > max { a, } there exists a number M > | s ( x, l ; R ) s (cid:48) ( a, l ; R ) | ≤ M x − , | s (cid:48) ( x, l ; R ) s ( a, l ; R ) | ≤ M x for R ≥ x ≥ b . Hence, it follows from (4.7) that (cid:90) ∞ b | δ R ( x, a ) φ ( x ) | dx ≤ M (cid:90) ∞ b x | φ ( x ) | x − a dx ≤ M (cid:90) ∞ b x − | φ ( x ) | dx, where M = 2 bM / ( b − a ). By hypothesis, the last integral is convergent; so forany ε > c > b such that(4.10) (cid:90) ∞ c | δ R ( x, a ) φ ( x ) | dx < ε R ≥ . To prove that there exists a number d > (cid:90) d | δ R ( x, a ) φ ( x ) | dx < ε R ≥ , we first need to demonstrate that(4.12) k s ( k , l ; r ) = O (1) as r → ∞ , (4.13) s (cid:48) ( k , l ; r ) = O (1) as r → ∞ , uniformly for all sufficiently small k ≥
0. (Recall: k = √ x .) This can be doneby considering two separate cases: (i) kr → ∞ , and (ii) kr bounded. In case (i),we first make the change of variable ρ = kr and set ω ( ρ ) = y ( ρ/k ) = y ( r ) so thatequation (4.1) becomes(4.14) d ωdρ + (cid:18) kρ − l ( l + 1) ρ (cid:19) ω = 0 , and then apply the Liouville-Green transformation given in [12, p.196] with f ( ρ ) =1 + (2 /kρ ) and g ( ρ ) = l ( l + 1) /ρ . The result is that equation (4.14) has a solution ω ( k ; ρ ) such that(4.15) ω ( k ; ρ ) ∼ (cid:18) kρ (cid:19) − sin (cid:26)(cid:90) (cid:18) kρ (cid:19) dρ (cid:27) as ρ → ∞ and(4.16) ddρ ω ( k ; ρ ) ∼ (cid:18) kρ (cid:19) cos (cid:26)(cid:90) (cid:18) kρ (cid:19) dρ (cid:27) as ρ → ∞ . With a suitable choice of the integration constant and for fixed k >
0, we have (cid:90) (cid:18) kρ (cid:19) dρ = ζ ∗ ( k, l ; ρ ) + O (cid:18) ρ (cid:19) as ρ → ∞ , where ζ ∗ ( k, l ; ρ ) = ρ + 1 k ln 2 ρ − lπ (cid:18) l + 1 − ik (cid:19) HE DIRAC DELTA FUNCTION 9 which is exactly equal to the function ζ ( k , l ; r ) given in (4.5). For fixed k >
0, wecan compare the behavior of s ( k , l ; r ) given in (4.3) and that given in (4.15). Theconclusion is(4.17) s ( k , l ; r ) = 1 √ πk ω ( k ; ρ ) , from which we also obtain(4.18) s (cid:48) ( k , l ; r ) = (cid:114) kπ dω dρ . Since ω ( k ; ρ ) = O (1) and ω (cid:48) ( k ; ρ ) = O ( k − ) for all small k and large ρ on accountof (4.15) and (4.16), the order estimates in (4.12) and (4.13) are established.In case (ii), we first recall the function f ( k , l ; r ) = ( i/k ) l +1 Γ(2 l + 2) M i/k,l + ( − ikr ) , where M κ,λ ( z ) is a Whittaker function; see Seaton [16, eqs. (14) & (22)]. Thisfunction is related to the Coulomb wave function s ( k , l ; r ) via s ( k , l ; r ) = (cid:20) A ( k , l )2(1 − e − π/k ) (cid:21) f ( k , l ; r ) , where A ( k , l ) is a polynomial of degree l in k ; see [16, eq.(114)]. In view of theconvergent expansion [4, §
