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2. Legendre functions as Laplace transforms

The Legendre functions of degree ν and order µ, which are solutions of the corresponding µ µ Legendre differential equation, include Pν (z) and Qν (z), which are defined on the com- µ µ plement of the cut z ∈ (−∞, 1]. They also include the Ferrers functions Pν (z) and Qν (z), sometimes called Legendre functions ‘on the cut,’ which are defined on the complement of z ∈ (−∞, −1] ∪ [1, ∞) and occur most often with z ∈ (−1, 1); especially, in physical applications. In the absence of cuts, all these functions would be multivalued. µ µ µ µ In the older English-language literature (e.g., [14]), Pν ,Qν and Pν , Qν were not typo- graphically distinguished, and this remains true in a few recent works (e.g., [4]). In the µ µ µ µ German literature (e.g., [6, 15]), they are denoted by Pν , Qν and Pν ,Qν . In the recent µ µ µ µ µ µ NIST handbook [8], Pν , Qν are denoted by Pν , Qν , to stress the dissimilarity to Pν ,Qν . µ The function Qν (z), defined on the complement of the cut z ∈ (−∞, 1], has historically been defined in three ways, differing only in normalization. The most recent (and perhaps the best) definition is that of the late FWJ Olver [16]. But for many decades, the standard definition in the literature has been that of EW Hobson [14]. There is also a definition originating with EW Barnes [17], which is no longer used but was employed in at least two major works that are still consulted [1, 18]. The three definitions are related by

− sin(νπ) e µπi [Qµ] = Γ(ν + µ + 1) [Qµ] = [Qµ] , ν H ν O sin[(ν + µ)π] ν B the subscript indicating the originator. In the NIST handbook [8], the function of Olver µ µ is denoted by Qν , the symbol Qν being reserved for the standard (Hobson) function. But the Barnes function, when in use, was never distinguished typographically from that of µ Hobson. It should be noted that alternative definitions of Pν as well have been employed µ (e.g., in [19, 20]). However, Qν is the chief potential source of confusion. The following are the complete set of Laplace transforms, employing Hobson’s now- µ standard definition of Qν . They are parametrized to agree with the circular- and hyperbolic-trigonometric versions given in [1]. (Except for [4], the other cited works give unnormalized, non-trigonometric forms, but their forms easily reduce to the following.) Each identity has been checked by numerical quadrature.

Theorem 2.1 The following identities hold, when θ ∈ (0, π/2) for (1) and (2), resp. ξ ∈ (0, ∞) for (3) and (4), and ξ ∈ R for (5).

∞ −t cos θ ν −µ e t Jµ(t sin θ)dt = Γ(ν + µ +1)Pν (cos θ), (1) Z0 ∞ −t cos θ ν −µ e t Yµ(t sin θ)dt = −(2/π) Γ(ν + µ +1)Qν (cos θ), (2) Z0 ∞ −t cosh ξ ν −µ e t Iµ(t sinh ξ)dt = Γ(ν + µ + 1) Pν (cosh ξ), (3) Z0 ∞ −t cosh ξ ν µπi −µ e t Kµ(t sinh ξ)dt = Γ(ν + µ + 1) e Qν (cosh ξ), (4) Z0

2 ∞ −t sinh ξ ν e t Kµ(t cosh ξ)dt = Z0 1/2 −1/2 −ν−1/2 (π/2) Γ(ν + µ + 1)Γ(ν − µ + 1) (cosh ξ) Pµ−1/2 (tanh ξ). (5)

It is assumed that the integrals converge, for which one needs Re(ν + µ) > −1 in (1) and (3), and Re(ν ± µ) > −1 in (2), (4) and (5).

The five identities of the theorem can be extended to the complex domain (see the cited references). Also, (3),(4) are equivalent to the following.

Theorem 2.2

∞ −t cosh ξ ν 1/2 −1/2 −(ν+1/2)πi ν+1/2 e t Iµ(t sinh ξ)dt = (2/π) (sinh ξ) e Qµ−1/2(coth ξ), (6) Z0 ∞ −t cosh ξ ν e t Kµ(t sinh ξ)dt = Z0 1/2 −1/2 −ν−1/2 (π/2) Γ(ν + µ + 1)Γ(ν − µ + 1) (sinh ξ) Pµ−1/2 (coth ξ). (7)

