arXiv:1703.06757v1 [math.CA] 14 Mar 2017 t ouingvnb h ent integral definite the by given solution Its rpitsbitdt Elsevier to submitted Preprint .Introduction 1. expansion hyp asymptotic Confluent integral, function, Gordeyev’s functions, Fresnel function, Keywords: functio these function. of hyperge hypergeometric expansions confluent asymptotic the plane the of complex hence, terms And in evaluated function. analytically are and function th tegral Fresnel Fried-Con function, physics (Jackson) integral), (Faddeeva’s mathematical function, function dispersion) in error functions complex the related clude and integral Dawson’s Abstract asnsitga n eae ucin nmathematical in functions related and integral Dawson’s colo ahmtc n ttsis altnUniversity Carleton Statistics, and Mathematics of School e scnie h rtodriiilvleproblem, value initial first-order the consider us Let mi address: Email nltcleauto n smttceauto of evaluation asymptotic and evaluation Analytical asnsitga,Cmlxerrfnto,Pam dispersion Plasma function, error Complex integral, Dawson’s C [email protected] r eie sn h smttcepnino h confluent the of expansion asymptotic the using derived are daw D ′ z 2 + = .Nijimbere V. zD D physics ( z 1 = = ) D , e − 0 0 = (0) z V Nijimbere) (V. 2 Z 0 z e η 2 . dη taa nai,Canada Ontario, Ottawa, , odyvsin- Gordeyev’s ac 1 2017 21, March ergeometric e(plasma te so the on ns ometric tin- at (2) (1) 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, . ··· 6 Proof. To prove (21), we use the asymptotic expansion of the confluent hypergeometric function valid for ξ 1 ([1], formula 13.5.1), | |≫

R 1 iπa a − 1F1 (a; b; ξ) e± ξ− (a)n(1 + a b)n n R = − ( ξ)− + O( ξ − ) Γ(b) Γ(b a) n! − | | − (n=0 ) XS 1 ξ a b − e ξ − (b a)n(1 a)n n S + − − (ξ)− + O( ξ − ) , (24) Γ(a) n! | | (n=0 ) X where a and b are constants, and the positive sign being taken if π 3π < arg(ξ) < , (25) − 2 2 and the negative sign if 3π π < arg(ξ) . (26) − 2 ≤ − 2 α 1 1 We now set ξ = z , a = α and b = α + 1 in (23), and obtain

1 1 α π R 1 1 i α − 1F1 α ; α + 1; z e ± α n α n α R − 1 = 1 (z )− + O (z ) Γ +1 α α n! α
(z ) (n=0
) X α S 1 1 z −
1 e (1)n 1 α n α n α S + − (z )− + O (z )− . (27) Γ 1 zα n! α (n=0
) X Then for z 1,
| |≫

i π 1 α Γ( α +1) π e± 1 R 1 1 1+ α + O 2α , if α is even i α − z z z e± α n α n α R | | 1 (z )− + O (z )−    α i π Γ 1 +1 (z ) α n! ∼  e± α ( α ) 1
(n=0
)  1+ α + O 2α , if α is odd X  z z z  
(28)  while 

α S 1 1 α 1 z − z 1 e (1)n α n α n α S 1 e Γ 2 α 1 (z )− + O (z )− 1+ − + O . Γ 1 zα n! ∼ Γ 1 zα zα z2α α (n=0
) α "
 # X (29)

7 Therefore for z 1, | |≫ 1 1 F ; + 1; zα 1 1 α α   π 1 1 i Γ +1 zα Γ 2 1 e± α ( α ) 1 e ( − α ) 1 Γ α +1 z 1+ zα + O z2α + αzα 1+ zα + O z2α , if α is even | |  π  1   1  i Γ +1 zα Γ 2 ∼  1
e± α ( α ) 1
e ( − α ) 1
Γ α +1 z 1+ zα + O z2α + αzα 1+ zα + O z2α , if α is odd     