7, eq.(16)] M κ,λ ( z ) = Γ(2 λ + 1)2 λ z λ + ∞ (cid:88) n =0 p (2 λ ) n ( z ) J λ + n (2 √ zκ )(2 √ zκ ) λ + n , where the p (2 λ ) n ( z ) are polynomials in z , we have for bounded krf ( k , l ; r ) = C ( kr ) r cos (cid:18) √ r − π l + 1) − π (cid:19) + O (cid:18) r / (cid:19) as r → ∞ , where C ( kr ) is a polynomial of kr . Hence, s ( k , l ; r ) = C ∗ ( kr ) k − (cid:20) A ( k , l )2(1 − e − π/k ) (cid:21) cos (cid:18) √ r − π l + 1) − π (cid:19) + O (cid:18) r / (cid:19) (4.19)and s (cid:48) ( k , l ; r ) = − C ( kr ) r − (cid:20) A ( k , l )(1 − e − π/k ) (cid:21) sin (cid:18) √ r − π l + 1) − π (cid:19) + O (cid:18) r / (cid:19) , (4.20)where C ∗ ( kr ) = ( kr ) C ( kr ), again proving (4.12) and (4.13).From (4.12) and (4.13), it follows that there are constants 0 < ρ < min { a, } , R > N > | x s ( x, l ; r ) | ≤ N and | s (cid:48) ( x, l ; r ) | ≤ N
10 Y. T. LI AND R. WONG for all r ≥ R and 0 ≤ x ≤ ρ . Furthermore, by (4.7), | δ R ( x, a ) | ≤ N | x − a | ( x − + a − ) ≤ N x − | x − a | for 0 < x ≤ ρ and R ≥ R , and (cid:90) ρ | δ R ( x, a ) φ ( x ) | dx ≤ N (cid:90) ρ x − | φ ( x ) | dx, where N = 2 N / ( a − ρ ). By hypothesis, the last integral is convergent, thusestablishing (4.11).Let us now consider the case when x lies in the interval ( d, c ). From (4.3), (4.4)and (4.7), we have π ( x − a ) δ R ( x, a ) = (cid:18) ax (cid:19) sin ζ ( x, l ; R ) cos ζ ( a, l ; R ) − (cid:18) xa (cid:19) cos ζ ( x, l ; R ) sin ζ ( a, l ; R ) + ε ( x, a ; R ) , (4.21)where ε ( x, a ; R ) / ( x − a ) is continuous in (0 , ∞ ) and ε ( x, a ; R ) = O (1 /R ) as R → ∞ uniformly for x ∈ ( d, c ). As a consequence, we obtain(4.22) lim R →∞ (cid:90) cd ε ( x, a ; R ) x − a φ ( x ) dx = 0 . For any A and B satisfying d ≤ A < B ≤ c , an integration by parts yields (cid:90) BA sin ζ ( k , l ; R ) dk = − cos ζ ( k , l ; R ) ζ k ( k , l ; R ) (cid:12)(cid:12)(cid:12)(cid:12) BA + (cid:90) BA ∂∂k (cid:18) ζ k ( k , l ; R ) (cid:19) cos ζ ( k , l ; R ) dk, where ζ k ( k , l ; R ) denotes the derivative of ζ ( k , l ; R ) with respect to k . From (4.5),it is readily seen that for d ≤ k ≤ c , ζ k ( k , l ; R ) → ∞ and ∂∂k (cid:18) ζ k ( k , l ; R ) (cid:19) → R → ∞ . Hence,(4.23) lim R →∞ (cid:90) BA sin ζ ( k , l ; R ) dk = 0 . Similarly, we also have(4.24) lim R →∞ (cid:90) BA cos ζ ( k , l ; R ) dk = 0 . Let η > d < a − η < a + η < c . A combinationof (4.23), (4.24) and the generalized Riemann-Lebesgue lemma gives(4.25) lim R →∞ (cid:18)(cid:90) a − ηd + (cid:90) ca + η (cid:19)(cid:18) ax (cid:19) φ ( x ) x − a sin ζ ( x, l ; R ) dx = 0and(4.26) lim R →∞ (cid:18)(cid:90) a − ηd + (cid:90) ca + η (cid:19)(cid:18) xa (cid:19) φ ( x ) x − a cos ζ ( x, l ; R ) dx = 0 . HE DIRAC DELTA FUNCTION 11
