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Analytical Evaluation and Asymptotic Evaluation of Dawson's Integral And Analytical evaluation and asymptotic evaluation of Dawson’s integral and related functions in mathematical physics V. Nijimbere School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada Abstract Dawson’s integral and related functions in mathematical physics that in- clude the complex error function (Faddeeva’s integral), Fried-Conte (plasma dispersion) function, (Jackson) function, Fresnel function and Gordeyev’s in- tegral are analytically evaluated in terms of the confluent hypergeometric function. And hence, the asymptotic expansions of these functions on the complex plane C are derived using the asymptotic expansion of the confluent hypergeometric function. Keywords: Dawson’s integral, Complex error function, Plasma dispersion function, Fresnel functions, Gordeyev’s integral, Confluent hypergeometric function, asymptotic expansion 1. Introduction Let us consider the first-order initial value problem, D′ +2zD =1,D(0) = 0. (1) arXiv:1703.06757v1 [math.CA] 14 Mar 2017 Its solution given by the definite integral z z2 η2 daw z = D(z)= e− e dη (2) Z0 Email address: [email protected] (V. Nijimbere) Preprint submitted to Elsevier March 21, 2017 is known as Dawson’s integral [1, 17, 24, 27]. Dawson’s integral is related to several important functions (in integral form) in mathematical physics that include Faddeeva’s integral (also know as the complex error function or Kramp function) [9, 10, 22, 18, 27] z z2 2i z2 z2 2i η2 w(z)= e− 1+ e daw z = e− 1+ e dη , (3) √π √π Z0 Fried-Conte function (or plasma dispersion function) [4, 11] z z2 2i z2 z2 2i η2 Z(z)= i√πw(z)= i√πe− 1+ e daw z = i√πe− 1+ e dη , √π √π Z0 (4) (Jackson) function [14] G(z)=1+ zZ(z)=1+ i√πzw(z) z2 2i z2 =1+ i√πze− 1+ e daw z √π z z2 2i η2 =1+ i√πze− 1+ e dη , (5) √π Z0 and Fresnel functions C(x) and S(x) [1] defined by the relation z iπz2 e 2 daw √iπz = eiπη dη = C(x)+ iS(x), (6) √iπ Z0 where z z C(x)= cos(πη2)dη and S(x)= sin (πη2)dη. (7) Z0 Z0 There is also Gordeyev’s integral [13] which is related to Dawson’s integral via the plasma dispersion (Fried-Conte) function Z, and is given ∞ iωt λ(1 cos t) νt2/2 Gν(ω,λ)= ω e − − − dt Z0 ∞ iω λ ω n = − e− In(λ)Z − , Re(ν) > 0, (8) √2ν √2ν n= X−∞ 2 where In is the Bessel function of the first kind [1], the real part of ω is the wave frequency of an electrostatic wave propagating in a hot magnetized plasma, and λ and ν are respectively the squares of the perpendicular and parallel components of the wave vector [21]. Another function related to Dawson’s integral was defined by Sitenko [25], and only takes real arguments. It is given by x x2 η2 ϕ(x)=2x daw x =2xe− e dη. (9) Z0 In that case, for real argument x, Jackson function G(x), given by (5), takes the form x2 G(x)=1 ϕ(x)+ i√πxe− . (10) − These functions find their applications in astronomy, celestial mechanics, optical physics, plasma physics, planetary atmosphere, radiophysics, spec- troscopy and so on [4, 6, 11, 13, 14, 18, 25, 26]. Therefore, it is important to adequately evaluate them. Having computed Dawson’s integral or Faddeeva’s integral, it is then straightforward to compute the other related integrals or functions such as Fried-Conte function, Jackson function, Fresnel integral and Gordeyev function. First, it is important to point out that Dawson’s integral and Faddeeva’s integral are non-elementary integrals. Being non-elementary means that they cannot neither be expressed in terms of elementary functions such polynomi- als of finite degree, exponentials and logarithms, nor in terms of mathematical expressions obtained by performing finite algebraic combinations involving elementary functions [16, 20, 23]. For this reason, it is not possible to evalu- ate analytically these integrals in closed form or, in other words, in terms of elementary functions [3, 16, 20, 23]. To this end, intensive works have mainly focused on numerical approximations [2, 3, 7, 8, 9, 10, 12, 17, 15, 22, 27, 28]. But numerical integrations do have drawbacks, they become very expensive and inaccurate very quickly as z becomes large or for some values of z, see for example the recent work