(30)  Hence for z 1, | |≫ 1 1 z F ; + 1; zα 1 1 α α   1 1 π α Γ( α +1) z 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     

(31)
 On the other hand, we observe that

1 1 1 1 arg(ξ) α i arg(ξ) α i ξ = z z = ξ α = ξ α e = ξ α e α . ⇒ | | | | Therefore, (25) gives   π 2kπ 3π 2kπ + < arg(z) < + , −2α α 2α α which is exactly (22), while (26) gives 3π 2kπ π 2kπ + < arg(z) < + , −2α α −2α α which is exactly (23), and where k =0, 1, 2, . ··· Theorem 2. For z 1 if π + kπ < arg(z) < 3π + kπ,k = 0, 1, 2, , | | ≫ − 4 4 ··· then Dawson’s integral is asymptotically given by

√π z2 √π 1 1 √π 1 daw z i e− 1+ + O + + + O . (32) ∼ 2 2z2 z4 2z 4z3 z5      8 While, on the other hand, it is given by

√π z2 √π 1 1 √π 1 daw z i e− 1+ + O + + + O (33) ∼ − 2 2z2 z4 2z 4z3 z5      if 3π + kπ < arg(z) < π + kπ,k =0, 1, 2, . − 4 − 4 ··· π 3π Therefore, for z 1, if 4 + kπ < arg(z) < 4 + kπ,k = 0, 1, 2, , then | | ≫ − ··· 1. Feddeeva’s integral w(z) is approximated by

z2 √π 1 i 1 √π 1 w(z) e− + O + + + O , (34) ∼ −2z2 z4 √π z 2z3 z5       2. Fried-Conte (plasma dispersion) function Z(z) by

z2 π 1 1 √π 1 Z(z) ie− + O + O (35) ∼ −2z2 z4 − 2z − 4z3 z5      3. and (Jackson) function G(z) by

z2 π 1 √π 1 G(z) ie− + O + O . (36) ∼ −2z z3 − 2z2 z4      While on the other hand, if 3π + kπ < arg(z) < π + kπ, then − 4 − 4 1. Feddeeva’s integral w(z) is approximated by

z2 √π 1 i 1 √π 1 w(z) e− 2+ + O + + + O , (37) ∼ 2z2 z4 √π z 2z3 z5       2. Fried-Conte (plasma dispersion) function Z(z) by

z2 √π 1 1 √π 1 Z(z) i√πe− 2+ + O + O , (38) ∼ 2z2 z4 − z − 2z3 z5      3. and (Jackson) function G(z) by

z2 √π 1 √π 1 G(z) i√πe− 2z + + O + O (39) ∼ 2z z3 − 2z2 z4     

9 Proof. For α = 2, and having in mind that α = 2 is even, (21) becomes

3 z2 3 1 3 2 3 i π z Γ 2 1 e Γ 2 1 z F ; ; z Γ e± 2 1+ + O + 1+ + O 1 1 2 2 ∼ 2 z z2 z4 2z z2 z4     | | "
 # "
 # √π z √π 1 ez2 √π 1 = i 1+ + O + 1+ + O , ± 2 z 2z2 z4 2z 2z2 z4       | | (40) where the positive sign is taken if π 3π + kπ < arg(z) < + kπ, (41) − 4 4 while the negative sign is taken if 3π π + kπ < arg(z) < + kπ, (42) − 4 − 4 k =0, 1, 2, . When substituting··· (40) in (12), the resulting equation together with (41) π 3π and (42) respectively gives (32) if 4 + kπ < arg(z) < 4 + kπ and (33) 3π π − if 4 + kπ < arg(z) < 4 + kπ. And hence, substituting (32) and (33) − − π in (13), (14) and (15) gives respectively (34), (35) and (36) if 4 + kπ < arg(z) < 3π + kπ and (37), (38) and (39) if 3π + kπ < arg(z) <− π + kπ. 4 − 4 − 4 This ends the proof.