By the same reasoning, we have(4.27) lim R →∞ (cid:90) a + ηa − η (cid:20)(cid:18) ax (cid:19) − (cid:21) φ ( x ) x − a sin ζ ( x, l ; R ) dx = 0and(4.28) lim R →∞ (cid:90) a + ηa − η (cid:20)(cid:18) xa (cid:19) − (cid:21) φ ( x ) x − a cos ζ ( x, l ; R ) dx = 0 . From (4.21), (4.22) and (4 . − (4 . R →∞ (cid:90) cd δ R ( x, a ) φ ( x ) dx = lim R →∞ (cid:90) a + ηa − η sin { ζ ( x, l ; R ) − ζ ( a, l ; R ) } x − a φ ( x ) dx = lim R →∞ (cid:18)(cid:90) aa − η + (cid:90) a + ηa (cid:19) sin { ζ ( x, l ; R ) − ζ ( a, l ; R ) } x − a φ ( x ) dx. (4.29)Since φ ( x ) is piecewise continuously differentiable in (0 , ∞ ), it is continuouslydifferentiable in ( a − η, a ) for sufficiently small η >
0, and φ ( x ) − φ ( a − ) / ( x − a ) isintegrable on ( a − η, a ). By (4.23), (4.24) and the generalized Riemann-Lebesguelemma, we have(4.30) lim R →∞ (cid:90) aa − η φ ( x ) − φ ( a − ) x − a sin { ζ ( x, l ; R ) − ζ ( a, l ; R ) } dx = 0 , or equivalentlylim R →∞ (cid:90) aa − η sin { ζ ( x, l ; R ) − ζ ( a, l ; R ) } π ( x − a ) φ ( x ) dx = φ ( a − ) lim R →∞ (cid:90) aa − η sin { ζ ( x, l ; R ) − ζ ( a, l ; R ) } π ( x − a ) dx. To obtain the value of the limit on the right-hand side of the last equation, weshall use the Cauchy residue theorem. Let ζ ( x, R ) = √ xR + 1 √ x ln R,θ ( x, l ) = 1 √ x ln(2 √ x ) − lπ (cid:18) l + 1 − i √ x (cid:19) (4.31)so that(4.32) ζ ( x, l ; R ) = ζ ( x ; R ) + θ ( x, l ) . Furthermore, let Γ denote the positively oriented closed contour depicted in Figure1 below. It consists of a horizontal line segment Γ , two vertical line segments Γ and Γ , a quarter-circle Σ centered at z = a with radius r , and the interval Γ onthe positive real-axis. The entire region bounded by Γ lies in the first quadrant { z ∈ C : Re z > z ≥ } . Consider the complex-value function(4.33) F R ( z ) = e i [ ζ ( z,R ) − ζ ( a,R )] z − a . Since ζ ( z, R ) is analytic in the right half-plane, by Cauchy’s theorem(4.34) (cid:90) Γ F R ( z ) dz = 0 . By Cauchy’s residue theorem, we also have(4.35) lim r → + (cid:90) Σ F R ( z ) dz = − iπ/ , where r is the radius of the quarter-circle Σ. D Γ ΓΓΓ Σ a η - ar
34 1 2
Figure 1.