by Abrarov and Quine [3]. None of the above integrals can be evaluated analytically in close form, or in terms of elementary functions, as pointed out by Abrarov and Quine [3] of course, but one can express them in terms of a special function, the confluent hypergeometric function 1F1 [1, 19]. Noting that Nijimbere [20] (Theorem 1) has evaluated the non-elementary integral b eλxα dx,α 2 for any constant a ≥ R 3 λ in terms of the confluent hypergeometric function 1F1, as a first objective of this paper, we use the results in Nijimbere [20] to obtain an analytical expression for Dawson’s integral in terms of the confluent hypergeometric function 1F1. And hence, we write Faddeeva’s integral and the above related integrals in terms of 1F1. On the other hand, the confluent hypergeometric function is an entire function on the whole complex plane C, and its properties are well known [1, 19]. For instance, its asymptotic expansion is given in [1, 19]. Therefore, as a second objective of this work, the asymptotic expansion of the confluent hypergeometric function is used to obtain the asymptotic expansions of the above functions (integrals) on the complex plane C. 2. Evaluation of Dawson’s integral and related function in terms of 1F1 In this section, Dawson’s integral is evaluated in terms of the confluent hypergeometric 1F1, and relations (3), (4), (5), (6) and (8) are used to express Faddeeva, Fried-Conte, Jackson, Fresnel and Gordeyev integrals respectively in terms of 1F1. 2.1. Evaluation of Dawson’s integral in terms of 1F1 We use Theorem 1 in Nijimbere [20] to express (2) in terms of the con- fluent hypergeometric functions 1F1, and then obtain z z2 η2 z2 1 3 2 daw z = D(z)= e− e dη = ze− F ; ; z . (11) 1 1 2 2 Z0 One may also solve (1) and obtain [20] the general solution z z2 η2 z2 1 3 2 z2 D(z)= e− e dη + C = ze− F ; ; z + Ce− . (12) 1 1 2 2 Z0 Therefore, applying the initial condition D(0) = 0 gives (11). 4 2.2. Evaluation of related functions in terms of 1F1 We now can evaluate the other functions related to Dawson’s integral (see section 1) in terms of the confluent hypergeometric function. Using (3) Faddeeva’s integral is now given by z z2 2i η2 z2 2iz 1 3 2 w(z)= e− 1+ e dη = e− 1+ F ; ; z . (13) √π √π 1 1 2 2 Z0 Using (4), the plasma dispersion function Z(z) also called Fried-Conte func- tion is given by z2 2iz 1 3 2 Z(z)= i√πw(z)= i√πe− 1+ F ; ; z . (14) √π 1 1 2 2 Using (5), the (Jackson) function, denoted by G(z) is given by z2 2iz 1 3 2 G(z)=1+zZ(z)=1+i√πzw(z)=1+i√πze− 1+ F ; ; z . √π 1 1 2 2 (15) Using (6) (one may also use Proposition 1 in [20]) gives z iπz2 iπη2 e 1 3 2 e dη = daw √iπz = z 1F1 ; ; iπz , (16) √iπ 2 2 Z0 which is equivalent to formula 7.3.25 in [1]. Now using formula (53) in The- orem 2 and formula (57) in Theorem 3 in [20] (or using formulas (50) and (51) in [20]), we obtain z 1 1 5 πz2 C(x)= cos(πη2)dη = z F ; , ; , (17) 1 2 4 2 4 − 4 Z0 and z πz3 3 3 7 πz2 S(x)= sin (πη2)dη = F ; , ; . (18) 3 1 2 4 2 4 − 4 Z0 5 ω n Setting z = − in Fried-Conte function and substituting in (8), Gordeyev’s √2ν integral can now be written in terms of 1F1 as ∞ iω λ ω n Gν(ω,λ)= − e− In(λ)Z − (19) √2ν √2ν n= X−∞ λ 2 2 ωe− ∞ ω n 2i(ω n) 1 3 ω n √− = e− 2ν In(λ) 1+ − 1F1 ; ; − . 2ν/π √2πν 2 2 √2ν n= ( " #) X−∞ p (20) 3. Asymptotic evaluation In this section, we derive the asymptotic expansion of Dawson’s integral and related functions that include the complex error function, Fried-Conte function, Jackson function, Fresnel functions and Gordeyev’s function, using the asymptotic expansion of the confluent hypergeometric 1F1. The results are summarized in Theorem 2, Theorem 3 and Theorem 4. 3.1. Asymptotic evaluation of Dawson’s integral Lemma 1. For z 1, | |≫ 1 1 z F ; + 1; zα 1 1 α α 1 1 π zα Γ( α +1) Γ(2 α ) 1 i α z 1 e − 1 Γ α +1 e± z 1+ zα + O z2α + αzα 1 1+ zα + O z2α , if α is even | | − 1 1 π zα Γ( α +1) Γ(2 α ) ∼ 1 i α 1 e − 1 Γ +1 e± 1+ α + O 2α + α 1 1+ α + O 2α , if α is odd α z z αz − z z (21) where α is a constant, and the positive sign is taken if π 2kπ 3π 2kπ + < arg(z) < + , (22) −2α α 2α α while the negative sign is taken if 3π 2kπ π 2kπ + < arg(z) < + , (23) −2α α −2α α k =0, 1, 2, .
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