Error Fun FIR

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Error functions

Nikolai G. Lehtinen

April 23, 2010

1 Error function erfx and complementary er-ror function erfc x

(Gauss) error function is

erfx =2

x0

et2

dt (1)

and has properties

erf () = 1, erf (+) = 1erf (

x) =

erf (x), erf (x) = [erf (x)]

where the asterisk denotes complex conjugation. Complementary errorfunction is defined as

erfc x =2

xet

2

dt = 1 erfx (2)

Note also that2

x

et2

dt = 1 + erfx

Another useful formula:

x

0e

t2

22 dt =

2

erf

x2

Some Russian authors (e.g., Mikhailovskiy, 1975; Bogdanov et al., 1976) callerfx a Cramp function.

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2 Faddeeva function w(x)

Faddeeva (or Fadeeva) function w(x) (Fadeeva and Terentev, 1954; Poppeand 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 lastname for Z(x), see below. Some Russian authors (e.g., Mikhailovskiy, 1975;Bogdanov et al., 1976) call it a (complex) Cramp function and denote asW(x). Faddeeva function is defined as

w(x) = ex2 1 + 2i

x

0

et2

dt = ex2 [1 + erf (ix)] = ex2erfc(ix) (3)Integral representations:

w(x) =i

et2

dt

x t =2ix

0

et2

dt

x2 t2 (4)

where x > 0. These integral representations can be converted to (3) using1

x + i t = 2i

0e2i(x+it)u du (5)

3 Plasma dispersion function Z(x)

Plasma dispersion function Z(x) (Fried and Conte, 1961) is also calledFried-Conte function (Baumjohann and Treumann, 1997, p. 268). In thebook by Mikhailovskiy (1975), notation is ZMikh(x) xZ(x), which may bea source of confusion. Plasma dispersion function is defined as:

Z(x) =1

+

et2

t x dt (6)

for

x > 0, and its analytic continuation to the rest of the complex x plane(i.e, to x 0).

Note that Z(x) is just a scaled w(x), i.e.

Z(x) iw(x)

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We see that

Z(x) = 2iex2

ix

et2 dt = iex2 [1 + erf (ix)] = iex2erfc(ix) (7)

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

Z(x) = Z(x) = Z(x) 2iex2 = 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

xw(x) = 1 + xZ(x) = Z(x)/2 (8)

Integral representation:

G(x) =1

tet2

dt

t x (9)

where again x > 0.

5 Other definitions

5.1 Dawson integral

Dawson integral F(x) (Abramowitz and Stegun, 1965, eq. 7.1.17), alsodenoted as S(x) by Stix (1962, p. 178) and as daw x by Weideman (1994),is defined as:

F(z) = ez2

z0

et2

dt = (Z(x) + iex2)/2 = xY(x)

(see also the definition of Y(x) below).

5.2 Fresnel functionsFresnel functions (Abramowitz and Stegun, 1965, ch. 7) C(x), S(x) are definedby

C(x) + iS(x) =x0

eit2/2 dt

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5.3 (Sitenko) function (x)

Another function (Sitenko, 1982, p. 24) is (x), defined only for real argu-ments:

(x) = 2xex2

x0

et2

dt (10)

so thatG(x) = 1 (x) + ixex2 (11)

5.4 Function Y(x) of Fried and Conte (1961)

Fried and Conte (1961) introduce

Y(x) = ex2

xx0

et2

dt (12)

so that for real argument

Z(x) = i

ex2 2xY(x) (13)

6 Asymptotic formulas

6.1 For |x| 1 (series expansion)See Abramowitz and Stegun (1965, 7.1.8):

w(x) =n=0

(ix)n

n2 + 1

(14)The even terms give ex

2

. To collect odd terms, note that for n 0:

n +3

2

=

2

(2n + 1)!!

2n(15)

n +1

2

=

(2n 1)!!2n

(16)

where we define (2n + 1)!! = 1

3

5

(2n + 1) and (

1)!! = 1. We have

w(x) = ex2

+2ix

n=0

(2x2)n(2n + 1)!!

(17)

Z(x) = i

ex2 2x

n=0

(2x2)n(2n + 1)!!

(18)

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The first few terms are

w(x) ex2 + 2ix

1 2x

2

3+ 4x

4

15 . . . (19)

Z(x) iex2 2x

1 2x2

3+

4x4

15 . . .

(20)

The Jackson function (8) also has a nice expansion

G(x) =

n=0

(ix)n

n+12

(21)

= i

xex2

+

n=0(2x2)n

(2n

1)!!

(22)

1 + ix 2x2 + . . . (23)

6.2 For |x| 1These formulas are valid for /4 < arg x < 5/4 (Abramowitz and Stegun,1965, 7.1.23), i.e., around positive imaginary axis:

w(x) =ix

m=0

(2m 1)!!(2x2)m

ix 1 +

1

2x2+

3

4x4+ . . . + e

x2 (24)

Z(x) = 1x

m=0

(2m 1)!!(2x2)m

1x

1 +

1

2x2+

3

4x4+ . . .