Contour Γ.For z ∈ Γ , we write z = u + iD . Since ζ ( a, R ) is real, a simple estimation gives(4.36) | F R ( z ) | ≤ D e − Im ζ ( z,R ) , z ∈ Γ . From (4.31), it is readily seen that lim R →∞ Im ζ ( z, R ) = + ∞ uniformly for z ∈ Γ .Hence, lim R →∞ e − Im ζ ( z,R ) = 0 uniformly for z ∈ Γ . From (4.36), it follows that(4.37) lim R →∞ (cid:90) Γ F R ( z ) dz = 0 . For z ∈ Γ , we write z = a − η + iv . Clearly(4.38) | F R ( z ) | ≤ η e − Im ζ ( z,R ) , z ∈ Γ . Let σ > R →∞ Im ζ ( z, R ) = + ∞ uniformly for z ∈ Γ and Im z = v ≥ σ . Thus, lim R →∞ e − Im ζ ( z,R ) = 0 uniformlyfor z ∈ Γ ∩ { z : Im z ≥ σ } andlim R →∞ (cid:90) Γ e − Im ζ ( z,R ) dz = lim R →∞ (cid:90) a − η + iσ a − η e − Im ζ ( z,R ) dz. From (4.31), it also follows that there is a constant M > e − Im ζ ( z,R ) ≤ M . Hence (cid:12)(cid:12)(cid:12)(cid:12) lim R →∞ (cid:90) Γ e − Im ζ ( z,R ) dz (cid:12)(cid:12)(cid:12)(cid:12) ≤ M σ . Since σ can be arbitrarily small, we obtain(4.39) lim R →∞ (cid:90) Γ e − Im ζ ( z,R ) dz = 0 . HE DIRAC DELTA FUNCTION 13
Coupling (4.38) and (4.39) gives(4.40) lim R →∞ (cid:90) Γ F R ( z ) dz = 0 . In a similar manner, one can establish(4.41) lim R →∞ (cid:90) Γ F R ( z ) dz = 0 . By a combination of (4.34), (4.35), (4.37) and (4 . − (4 . R →∞ (cid:90) aa − η sin { ζ ( x, R ) − ζ ( a, R ) } π ( x − a ) dx = − π Im (cid:26) lim R →∞ lim r → + (cid:90) Σ F R ( z ) dz (cid:27) = 12 . (4.42)Using (4.42), we are now ready to handle the limit on the right-hand side of theequation following (4.30).For simplicity, let us write θ ( x ) = θ ( x, l ); cf. (4.31). By the mean-value theorem,cos[ θ ( x ) − θ ( a )] = 1 + O { ( x − a ) } for x near a . Hence, by the generalized Riemann-Lebesgue lemma, we have from (4.42)(4.43) lim R →∞ (cid:90) aa − η sin { ζ ( x, R ) − ζ ( a, R ) } cos[ θ ( x ) − θ ( a )] π ( x − a ) dx = 12 , and also from (4.23) and (4.24)(4.44) lim R →∞ (cid:90) aa − η cos { ζ ( x, R ) − ζ ( a, R ) } sin[ θ ( x ) − θ ( a )] π ( x − a ) dx = 0 . Upon using an addition formula, it follows from (4.43) and (4.44) that(4.45) lim R →∞ (cid:90) aa − η sin { ζ ( x, l ; R ) − ζ ( a, l ; R ) } π ( x − a ) dx = 12 . Coupling (4.30) and (4.45), we obtain(4.46) lim R →∞ (cid:90) aa − η sin { ζ ( x, l ; R ) − ζ ( a, l ; R ) } π ( x − a ) φ ( x ) dx = 12 φ ( a − ) . In a similar manner, we also have(4.47) lim R →∞ (cid:90) a + ηa sin { ζ ( x, l ; R ) − ζ ( a, l ; R ) } π ( x − a ) φ ( x ) dx = 12 φ ( a + ) . A combination of (4.29), (4.46) and (4.47) yields(4.48) lim R →∞ (cid:90) cd δ R ( x, a ) φ ( x ) dx = 12 [ φ ( a − ) + φ ( a + )] . Since ε in (4.10) and (4.11) can be arbitrarily small, (4.8) now follows from (4.48).This completes the proof of the theorem. (cid:3) Airy and parabolic cylinder functions
We now turn our attention to the integral representation (1.7). Here, the intervalof concern is the whole real line. However, the argument for this result remainssimilar to that for Theorems 1 & 2, and we will keep it brief. As before, we let b and η be positive numbers such that b > max { , | a |} . The Airy function Ai( t − x )is a solution of the equation(5.1) d ydt + ( x − t ) y = 0 , −∞ < t < ∞ , and satisfies(5.2) Ai( ∞ ) = Ai (cid:48) ( ∞ ) = 0 , (5.3) Ai( − t − x ) = 1 √ π ( t + x ) − / [sin ζ ( x, t ) + ε ( x, t )] , and(5.4) Ai (cid:48) ( − t − x ) = − √ π ( x + t ) / [cos ζ ( x, t ) + ε ( x, t )] , where ε ( x, t ) = O ( t − / ) and ε ( x, t ) = O ( t − / ), as t → ∞ , uniformly for x > − b ,and where(5.5) ζ ( x, t ) = 23 ( t + x ) / + π . Define(5.6) δ R ( x, a ) = (cid:90) ∞− R Ai( t − x )Ai( t − a ) dt. From (5.2), we have(5.7) δ R ( x, a ) = Ai( − R − x )Ai (cid:48) ( − R − a ) − Ai( − R − a )Ai (cid:48) ( − R − x ) x − a ;see (3.6). Theorem 3.