3.2. Asymptotic evaluation of Fresnel functions In this section, we use the asymptotic expansion of the confluent hy- pergeometric function 1F1 to derive the asymptotic expansion of Fresnel’s functions C(z) and S(z). And the results are described in Theorem 3.

Theorem 3. For z 1, Fresnel’s functions C(z) and S(z) are asymptot- ically given by | |≫

√2 2 1 sin(πz2) cos (πz2) 1 π π 2 3/2 3 + O 4 , if + kπ < arg(z) < + kπ C(z) 4 − π 8z − 2πz − 4(π) z z − 2 2 , ∼  √2 q 2 1 sin(πz2) cos (πz2) 1 π + 2 3/2 + O
4 , if π + kπ < arg(z) < + kπ − 4 π 8z − 2πz − 4(π) z3 z − − 2 q (43) 

10 and

√2 2 1 sin(πz2) cos (πz2) 1 π π + 2 + + 3/2 3 + O 4 , if + kπ < arg(z) < + kπ S(z) 4 π 8z 2πz 4(π) z z − 2 2 , ∼  √2 q 2 1 sin(πz2) cos (πz2) 1 π 2 + + 3/2 + O
4 , if π + kπ < arg(z) < + kπ − 4 − π 8z 2πz 4(π) z3 z − − 2 q
(44) where k=0, 1, 2, . ··· Proof. Let us first assume that for z 1, Fresnel’s functions are approx- | |≫ imated by C(z) C˜(z) and S(z) S˜(z). Then using (16), (17) and (18) yields ∼ ∼

z iπz2 iπη2 e 1 3 2 e dη = daw √iπz = z 1F1 ; ; iπz (45) √iπ 2 2 Z0     z z = cos(πη2)dη + i sin (πη2)dη (46)

Z0 Z0 1 1 5 πz2 πz3 3 3 7 πz2 = z F ; , ; + i F ; , ; (47) 1 2 4 2 4 − 4 3 1 2 4 2 4 − 4     = C(z)+ iS(z) C˜(z)+ iS˜(z), z 1. (48) ∼ | |≫ And we observe that the asymptotic expansion of (45) is a sum of two parts C˜(z) and S˜(z). 2 1 1 1 3 Now, setting ξ = iπz , a = α = 2 and b = α +1= 2 in (24), and taking account that α = 2 is even yields

1 3 2 π R 1 1 i 2 − 1F1 2 ; 2 ; iπz e± 2 n 2 n 2 R 3 = 1 ( iπz )− + O( z − ) Γ 2 2 n! − | | 2
(iπz ) (n=0
) X 2 S 1 1 iπz −
1 e (1)n 2 n n S + (ξ)− + O( ξ − ) . (49) Γ 1 iπz2 n! | | 2 (n=0
) X Moreover,
1 1 2 2 2 ξ ξ i( arg(ξ) π ) ξ = iπz z = = | | e 2 − 4 ⇒ iπ π     that gives arg(ξ) π arg(z)= + kπ,k =0, 1, 2, . (50) 2 − 4 ··· 11 Rearranging terms on one hand, while neglecting higher order terms on another hand, yields

z 2 2 2 1 3 √2 2 1 sin (πz ) cos(πz ) 1 eiπη dη = z F ; ; iπz2 = +O 1 1 2 2 ± 4 ∓ π 8z2 − 2πz −4(π)3/2z3 z 4 Z0   r | |  √2 2 1 sin (πz2) cos(πz2) 1 + i + + + O . (51) ± 4 ± π 8z2 2πz 4(π)3/2z3 z 4 " r | | # where the positive sign is now taken if π π + kπ < arg(z) < + kπ, (52) − 2 2 and the negative sign if π π + kπ < arg(z) < + kπ. (53) − − 2 Hence, comparing (48) with (51) gives (43) and (44).