These two supplementary identities are obtained by rewriting the right sides of (3),(4) ±µ ±µ with the aid of Whipple’s formula [16, Ex. 12.2] and the fact that Qν , i.e., [Qν ]O, are equal to each other. Of the seven Laplace transform evaluations listed in Theorems 2.1 and 2.2, the number appearing, in whatever form, in [1],[2],[3],[6],[5],[21],[7],[8],[4] is respectively 6,6,6,4,4,7,5,4,6. The number appearing incorrectly is respectively 1,4,2,0,1,0,0,0,3. (The ordering used here is chronological.) In some cases the right sides are given as expressions involving the Gauss hypergeometric function 2F1, but can be rewritten using Legendre functions so as to agree with entries in the foregoing list. The identities (1)–(4), in particular (1) and (3), are proved by series expansion and term-by-term integration; which is a procedure that can be traced to Hankel’s 1875 proof of (1), and extended to more general integral transforms (see, e.g., [22]). The most ν troublesome of them may be (2). Only in [5] is the Laplace transform of t Yµ(t) given in −µ the form based on the second Ferrers function Qν that appears in (2). In many works this µ −µ transform is given as a combination of the Ferrers functions Pν , Pν , but by standard Legendre identities [8, 16] this can be seen to be equivalent to (2). The little-known evaluation (5) appears only in [6, 21], where it is given correctly, in a non-trigonometric parametrization. (See p. 92, fourth equation down; resp. p. 151, eq. 1.15.16.) −µ −µ It is evident from (1)–(4) that the interchange symmetry between the pairs Pν ,Qν −µ −µ and Pν , Qν is not reflected in a symmetry between the pairs Jµ,Yµ and Iµ, Kµ. The difficulty is the prefactor 2/π in (2): there is no such prefactor in (4). This asymmetry is attributable to the definition of the second modified Bessel function Kµ, ‘MacDonald’s function,’ which includes a factor equal to π/2. It has been suggested that using a differ- ently normalized function Khµ, equal to 2/π times Kµ, would be more reasonable [23]. (The ‘h’ is for Heaviside, who preferred this normalization.) Rewriting (4),(5) and (7) in terms of Khµ would exhibit a symmetry between Jµ,Yµ and Iµ, Khµ.

3 3. Some consequences

The incomplete Lipschitz–Hankel integral Ceν,µ(T ; s), for C = J,Y,I,K, is defined for real s, when T > 0, by

T −st ν Ceν,µ(T ; s)= e t Cµ(t) dt, (8) Z0 provided that the integral converges. For convergence, one needs Re(ν + µ) > −1 for C = J, Y , and Re(ν ± µ) > −1 for C = I, K. The limit as T → ∞ of Ceν,µ(T ; s) is ν the Laplace transform L{t Cµ(t)}(s), provided that this transform exists. The transform exists if s> 0 for C = J, Y , if s> 1 for C = I, and if s> −1 for C = K. For comparison with previous work on ILHIs [5, 12, 13], the following expressions for ν L{t Cµ(t)}(s) with C = J, I, K may be useful. They are non-trigonometric versions of µ µ (1),(3),(5),(7), respectively, and none involves the troublesome functions Qν , Qν .

L ν 2 −(ν+1)/2 −µ 2 {t Jµ(t)}(s)=Γ(ν + µ + 1) (1 + s ) Pν (s/ 1+ s ), s ∈ (0, ∞); (9) p L ν 2 −(ν+1)/2 −µ 2 {t Iµ(t)}(s)=Γ(ν + µ + 1) (s − 1) Pν (s/ s − 1), s ∈ (1, ∞); (10) p ν 1/2 L{t Kµ(t)}(s) = (π/2) Γ(ν + µ + 1)Γ(ν − µ + 1)

2 −(2ν+1)/4 −ν−1/2 (1 − s ) Pµ−1/2 (s), s ∈ (−1, 1), ×  (11)  2 −(2ν+1)/4 −ν−1/2 (s − 1) Pµ−1/2 (s), s ∈ (1, ∞).  The conditions on ν,µ for convergence are as stated. The standard reference on ILHIs [5] includes versions of (9) and (10). But as is noted in § 4 below, its version of (10) includes an erroneous −(2/π) factor. Also, the Laplace ν −µ transform of t Kµ(t) given in [5] is valid only when s ∈ (1, ∞); and involving Qν rather −ν−1/2 than Pµ−1/2 , it is a non-trigonometric version not of (7), but rather of (4). ν The duality between the expressions for L{t Kµ(t)}(s) on s ∈ (−1, 1) and s ∈ (1, ∞), evident in (11), deserves to be better known. The value at s = 1 must be supplied separately.

ν Proposition 3.1 At s = 1, the Laplace transform of t Kµ(t) is given by

π1/2 Γ(ν + µ + 1)Γ(ν − µ + 1) L{tν K (t)}(s =1)= , (12) µ 2ν+1 Γ(ν + 3/2) where one needs Re(ν ± µ) > −1 for convergence.

−ν−1/2 − This comes from (11) by considering the asymptotic behavior of Pµ−1/2 (s) as s → 1 , −ν−1/2 + or of Pµ−1/2 (s) as s → 1 , which are well known [16]. Viewed as a function of ν, the −t right side of (12) is the Mellin transform of e Kµ(t), and it has been tabulated as such [2, eq. 6.8(28)].

