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Error Functions Error functions Nikolai G. Lehtinen April 23, 2010 1 Error function erf x and complementary er- ror function erfc x (Gauss) error function is 2 x 2 erf x = e−t dt (1) √π Z0 and has properties erf ( )= 1, erf (+ ) = 1 −∞ − ∞ erf ( x)= erf (x), erf (x∗) = [erf(x)]∗ − − where the asterisk denotes complex conjugation. Complementary error function is defined as ∞ 2 2 erfc x = e−t dt = 1 erf x (2) √π Zx − Note also that 2 x 2 e−t dt = 1 + erf x −∞ √π Z Another useful formula: 2 x − t π x e 2σ2 dt = σ erf Z0 r 2 "√2σ # Some Russian authors (e.g., Mikhailovskiy, 1975; Bogdanov et al., 1976) call erf x a Cramp function. 1 2 Faddeeva function w(x) Faddeeva (or Fadeeva) function w(x)(Fadeeva and Terent’ev, 1954; Poppe and Wijers, 1990) does not have a name in Abramowitz and Stegun (1965, ch. 7). It is also called complex error function (or probability integral)(Weide- man, 1994; Baumjohann and Treumann, 1997, p. 310) or plasma dispersion function (Weideman, 1994). To avoid confusion, we will reserve the last name for Z(x), see below. Some Russian authors (e.g., Mikhailovskiy, 1975; Bogdanov et al., 1976) call it a (complex) Cramp function and denote as W (x). Faddeeva function is defined as 2 2i x 2 2 2 w(x)= e−x 1+ et dt = e−x [1+erf(ix)] = e−x erfc ( ix) (3) √π Z0 ! − Integral representations: 2 2 i ∞ e−t dt 2ix ∞ e−t dt w(x)= = (4) π −∞ x t π 0 x2 t2 Z − Z − where x> 0. These integral representations can be converted to (3) using ℑ 1 ∞ = 2i e2i(x+i∆−t)u du (5) x + i∆ t − 0 − Z 3 Plasma dispersion function Z(x) Plasma dispersion function Z(x) (Fried and Conte, 1961) is also called Fried-Conte function (Baumjohann and Treumann, 1997, p. 268). In the book by Mikhailovskiy (1975), notation is ZMikh(x) xZ(x), which may be a source of confusion. Plasma dispersion function is≡ defined as: 2 1 +∞ e−t Z(x)= dt (6) √π −∞ t x Z − for x> 0, and its analytic continuation to the rest of the complex x plane (i.e,ℑ to x 0). Noteℑ that≤ Z(x) is just a scaled w(x), i.e. Z(x) i√πw(x) ≡ 2 We see that 2 ix 2 2 2 Z(x) = 2ie−x e−t dt = i√πe−x [1+erf(ix)] = i√πe−x erfc ( ix) (7) −∞ Z − One can define Z¯ which is given by the same equation (6), but for x < 0, and its analytic continuation to x 0. It is related to Z(x) as ℑ ℑ ≥ 2 Z¯(x)= Z∗(x∗)= Z(x) 2i√πe−x = Z( x) − − − 4 (Jackson) function G(x) Another function useful in plasma physics was introduced by Jackson (1960) and does not (yet) have a name (to my knowledge): G(x)=1+ i√πxw(x)=1+ xZ(x)= Z′(x)/2 (8) − Integral representation: 2 1 ∞ te−t dt G(x)= (9) √π −∞ t x Z − where again x> 0. ℑ 5 Other definitions 5.1 Dawson integral Dawson integral F (x) (Abramowitz and Stegun, 1965, eq. 7.1.17), also denoted as S(x) by Stix (1962, p. 178) and as daw x by Weideman (1994), is defined as: 2 z 2 2 F (z)= e−z et dt =( Z(x)+ i√πe−x )/2= xY (x) Z0 − (see also the definition of Y (x) below). 5.2 Fresnel functions Fresnel functions (Abramowitz and Stegun, 1965, ch. 7) C(x), S(x) are defined by x 2 C(x)+ iS(x)= eπit /2 dt Z0 3 5.3 (Sitenko) function ϕ(x) Another function (Sitenko, 1982, p. 24) is ϕ(x), defined only for real argu- ments: 2 x 2 ϕ(x) = 2xe−x et dt (10) Z0 so that 2 G(x) = 1 ϕ(x)+ i√πxe−x (11) − 5.4 Function Y (x) of Fried and Conte (1961) Fried and Conte (1961) introduce −x2 e x 2 Y (x)= et dt (12) x Z0 so that for real argument 2 Z(x)= i√πe−x 2xY (x) (13) − 6 Asymptotic formulas 6.1 For x 1 (series expansion) | | ≪ See Abramowitz and Stegun (1965, 7.1.8): ∞ (ix)n w(x)= (14) Γ n + 1 nX=0 2 2 The even terms give e−x . To collect odd terms, note that for n 0: ≥ 3 √π (2n + 1)!! Γ n + = n (15) 2 2 2 1 (2n 1)!! √ Γ n + = π −n (16) 2 2 where we define (2n + 1)!! = 1 3 5 (2n +1) and ( 1)!! = 1. We have · · ··· − ∞ 2 n − 2 2ix ( 2x ) w(x) = e x + − (17) √π (2n + 1)!! nX=0 ∞ 2 n − 2 ( 2x ) Z(x) = i√πe x 2x − (18) − (2n + 1)!! nX=0 4 The first few terms are 2 4 2 2ix 2x 4x w(x) e−x + 1 + . (19) ≈ √π − 3 15 − ! 