3.3. Asymptotic evaluation of Gordeyev’s integral In this section, we derive the asymptotic expansion of Gordeyev’s integral using the asymptotic expansion of the confluent hypergeometric function 1F1 and present the results in Theorem 4. We first observe that for complex ω = ωr + iωi and complex ν = νr + iνi, if we set √νr + iνi =ν ˜r + iν˜i, where the subscripts r and i stand for real and imaginary parts respectively, then ω n (ω + n)˜ν + ω ν˜ ω ν˜ (ω + n)˜ν − = r i i i + i i r − r i . (54) 2 2 2 2 √2ν √2(˜νr +˜νi ) √2(˜νr +˜νi ) And so ω n ω ν˜ (ω + n)˜ν arg − = arctan i r − r i . (55) √2ν (ωr + n)˜νi + ωiν˜i    

Theorem 4. Let λ = λr + iλi, ω = ωr + iωi, ν = νr + iνi and √ν = √νr + iνi =ν ˜r +iν˜i, where the subscripts r and i stand for real and imaginary parts respectively, and let ω n ω ν˜ (ω + n)˜ν θ = arg − = arctan i r − r i . (56) √ ν (ωr + n)˜νi + ωiν˜i  2    12 1. Then for any fixed ν and any fixed ω, Gordeyev’s integral Gν(ω,λ) is asymptotically given by

iω ∞ 4n2 1 1 ω n Gν(ω,λ) − 1 − + O Z − , λ 1, ∼ 2√πνλ − 8λ λ2 √2ν | |≫ n=      X−∞ (57) where Z is the plasma dispersion (Fried-Conte) function, see equation (4), and π < arg(λ) < π . − 2 2 2. For any fixed λ, if ω 1 and ν is fixed, or if ν 0 and ω is fixed, then | |≫ | |→

2 4 2 ∞ ω n iω λ − π √2ν √2ν G (ω,λ) − e− I (λ) ie− √2ν  + O ν ∼ √ n − 2 ω n ω n  2ν n= ! ! X−∞ n − − 3 5  1 √2ν √π √2ν √2ν + O , (58) − 2 ω n! − 4 ω n! ω n! − − − o if π + kπ<θ< 3π + kπ, k =0, 1, 2, . And − 4 4 ··· 2 4 2 ∞ ω n iω λ − √π √2ν √2ν G (ω,λ) − e− I (λ) i√πe √2ν  2+ + O ν ∼ √ n  2 ω n ω n  2ν n= ! ! X−∞ n − − 3  5  1 √2ν √π √2ν √2ν + O , (59) − 2 ω n! − 4 ω n! ω n! − − − o 3π π if 4 + kπ<θ< 4 + kπ. − π − π 3. If λ 1 and 2 < arg(λ) < 2 , and ω 1 while ν is fixed, or and ν | |≫0 while ω−is fixed, then | |≫ | |→ 2 2 2 ∞ ω n 2 iω − π √2ν (4n 1)π √2ν G (ω,λ) − ie− √2ν  + − ν ∼ √ − 2 ω n 16λ ω n 2 πνλ n= ! ! −∞n h − − 2 X 3 √2ν 1 √2ν (4n2 1) √2ν √π √2ν +O + − λ(ω n) −2 ω n 16λ ω n − 4 ω n − ! − ! − ! − ! i 3 (4n2 1)√π √2ν √2ν + − + O 2 , (60) 32λ ω n! λ (ω n)! − − o

