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Integral Formula for the Bessel Function of the First Kind

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Integral Formula for the Bessel Function of the First Kind INTEGRAL FORMULA FOR THE BESSEL FUNCTION OF THE FIRST KIND ENRICO DE MICHELI ABSTRACT. In this paper, we prove a new integral representation for the Bessel function of the first kind Jµ (z), which holds for any µ,z C. ∈ 1. INTRODUCTION In this note, we prove the following integral representation of the Bessel function of the first kind Jµ (z), which holds for unrestricted complex values of the order µ: z µ π 1 izcosθ 1 iθ C C (1.1) Jµ (z)= e γ∗ µ, 2 ize dθ (µ ;z ), 2π 2 π ∈ ∈ Z− where γ∗ denotes the Tricomi version of the (lower) incomplete gamma function. Lower and upper incomplete gamma functions arise from the decomposition of the Euler integral for the gamma function: w t µ 1 (1.2a) γ(µ,w)= e− t − dt (Re µ > 0), 0 Z ∞ t µ 1 (1.2b) Γ(µ,w)= e− t − dt ( argw < π). w | | Z Analytical continuation with respect to both parameters can be based either on the integrals in (1.2) or on series expansions of γ(µ,w), e.g. [4, Eq. 9.2(4)]: ∞ ( 1)n wµ+n (1.3) γ(µ,w)= ∑ − , n=0 n! µ + n which shows that γ(µ,w) has simple poles at µ = 0, 1, 2,... and, in general, is a multi- valued (when µ is not a positive integer) function with− a− branch point at w = 0. Similarly, even Γ(µ,w) is in general multi-valued but it is an entire function of µ. Inconveniences re- lated to poles and multi-valuedness of γ can be circumvented by introducing what Tricomi arXiv:2012.02887v1 [math.CA] 4 Dec 2020 called the fundamental function of the incomplete gamma theory [11, §2, p. 264]: µ w w w− t µ 1 e− (1.4) γ∗(µ,w)= e− t − dt = M(1,1 + µ;w), Γ(µ) Γ(1 + µ) Z0 where M(a,b;z) is the Kummer confluent hypergeometric function [8, Eq. 13.2.2]. The function γ (µ,w) is entire in both w and µ, real-valued for real w and µ and γ ( n,w)= wn ∗ ∗ − for n = 0,1,2,... B¨ohmer [1] and later Tricomi [10, 11] (see also [5]) assumed γ∗(µ,w) as base point for the theory, defining the incomplete gamma functions by the relations: µ (1.5) γ(µ,w)= Γ(µ)w γ∗(µ,w), µ (1.6) Γ(µ,w)= Γ(µ)[1 w γ∗(µ,w)]. − 2010 Mathematics Subject Classification. 33C10,40C10,33B20. Key words and phrases. Bessel functions, Incomplete gamma functions, Integral representations. 1 2 ENRICO DE MICHELI For a detailed account of the incomplete gamma functions theory, see [5], [9, Chapter 11] and [4, 8], where massive compilations of results are stated. For what concerns details on the Bessel function, the reader is referred to the classical treatise of Watson [12]. The strong link between Bessel functions and incomplete gamma functions has long been known. Examples of this connection are represented, e.g., by the integral representa- tions of γ(µ,w) in terms of Jµ (z) for Re µ > 0 [4, Eq. 9.3(4)] and of Γ(µ,w) in terms of modified Bessel function of the second kind Kµ (z) for Re µ < 1 [8, Eq. 8.6.6]. In this re- spect, it is worth mentioning also the expansions of incomplete gamma functions in series of Bessel functions: [5, Eqs. (2.8), (2.9)], [11, Eqs. (40)], and of modified Bessel func- tions: [8, Eqs. 8.7.4, 8.7.5], [10, Eqs. (48), (49)]. Formula (1.1), whose validity extends to (µ,z) C2, makes even more explicit the intimate link between these two classes of functions.∈ Classical formulae of the Bessel function of the first kind Jµ (z) given by integrals on the real line can be found in [8, Chapter 10.9], [4, Sect. 7.3] and [6, Sect. 8.41]. Many of them hold only for restricted values of µ: for instance, the classical Bessel integral [7, p. 70], µ π ( i) izcosθ (1.7) Jµ (z)= − e cos µθ dθ (µ Z), π 0 ∈ Z and Poisson’s integral [7, Eq. (14.5), p. 38], µ π 1 z izcosθ 2µ 1 (1.8) Jµ (z)= 1 e (sinθ) dθ (Re µ > 2 ), Γ(µ + )√π 2 0 − 2 Z 1 hold only for m Z and Re µ > 2 , respectively. Additional examples are the Gubler ∈ − 1 representations [4, Eq. 7.3(11)], which holds for Re µ < 2 , the Mehler-Sonine integral 1 formulae for Re µ < 2 [4, Eq. 7.12(12)] or for Re µ < 1 [4, Eq. 7.12(14)]. Only the well-known Schl¨afli’s| | and Heine’s representations| hold fo| r unrestricted complex values of the index µ. Schl¨afli’s representation [4, Eq. 7.3(9)] holds for µ C, with Rez > 0 (it holds also in case Rez = 0 provided that Re µ > 0), while the two Heine’s∈ expressions [4, Eqs. 7.3(31) & (32)] hold for µ C, but separately in the upper and lower half-plane of the complex z-plane, respectively.