For any a ∈ R and any piecewise continuously differentiable function φ ( x ) on ( −∞ , ∞ ) , we have (5.8) lim R →∞ (cid:90) ∞−∞ φ ( x ) (cid:90) ∞− R Ai( t − x )Ai( t − a ) dt dx = 12 [ φ ( a − ) + φ ( a + )] , provided that the two integrals (5.9) (cid:90) − −∞ | x | − | φ ( x ) | dx and (cid:90) ∞ x − | φ ( x ) | dx are convergent.Proof. Using the asymptotic formulas (5.3) and (5.4), one can show from (5.7) thatthere are positive constants M and M such that (cid:90) ∞ b | δ R ( x, a ) φ ( x ) | dx ≤ M (cid:90) ∞ b x − | φ ( x ) | dx and (cid:90) − b −∞ | δ R ( x, a ) φ ( x ) | dx ≤ M (cid:90) − b −∞ | x | − | φ ( x ) | dx HE DIRAC DELTA FUNCTION 15 for
R > | a | + 1. On account of the convergence of the two integrals in (5.9), forany ε > c > b such that(5.10) (cid:18)(cid:90) − c −∞ + (cid:90) ∞ c (cid:19) | δ R ( x, a ) φ ( x ) | dx < ε for all R > | a | + 1.In view of the asymptotic formulas (5.3) and (5.4), equation (5.7) also gives π ( x − a ) δ R ( x, a ) = (cid:18) R + aR + x (cid:19) sin ζ ( x, R ) cos ζ ( a, R ) − (cid:18) R + xR + a (cid:19) cos ζ ( x, R ) sin ζ ( a, R ) + ε ( x, a ; R ) , (5.11)where ε ( x, a ; R ) = O ( R − / ), as R → ∞ , uniformly for x, a ∈ ( − b, b ). In a mannersimilar to (4.29), by using the generalized Riemann-Lebesgue lemma we have(5.12) lim R →∞ (cid:90) c − c δ R ( x, a ) φ ( x ) dx = lim R →∞ (cid:90) a + ηa − η sin { ζ ( x, R ) − ζ ( a, R ) } π ( x − a ) φ ( x ) dx, for any piecewise continuously differentiable function φ ( x ) and any η >
0. Since ζ ( x, R ) − ζ ( a, R ) = √ R ( x − a ) + O (1 / √ R ) for large R and bounded x and a , we alsohave sin { ζ ( x, R ) − ζ ( a, R ) } = sin( √ R ( x − a )) + O (1 / √ R ) as R → ∞ for bounded x and a . Thus, equation (5.12) gives(5.13) lim R →∞ (cid:90) c − c δ R ( x, a ) φ ( x ) dx = lim R →∞ (cid:90) a + ηa − η sin √ R ( x − a ) π ( x − a ) φ ( x ) dx. By Jordan’s theorem on the Dirichlet kernel [2, p.473], the value of the last limit is[ φ ( a − ) + φ ( a + )] /
2. The final result (5.8) now follows from (5.10) and (5.13). (cid:3)
The parabolic cylinder function W ( a, x ) is a solution of Weber’s equation(5.14) d ydx + (cid:18) x − a (cid:19) y = 0with boundary conditions(5.15) y ( x ) = (cid:114) k a x (cid:20) cos ζ ( a, x ) + O (cid:18) x (cid:19)(cid:21) , x → ∞ , and(5.16) y ( − x ) = (cid:114) k a x (cid:20) sin ζ ( a, x ) + O (cid:18) x (cid:19)(cid:21) , x → ∞ , where k a = √ e πa − e πa and(5.17) ζ ( a, x ) = 14 x − a ln x + 12 arg Γ (cid:18)