13 if π + kπ<θ< 3π + kπ, k =0, 1, 2, . And − 4 4 ··· 2 2 ∞ ω n 2 iω − (4n 1) √π √2ν G (ω,λ) − i√πe− √2ν  2 − + ν ∼ √ − 4λ 2 ω n 2 πνλ n= ! −∞n h − X2 2 (4n2 1)√π √2ν √2ν 1 √2ν (4n2 1) √2ν − +O + − − 16λ ω n λ(ω n) −2 ω n 16λ ω n − ! − ! − ! − ! 3 i 3 √π √2ν (4n2 1)√π √2ν √2ν + − + O 2 , − 4 ω n! 32λ ω n! λ (ω n)! − − − o (61)

if 3π + kπ<θ< π + kπ, k =0, 1, 2, . − 4 − 4 ···

Proof. 1. For λ 1, we have using formula 9.7.1 in [1] that | |≫ eλ 4n2 1 1 π π I (λ)= 1 − + O , < arg(λ) < . (62) n √ − 8λ λ2 − 2 2 2πλ    Substituting in (19) gives (57). ω n 2. Setting z = − in (35) and (38), and letting ω 1 while ν is fixed, √2ν | |≫ or and ν 0 while ω is fixed, yields respectively | |→ 2 4 2 ω n ω n − π √2ν √2ν Z − ie− √2ν  + O √2ν ∼ − 2 ω n ω n    − ! − !  3  5 1 √2ν √π √2ν √2ν + O , (63) − 2 ω n − 4 ω n ω n − ! − ! − !

π ω n 3π if + kπ <θ = arg − < + kπ, k =0, 1, 2, . And − 4 √2ν 4 ···   2 4 2 ω n ω n − √π √2ν √2ν Z − i√πe √2ν  2+ + O √2ν ∼  2 ω n ω n    − ! − !  3 5  1 √2ν √π √2ν √2ν + O , (64) − 2 ω n − 4 ω n ω n − ! − ! − !

14 3π ω n π if + kπ < θ = arg − < + kπ, k = 0, 1, 2, . Hence, − 4 √2ν − 4 ··· substituting (63) and (64) in (20) respectively gives (58) and (59). 3. On the other hand, combining (58) and (59) with (57) respectively gives (60) and (61).

4. Discussion and conclusions

Having evaluated Dawson’s integral in terms of the confluent hypergeo- metric function, other related functions including the complex error (Fad- deeva’s integral), Fried-Conte (plasma dispersion) function, (Jackson) func- tion, Fresnel functions and Gordeyev’s integral were also evaluated in terms of the confluent hypergeometric function . Using the asymptotic expansions of the confluent hypergeometric func- tion, the asymptotic expansion for z 1 of Dawson’s integral were derived and consequently the asymptotic expansions| |≫ of the complex error function (Faddeeva’s integral), Fried-Conte (plasma dispersion) function, (Jackson) function, Fresnel functions and Gordeyev’s integral were evaluated (Theorem 2, Theorem 3 and Theorem 4). To obtain, on the other hand, the asymptotic expansion of these functions for small arguments z 1 one should keep the first few terms in the Taylor series representing the| |≪ confluent hypergeometric function. It is also important to point that asymptotic expansions of Gordeyev’s integral that takes into account the properties of an electromagnetic wave propagating in a hot plasma, which are the wave frequency, the perpendicular and parallel components of the wave vector, were carefully derived (Theorem 4). Moreover, writing these functions in terms confluent hypergeometric func- tion confirms once again that these functions are entire on the whole complex plane C since the confluent hypergeometric function is entire on the whole complex plane C.

References

[1] M. Abramowitz, I.A. Stegun, Handbook of mathematical functions with formulas, graphs and mathematical tables. Nat. Bur. Stands. 1964. 1046 pp.

15 [2] S.M. Abrarov, B.M. Quine, Efficient algorithimic implementation of the Voigt/complex error functions, Appl. Math. Comput., 218(5) (2012), pp. 1894 – 1902.

[3] S.M. Abrarov, B.M. Quine, Accurate approximations for the complex er- ror function with small imaginary argument, J. Math. Res., 7(1) (2015), pp. 44 – 53.

[4] W. Baumjohann, R.A. Treumann, Basic space plasma physics. Imperial College Press, 1997.

[5] E.P. Bogdanov, Y.A. Ramanov, S.A. Shamov, Kinetic theory of beam instability in periodic systems, Radiophys. Quant. Electronics 19(2) (1976), pp. 212 – 217.