∈ In both cases, Schl¨afli’s and Heine’s, the representations are made up of the sum of two integrals. Standard generalizations of Poisson’s integral are obtained via its extension to complex contour integrals, examples of which are the classical Hankel’s representations [7, Eqs. (19.7), (19.8)]. As Poisson’s representation of Jµ (z) can be seen as the generalization of 1 Bessel’s integral to complex values of µ with Re µ > 2 , mere visual inspection shows that formula (1.1) can be viewed as the generalization− of Poisson’s integral to represent 2µ Jµ (z) at values of µ belonging to the entire complex plane, the role of (sinθ) in (1.8) 1 iθ being now taken by the incomplete gamma function γ∗(µ, 2 ize ). The proof of formula (1.1) is given in the next section and, essentially, makes use of elementary properties of the special functions involved. A preliminary result was given in [3]. 2. INTEGRAL REPRESENTATION FOR Jµ (z) We denote by N0 = 0,1,2,... the set of non-negative integers. { } Theorem 2.1. The following integral representation of the Bessel function of the first kind holds true for any complex order µ C and for z C (slit along the real negative axis ∈ ∈ INTEGRALFORMULAFORBESSELFUNCTION 3 when µ Z): 6∈ µ π 1 z izcosθ 1 iθ C C (2.1) Jµ (z)= e γ∗ µ, 2 ize dθ (µ ;z ). 2π 2 π ∈ ∈ Z− Proof. Our starting point is the Gegenbauer generalization of Poisson’s integral represen- tation of the Bessel functions of the first kind [12, §3.32, Eq. (1), p. 50], which holds for 1 ℓ N0 and Reν > : ∈ − 2 ℓ ν π ( i) Γ(2ν)ℓ! z izcosu 2ν ν (2.2) Jν+ℓ(z)= −1 1 e (sinu) Cℓ (cosu)du, Γ(ν + )Γ( )Γ(2ν + ℓ) 2 0 2 2 Z ν where Cℓ (t) is the Gegenbauer polynomial of order ν and degree ℓ. Now, we recall the ν following representation of the Gegenbauer polynomials Cℓ (cosu), holding for Reν > 0 and ℓ N0, that we proved in [2, Proposition 1]: (2.3) ∈ iπν 2π u ν Γ(ℓ + 2ν) e− − i(ℓ+ν)t ν 1 Cℓ (cosu)= ν 2 e (cosu cost) − dt (u [0,π]). 2 ℓ![Γ(ν)] sinu (2ν 1) u − ∈ ( ) − Z Plugging (2.3) into (2.2) and using the Legendre duplication formula for the gamma func- tion, we obtain: ℓ iπν ν π 2π u ( i) e− z izcosu − i(ℓ+ν)θ ν 1 (2.4) Jν+ℓ(z)= − du sinue e (cosu cosθ) − dθ. 2π Γ(ν) 0 u − Z Z Interchanging the order of integration, (2.4) can be written as follows: ℓ iπν ν π θ ( i) e− z i(ℓ+ν)θ izcosu ν 1 Jν+ℓ(z)= − dθ e e (cosu cosθ) − sinudu 2π Γ(ν) 0 0 − (2.5) Z Z 2π 2π θ i(ℓ+ν)θ − izcosu ν 1 + dθ e e (cosu cosθ) − sinudu . π 0 − Z Z Next, changing the integration variables: θ 2π θ and u u, the second integral on the r.h.s. of (2.5) becomes: − → →− 0 θ iν2π i(ℓ+ν)θ izcosu ν 1 e dθ e e (cosu cosθ) − sinudu, π 0 − Z− Z which, inserted in (2.5), yields: ℓ ν 0 θ ( i) z iνπ i(ℓ+ν)θ izcosu ν 1 Jν+ℓ(z)= − e dθ e e (cosu cosθ) − sinudu 2π Γ(ν) π 0 − (2.6) Z− Z π θ iνπ i(ℓ+ν)θ izcosu ν 1 +e− dθ e e (cosu cosθ) − sinudu . 0 0 − Z Z Formula (2.6) allows us to write Jν+ℓ(z) as the Fourier coefficient of a suitable 2π-periodic function, i.e.: π ℓ iℓθ (2.7) Jν+ℓ(z) = ( i) Aν,z(θ) e dθ (ℓ N0,Reν > 0), − π ∈ Z− where Aν,z(θ) denotes the Abel-type integral ν 1 z iν[θ π sgn(θ)] izt ν 1 (2.8) A (θ)= e − e (t cosθ) − dt (Reν > 0), ν,z 2πΓ(ν) Zcosθ − 4 ENRICO DE MICHELI and sgn( ) denotes the sign function. Changing in (2.8) the integration variable iz(t · − − cosθ) t and recalling (1.2a), Aν,z(θ) can be rewritten as follows: → ν i iν[θ π sgn(θ)] izcosθ (2.9) Aν,z(θ)= e − e P(ν, iz(1 cosθ)) (Reν > 0), 2π − − .
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