4 4. Table errata

The following are the errata. None seems to have been previously reported, though the incompatibility between the definitions used in [1, 4] has been noted [10].

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, Cambridge, UK, 1944. Cited here as Ref. [1]. µ µ µ The Barnes definition of Qν is used, and Pν , Qν are not visually distinguished from µ µ Pν ,Qν . On pp. 387–388, in § 13.21, versions of identities (1),(2),(3),(4),(6),(7) appear as 1 νπi − 1 νπi eqs. (3),(4),(1),(2),(6),(7). But in eq. (4), the factors e 2 , e 2 should be interchanged. µ µ If reference is made to the definition of the Ferrers function Qν in terms of Qν [16, § 5.15], µ and to the relation between the Barnes and Hobson definitions of Qν , it will be seen that this interchange makes eq. (4) equivalent to the simpler identity (2) of Theorem 2.1.

A. Erdelyi,´ W. Magnus, F. Oberhettinger & F. Tricomi, Tables of Integral Transforms, Volume I, McGraw–Hill, New York, 1954. Cited here as Ref. [2]. µ µ µ The standard (Hobson) definition of Qν is used, and Pν , Qν are typographically distinct µ µ from Pν ,Qν . On pp. 182,187,196,198, versions of identities (1),(2),(3),(4),(6),(7) appear as eqs. 4.14(9),4.14(48),4.16(8),4.16(28),4.16(9),4.16(27). ν ν In 4.14(9), Pµ should be read as Pµ, making the identity consistent with (1). ν −ν ν −ν In 4.14(28), Pµ , Pµ should be read as Pµ, Pµ . It follows from the expressions for the second Ferrers function in [16, § 5.15] that these changes yield consistency with (2). In 4.16(28), the sine factors in numerator and denominator should be deleted, and ν νπi ν µ Qµ should be read as e Qµ. In 4.16(9), the sine factors should be deleted, and Qν µπi µ should be read as e Qν . With these emendations, 4.16(28),4.16(9) will agree with the identities (4),(6). As they stand, they inconsistently employ the Barnes and not the Hobson definition; they may have been taken from [1]. The corresponding inverse Laplace transforms, 5.13(9) and 5.13(3) on pp. 270–271, should be emended similarly. It has been found that there are many entries in these tables, besides 4.14(9) and 4.14(28), in which Legendre functions are typographically confused. A list is available from the present author.

G. E. Roberts & H. Kaufman, Table of Laplace Transforms, W. B. Saunders, Philadelphia, 1966. Cited here as Ref. [3]. µ µ µ The standard (Hobson) definition of Qν is used, but Pν , Qν are not visually distin- µ µ guished from Pν ,Qν . Versions of identities (1),(2),(3),(4),(6),(7) appear in this table, but its entries 12.2.4 (p. 73) and 13.2.2 (p. 79) are incorrect, since they incorporate without change the just-noted versions of (4),(6) found in [2].

M. M. Agrest & M. S. Maksimov, Theory of Incomplete Cylindrical Functions and Their Applications, Springer-Verlag, New York/Berlin, 1971. Cited here as Ref. [5]. µ µ µ The standard (Hobson) definition of Qν is used, but Pν , Qν are not visually distin- µ µ guished from Pν ,Qν . On pp. 41–42, versions of identities (1),(2),(3),(4) appear as eqs. (5.3),(5.4),(5.5),(5.6). In eq. (5.5), the prefactor −(2/π) should be deleted from the right side. The domain of validity of eq. (5.6) is Re a> 1, not Re a> −1 as stated; although, the Laplace transform on the left side is defined if Re a> −1. Also, in eqs. (5.4),(5.5), the representations given for Legendre functions in terms of the Gauss hypergeometric function F = 2F1 are incorrect: they should be replaced by the representations given in [8, 16].

5 In each of eqs. (5.8),(5.9), the exponent ν + 1/2 should be read as (ν + 1)/2.

I. S. Gradshteyn & I. M. Ryzhik, Table of Integrals, Series, and Products, 8th ed., Elsevier/Academic Press, Amsterdam, 2015. Cited here as Ref. [4]. µ µ µ The standard (Hobson) definition of Qν is used, but Pν , Qν are not visually distin- µ µ guished from Pν ,Qν . On pp. 712–713, in § 6.628, versions of (1),(2),(3),(4),(6),(7) appear as 6.628(1),(2),(4),(5),(6),(7). All were taken from [1] without change, so the three con- µ taining the function Qν inconsistently employ the Barnes definition, rather than that of Hobson. (Equation 6.628(6) has been altered since the seventh edition, without indica- tion, by correcting a typographical error that caused it to disagree with [1].) The three requiring emendation are 6.628(2),(5),(6). Their respective prefactors,