2 4 2 2x 4x Z(x) i√πe−x 2x 1 + . (20) ≈ − − 3 15 − ! The Jackson function (8) also has a nice expansion ∞ (ix)n G(x) = √π (21) Γ n+1 nX=0 2 ∞ 2 n − 2 ( 2x ) = i√πxe x + − (22) (2n 1)!! nX=0 − 1+ i√πx 2x2 + . (23) ≈ − 6.2 For x 1 | | ≫ These formulas are valid for π/4 < arg x< 5π/4 (Abramowitz and Stegun, 1965, 7.1.23), i.e., around positive− imaginary axis: i ∞ (2m 1)!! w(x) = − √πx (2x2)m mX=0 i 1 3 −x2 1+ 2 + 4 + . + e (24) ≈ √πx 2x 4x 1 ∞ (2m 1)!! Z(x) = − −x (2x2)m mX=0 1 1 3 2 √ −x 1+ 2 + 4 + . + i πe (25) ≈ −x 2x 4x ∞ (2m 1)!! G(x) = − − (2x2)m mX=1 1 3 2 . + i√πxe−x (26) ≈ −2x2 − 4x4 − 7 Gordeyev’s integral Gordeyev’s integral Gν(ω,λ) (Gordeyev, 1952) is defined as ∞ νt2 Gν(ω,λ)= ω exp iωt λ(1 cos t) dt, ν > 0 (27) Z0 " − − − 2 # ℜ 5 and is calculated in terms of the plasma dispersion function Z (Paris, 1998): ∞ iω −λ ω n Gν(ω,λ)= − e In(λ)Z − (28) √2ν n=−∞ √2ν ! X 8 Plasma permittivity The dielectric permittivity of hot (Maxwellian) plasma is (Jackson, 1960) 1 ǫ(ω, k)=1+ ∆ǫs =1+ 2 2 G(xs) (29) s s k λs X X The summation is over charged species. For each species, we have introduced the Debye length ǫ T v λ = 0 = sNq2 Π 2 where v = T/m is the thermal velocity and Π = Nq /(mǫ0) is the plasma frequency;q and q ω (k u) x = − · √2kv where u is the species drift velocity. For warm components, x 1, we have ≫ Π2 3k2v2 ∆ǫ = 2 1+ 2 −[ω (k u)] [ω (k u)] ! − · − · The dispersion relation for plasma oscillations is obtained by equating ǫ = 0. For example, for warm electron plasma at rest, Π2 3k2v2 ǫ = 1 1+ − ω2 ω2 ! and we have the dispersion relation (for ω Π): ≈ ω2 = Π2 + 3k2v2 = Π2 + k2 v2 h i 6 9 Ion acoustic waves Assume that for electrons, ω kv but for ions still ω kV (we’ll see later ≪ ≫ that in means T T ). For x 1, we use G(x) 1+ i√πx: i ≪ e ≪ ≈ Π2 ω Π2 3k2V 2 ǫ =1+ e 1+ i√π i 1+ (30) k2v2 √2kv ! − ω2 ω2 ! where v and V are thermal velocities of electrons and ions, respectively. From ǫ = 0 we have 2 2 2 3k V ω 2 2 π ω 1+ 2 = 2 2 1+ k λe + i ω k vs r 2 Πekλe where vs = λeΠi = ZTe/M. If we neglect V and imaginary part, then we get q kv ω = s (31) 2 2 1+ k λe q For long wavelengths, it reduces to the usual relation ω = kvs. If we substi- tute this into the imaginary part, we get kv ω s (32) ≈ πZm 1+ i 2M r q The attenuation coefficient for the ion-acoustic waves is small: πZm γ = ω = kvs ω −ℑ s 8M ≪ ℜ Neglecting V is equivalent to kV ω, i.e., V vs or Ti Te. Other- wise, the ion-acoustic waves do not exist.≪ ≪ ≪ 10 Fourier transform of Z(x) In the plasma dielectric permittivity expression, the argument of G(x) = Z′(x) is x = [ω (k u)]/[√2kv]. Thus, the plasma dispersion function is usually− applied in− the frequency· domain. The argument is dimensionless, but is proportional to ω. Let us consider an inverse Fourier transform of F˜(ω)= 7 Z(ω/[√2Ω]), where Ω is a parameter which has the same dimensionality as ω. (In the plasma dielectric permittivity expression, we have Ω kv.) Quick reminder of the time-frequency and space-wavevector Fourier≡ trans- forms: X˜(ω, k) = X(t, r)eiωt−ik·r dtd3r ZZ 3 ˜ −iωt+ik·r dω d k X(t, r) = X(ω, k)e 3 ZZ 2π (2π) where integrals are from to + . Using expression (6),−∞ we have ∞ ′2 − ω ′ +∞ 2Ω2 ˜ ω 2π e dω F (ω)= Z = ′ √2Ω! √π −∞ ω ω i∆ 2π Z − − where the condition that ω > 0 is implemented by adding a small imaginary part to ω, i.e., ω ω ℑ+ i∆. We notice that the above expression is a convolution which simply→ gives a product in the t-domain: +∞ ′ ′ ′ dω F˜(ω)= F˜1(ω )F˜2(ω ω ) = F (t)= F1(t)F2(t) −∞ Z − 2π ⇒ where 2 2 2 1 − ω − Ω t 2Ω2 2 F˜1(ω) = 2π e = F1(t)= e √2πΩ ⇒ and i F˜ (ω)= √2iΩ = F (t)= √2iΩH(t) 2 ω + i∆ ⇒ 2 where H(t) is the Heaviside (step) function, defined to be H(t) = 0 for t< 0 and H(t) = 1 for t> 0.
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