12 + ia (cid:19) + π W (cid:48) ( a, x ) = − (cid:114) k a x (cid:20) sin ζ ( a, x ) + O (cid:18) x (cid:19)(cid:21) , x → ∞ , and(5.19) W (cid:48) ( a, − x ) = − (cid:114) x k a (cid:20) cos ζ ( a, x ) + O (cid:18) x (cid:19)(cid:21) , x → ∞ . This function is also related to the parabolic cylinder function U ( a, x ) via the con-nection formulas [1, p.693](5.20) W ( a, x ) = (2 k a ) e πa Re (cid:8) e i ( φ + π ) U ( ia, xe − πi ) (cid:9) , (5.21) W ( a, − x ) = (2 /k a ) e πa Im (cid:8) e i ( φ + π ) U ( ia, xe − πi ) (cid:9) , where x > φ = arg Γ( + ia ). From the integral representation [12, p.208](5.22) U ( a, z ) = e − z Γ( + a ) (cid:90) ∞ e − zs − s s a − ds, Re a > − , one can easily show that | U ( ia, xe − πi ) | ≤ √ π | Γ( + ia ) | x − . Since (cid:12)(cid:12) Γ( + ia ) (cid:12)(cid:12) = √ π (cosh πa ) / ∼ √ πe − πa and (cid:112) k a ∼ e − πa , (cid:114) k a ∼ e πa as a → + ∞ , it follows that for large positive a , there is a constant M such that | W ( a, x ) | ≤ M e πa x − , x > , | W ( a, − x ) | ≤ M e πa x − , x > . As a → −∞ , we have (cid:12)(cid:12) Γ( + ia ) (cid:12)(cid:12) ∼ √ πe πa = √ πe − π | a | and (cid:112) k a ∼ √ , (cid:114) k a ∼ √ . Hence, there exists a constant M (cid:48) > a , | W ( a, x ) | ≤ M (cid:48) e π | a | x − , x > , | W ( a, − x ) | ≤ M (cid:48) e π | a | x − , x > . By using (5.20), (5.21) and (5.22), it can also be shown that there are positiveconstants M and M (cid:48) such that for large positive a , | W (cid:48) ( a, x ) | ≤ M e πa x , x > , | W (cid:48) ( a, − x ) | ≤ M e πa x , x > , and for large negative a , | W (cid:48) ( a, x ) | ≤ M (cid:48) e π | a | x , x > , | W (cid:48) ( a, − x ) | ≤ M (cid:48) e π | a | x , x > . We define(5.23) δ R ( a, b ) = (cid:90) R − R W ( a, x ) W ( b, x ) dx. HE DIRAC DELTA FUNCTION 17
From (5.14), one can derive δ R ( a, b ) = W ( a, x ) W (cid:48) ( b, x ) − W (cid:48) ( a, x ) W ( b, x ) b − a (cid:12)(cid:12)(cid:12)(cid:12) R − R . Using the asymptotic formulas (5.15) and (5.18), we obtain( b − a ) δ R ( a, b ) = (cid:18)(cid:112) k a k b + 1 √ k a k b (cid:19) sin { ζ ( a, R ) − ζ ( b, R ) } + O (cid:18) R (cid:19) . By an argument similar to that for Theorem 3, one can establish the followingresult.
Theorem 4.
For any a ∈ R and any piecewise continuously differentiable function φ ( x ) on ( −∞ , + ∞ ) , we have lim R →∞ (cid:90) + ∞−∞ φ ( b ) (cid:90) R − R W ( a, x ) W ( b, x ) dx db = π (cid:112) e πa [ φ ( a − ) + φ ( a + )] , provided that two integrals (cid:90) − −∞ | φ ( x ) | e π | x | dx | x | and (cid:90) ∞ | φ ( x ) | e πx dxx are convergent. Series representations
To prove the representations in (1 . − (1 . P n ( x ), Theorem 1 in [9, p.55] can be stated in thefollowing form. Theorem 5.