sin(µπ) sin(µπ) cos(νπ) − , , , sin[(µ + ν)π] sin[(ν + µ)π] sin[(µ + ν)π] should be replaced by −e−νπi, e−νπi, e−(µ−1/2)πi. When 6.628(6) is emended as described, it will become identical to the (correct) integral 6.622(3), so that one or the other will become redundant. 1 νπi As was noted above, there is an error in [1], in the original version of eq. 6.628(2): e 2 − 1 νπi and e 2 should be interchanged. If this change too is made in eq. 6.628(2), eq. 6.628(2) will become equivalent to the identity (2) listed in Theorem 2.1, which involves the Ferrers −µ µ −µ function Qν rather than Qν ,Qν , and is much simpler.

References

[1] Watson GN. A treatise on the theory of Bessel functions. 2nd ed. Cambridge, UK: Cambridge Univ. Press; 1944. [2] Erd´elyi A, Magnus W, Oberhettinger F, Tricomi FG. Tables of integral transforms. New York: McGraw-Hill; 1954. [3] Roberts GE, Kaufman H. Table of Laplace transforms. Philadelphia, PA: W. B. Saunders; 1966. [4] Gradshteyn IS, Ryzhik IM. Table of integrals, series, and products. 8th ed. Amsterdam: Elsevier/Academic Press; 2015. [5] Agrest MM, Maksimov MS. Theory of incomplete cylindrical functions and their applica- tions. Vol. 160 of Die Grundlehren der mathematischen Wissenschaften. New York/Berlin: Springer-Verlag; 1971. [6] Magnus W, Oberhettinger F, Soni RP. Formulas and theorems for the special functions of mathematical physics. 3rd ed; Vol. 52 of Die Grundlehren der mathematischen Wis- senschaften. New York/Berlin: Springer-Verlag; 1966. [7] Prudnikov AP, Brychkov IA, Marichev OI. Integrals and series. New York: Gordon and Breach; 1986–92. [8] Olver FWJ, Lozier DW, Boisvert RF, Clark CW, editors. NIST handbook of mathematical functions. Washington, DC: U.S. Department of Commerce, National Institute of Standards and Technology; 2010. [9] Taylor ME. Partial differential equations, II. Qualitative studies of linear equations. 2nd ed; Vol. 116 of Applied Mathematical Sciences. New York: Springer; 2011. [10] Matsumoto H, Nguyen L, Yor M. Subordinators related to the exponential functionals of Brownian bridges and explicit formulae for the semigroups of hyperbolic Brownian motions. In: Buckdahn R, Engelbert HJ, Yor M, editors. Stochastic processes and related topics. London: Taylor & Francis; 2002. p. 213–235.

6 [11] Grosche C. Path integration on hyperbolic spaces. J Phys A. 1992;25(15):4211–4244. [12] Dvorak SL. Applications for incomplete Lipschitz–Hankel integrals in electromagnetics. IEEE Antennas and Propagation Mag. 1994;36(6):26–32. [13] Sofotasios PC, Tsiftsis TA, Brychkov YA, Freear S, Valkama M, Karagiannidis GK. Analytic expressions and bounds for special functions and applications in communication theory. IEEE Trans Info Theory. 2014;60(12):7798–7823. Available on-line as arXiv:1403.5326. [14] Hobson EW. The theory of spherical and ellipsoidal harmonics. Cambridge, UK: Cambridge Univ. Press; 1931. [15] Lense J. Kugelfunktionen. 2nd ed. Leipzig, East Germany: Geest & Portig; 1954. [16] Olver FWJ. Asymptotics and special functions. AKP Classics ed. Wellesley, MA: A. K. Peters; 1997. [17] Barnes EW. On generalized Legendre functions. Quart J Pure Appl Math. 1908;39:97–204. [18] Bateman H. Partial differential equations of mathematical physics. Cambridge, UK: Cam- bridge Univ. Press; 1932. [19] Snow C. Hypergeometric and Legendre functions with applications to integral equations of potential theory. 2nd ed. No. 19 in National Bureau of Standards Applied Mathematics Series; Washington, DC: U.S. Government Printing Office; 1952. [20] Frolov V, Singh D. Quantum radiation of uniformly accelerated spherical mirrors. Classical Quantum Gravity. 2001;18(15):3025–3038. Available on-line as arXiv:gr-qc/0103071. [21] Oberhettinger F, Badii L. Tables of Laplace transforms. New York/Berlin: Springer-Verlag; 1973. [22] Srivastava HM. The Laplace transform of the modified Bessel function of the second kind. Publ Inst Math (Beograd) (NS). 1979;26(40):273–282. [23] Jeffreys H, Jeffreys BS. Methods of mathematical physics. 3rd ed. Cambridge, UK: Cam- bridge Univ. Press; 1956.

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