[6] H. Borchert, D.V. Talapin, N. Gaponik, C. McGniley, S. Adam, A. Lobo, T. M¨oller, H. Weller, Relations between the photoluminescence efficiency of CdTe nanocrystals and their surface properties revealed by synchrotron XPS, J. Phys. Chem. B, 107(36) (2003), pp. 9662 – 9668.

[7] I. Charpentier, J. Gustedt, Arbogast Higher order AD for special functions with Modular C., Inria Nancy-Grand Est (Villers-l`es-Nancy, France), Research Report 8907, 2016.

[8] W.J. Cody, K.A. Paciorek, Chebyshev approximations for Dawson’s in- tegral, Math. Comp. 24 (1970), pp. 171 – 178.

[9] V.N. Fadeeva, N.M. Terent’ev, Tables of the probabilistic integral z2 2i z t2 w(z) = e− 1+ √π 0 e dt for complex arguments. Oxford Parga- mon Press, 1961. R  [10] V.N. Fadeeva, N.M. Terent’ev, Tables of values of the probabilistic inte- gral for complex arguments. Moscow State Publishing House for Tech- nical Theoritical Literature, 1954.

[11] B.D. Fried, S.D. Conte, The plasma dispersion function. New York Aca- denic Press, 1961.

[12] W. Gautschi, Efficient computation of the complex error function, SIAM J. Numer. Anal. 7 (1970), pp. 187 – 198.

16 [13] G.V. Gordeyev, Plasma oscillations in a magnetic field, Sov. Phys. JETP 6 (1952), pp. 660 – 669.

[14] E.A. Jackson, Drift instabilities in a maxwellian plasma, Phys. Fluids 3(5)(1960), pp. 786 – 792.

[15] S.J. McKenna, A method of computing the complex probability function and other related functions over the whole complex plane Astrophys. Space Sci., 107(1) (1984), pp. 71 – 83.

[16] E.A. Marchisotto, G.-A. Zakeri, An invitation to integration in finite terms, College Math. J., 25(4) (1994), pp. 295-308.

[17] J.H. McCabe, A continued fraction expansion with a truncation error estimate for Dawson’s integral, Math. Comp. 28(127) (1974), pp. 811 – 816.

[18] A.B. Mikhailovskiy, 1975 Theory of plasma instabilities. Russia Atom- izdat (in Russian), 1975

[19] NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/16.11.

[20] V. Nijimbere, Evaluation of the non-elementary integral eλxα dx, α 2 and related integrals, arXiv: 1702.08438 ≥ R [21] R.B. Paris, The asymptotic evaluation of Goldyev’s integral, Zeitschrift f¨ur Angewandte Mathematik and Physik, 49(2) (1998), pp. 322 – 338.

[22] G.P.M. Poppe, C.M.J Wijers, More efficient computation of the complex error function, ACM Transact. Math. Software 16(1) (1990), pp. 38 – 46.

[23] M. Rosenlicht, Integration in finite terms, Amer. Math. Monthly 79(9) (1972), pp. 963 – 972.

[24] F. Schreier, The Voigt and complex error functions: A comparison of computational methods, J. Quant. Spectrosc. Radiat. Transfer, 48(5) (1992), pp. 743 – 762.

[25] A.G. Sitenko, Fluctuations and nonlinear wave interactions in plasmas. New York Pergamon Press, 1982.

17 [26] T.H. Stix, The theory of plasma waves. McGraw-Hill Book Company, New York, 1962.

[27] J.A.C Weideman, Computation of the complex error function, SIAM J. Numer. Anal. 31(5) (1994), pp. 1497 – 1518.

[28] M.R. Zaghloul, A.N. Ali, Algorithm 916: computing the Faddeyeva and Voigt functions, ACM Transact. Math. Software 38(2) (2011), pp. 15:1 – 15:22.

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