Let f ( x ) be a piecewise continuously differentiable function in ( − , ,and put (6.1) δ n ( t, x ) = n (cid:88) k =0 (cid:18) k + 12 (cid:19) P k ( t ) P k ( x ) . If the integral (cid:90) − f ( x ) dx is finite, then (6.2) lim n →∞ (cid:90) − δ n ( t, x ) f ( t ) dt = 12 [ f ( x + ) + f ( x − )] . The statement in (6.2) is equivalent to that in (1.8); i.e., the finite sum in (6.1)defines a delta sequence. In a similar manner, one can restate Theorems 2 and 3 in[9, p.71 and p.88] as follows.
Theorem 6.
Let f ( x ) be a piecewise continuously differentiable function in ( −∞ , ∞ ) ,and put (6.3) δ n ( t, x ) = e − t n (cid:88) k =0 k k ! √ π H k ( t ) H k ( x ) . If the integral (cid:90) ∞−∞ e − x f ( x ) dx is finite, then (6.4) lim n →∞ (cid:90) ∞−∞ δ n ( t, x ) f ( t ) dt = 12 [ f ( x + ) + f ( x − )] . Theorem 7.
Let f ( x ) be a piecewise continuously differentiable function in (0 , ∞ ) ,and put (6.5) δ n ( t, x ) = e − t t α n (cid:88) k =0 k !Γ( k + α + 1) L ( α ) k ( t ) L ( α ) k ( x ) . If the integral (cid:90) ∞ e − t t α f ( t ) dt, α > − , is finite, then (6.6) lim n →∞ (cid:90) ∞ δ n ( t, x ) f ( t ) dt = 12 [ f ( x + ) + f ( x − )] . To demonstrate (1.11), we recall the Laplace series expansion(6.7) f ( θ , φ ) = ∞ (cid:88) k =0 k + 14 π (cid:90) π − π (cid:90) π f ( θ , φ ) P k (cos γ ) sin θ dθ dφ , where cos γ = cos θ cos θ − sin θ sin θ cos( φ − φ ) and P k ( x ) is a Legendre poly-nomial; see [6, p.147]. By the addition formula [3, p.797](6.8) P k (cos γ ) = 4 π k + 1 k (cid:88) l = − k Y kl ( θ , φ ) Y ∗ kl ( θ , φ ) , we can rewrite (6.7) in the form(6.9) f ( θ , φ ) = ∞ (cid:88) k =0 (cid:90) π − π (cid:90) π f ( θ , φ ) sin θ k (cid:88) l = − k Y kl ( θ , φ ) Y ∗ kl ( θ , φ ) dθ dφ . For f ∈ C ([0 , π ] × [ − π, π ]), the series on the right converges pointwise to the functionon the left; see [6, p.344]. This result can be expressed as follows. Theorem 8.
Let f ( θ , φ ) be a continuous function on [0 , π ] × [ − π, π ] , and put (6.10) δ n ( θ , θ ) δ n ( φ , φ ) := sin θ n (cid:88) k =0 k (cid:88) l = − k Y kl ( θ, φ ) Y ∗ kl ( θ , φ ) . Then, we have (6.11) f ( θ , φ ) = lim n →∞ (cid:90) π − π (cid:90) π δ n ( θ , θ ) δ n ( φ , φ ) f ( θ , φ ) dθ dφ . Equation (6.11) is equivalent to saying that δ n ( θ , θ ) δ n ( φ , φ ) is a delta se-quence of δ ( θ − θ ) δ ( φ − φ ) for θ , θ ∈ [0 , π ] and φ , φ ∈ [ − π, π ]. The use ofthe identity δ (cos θ , cos θ ) = θ δ ( θ − θ ) gives (1.11); see [8, p.49]. HE DIRAC DELTA FUNCTION 19
Acknowledgements
The authors would like to thank Professor W. Y. Qiu of Fudan University formany helpful discussions. His unfailing assistance is greatly appreciated.
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