+ i

ex

2

(25)

G(x) =

m=1

(2m 1)!!(2x2)m

12x2

34x4

. . . + ixex2 (26)

7 Gordeyevs integralGordeyevs integral G(, ) (Gordeyev, 1952) is defined as

G(, ) =

0exp

it (1 cos t) t

2

2

dt, > 0 (27)

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and is calculated in terms of the plasma dispersion function Z (Paris, 1998):

G(, ) =i

2e

n=

In()Z

n

2

(28)

8 Plasma permittivity

The dielectric permittivity of hot (Maxwellian) plasma is (Jackson, 1960)

(, k) = 1 +s

s = 1 +s

1

k22sG(xs) (29)

The summation is over charged species. For each species, we have introducedthe Debye length

=

0T

N q2=

v

where v =

T /m is the thermal velocity and =

N q2/(m0) is the plasmafrequency; and

x = (k u)

2kv

where u is the species drift velocity.For warm components, x

1, we have

= 2

[ (k u)]2

1 +3k2v2

[ (k u)]2

The dispersion relation for plasma oscillations is obtained by equating = 0. For example, for warm electron plasma at rest,

= 1 2

2

1 +

3k2v2

2

and we have the dispersion relation (for ):

2 = 2 + 3k2v2 = 2 + k2v2

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9 Ion acoustic waves

Assume that for electrons, kv but for ions still kV (well see laterthat in means Ti Te). For x 1, we use G(x) 1 + i

x:

= 1 +2e

k2v2

1 + i

2kv

2i

2

1 +

3k2V2

2

(30)

where v and V are thermal velocities of electrons and ions, respectively. From = 0 we have

1 +3k2V2

2=

2

k2v2s

1 + k22e + i

2

eke

where vs = ei =

ZTe/M. If we neglect V and imaginary part, then weget

=kvs

1 + k22e(31)

For long wavelengths, it reduces to the usual relation = kvs. If we substi-tute this into the imaginary part, we get

kvs1 + iZm

2M

(32)

The attenuation coefficient for the ion-acoustic waves is small:

= = kvs

Zm

8M

Neglecting V is equivalent to kV , i.e., V vs or Ti Te. Other-wise, the ion-acoustic waves do not exist.

10 Fourier transform ofZ(x)

In the plasma dielectric permittivity expression, the argument of G(x) =Z(x) is x = [ (k u)]/[2kv]. Thus, the plasma dispersion function isusually applied in the frequency domain. The argument is dimensionless, butis proportional to . Let us consider an inverse Fourier transform ofF() =

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Z(/[

2]), where is a parameter which has the same dimensionality as

. (In the plasma dielectric permittivity expression, we have kv.)Quick reminder of the time-frequency and space-wavevector Fourier trans-forms:

X(, k) =

X(t, r)eitikr dt d3r

X(t, r) =

X(, k)eit+ikrd

2

d3k

(2)3

where integrals are from to +.Using expression (6), we have

F() = Z

2

=

2

+

e

2

22

id

2

where the condition that > 0 is implemented by adding a small imaginarypart to , i.e., + i. We notice that the above expression is aconvolution which simply gives a product in the t-domain:

F() =+

F1()F2( ) d

2= F(t) = F1(t)F2(t)

where

F1() = 21

2e

2

22 = F1(t) = e2t2

2

and

F2() =

2ii

+ i= F2(t) =

2iH(t)

where H(t) is the Heaviside (step) function, defined to be H(t) = 0 for t < 0and H(t) = 1 for t > 0. (The value at t = 0 is not important, but most oftenis assumed to be 1/2.) The last inverse Fourier trasform is accomplished byusing the usual technique of integrating over a closed contour in the planeof complex around the pole at i and taking a residue. Note that theFourier transform between F2(t) and F2() illuminates the physical sense of

the trick used in equation (5).Thus,

F() = Z

2

= F(t) =

2iH(t)e

2t2

2

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The derivative G(x) = Z(x)/2 is found by using d/d it:

FG() = G

2

=

2

d

dF() = FG(t) = 2tH(t)e

2t2

2

The shift by = (k u) is accomplished using the property

F( ) = eitF(t)

Finally, we have the dielectric permittivity in time-wavevector domain:

(t, k) = (t) +

ss(t, k) = (t) + tH(t)s

2se

k2v2st2

2i(kus)t (33)

where the delta function is obtained from transforming 1. The same an-swer may be obtained from the first principles by calculating the polarizationcreated by a delta-function electric field in the time-space domain and con-verting r k. But this is a completely different topic.

References

Abramowitz, M., and I. A. Stegun (1965), Handbook of Mathematical Func-tions with Formulas, Graphs, and Mathematical Tables, Dover.

Baumjohann, W., and R. A. Treumann (1997), Basic Space Plasma Physics,Imperial College Press, London.

Bogdanov, E. P., Y. A. Romanov, and S. A. Shamov (1976), Kinetic the-ory of beam instability in periodic systems, Radiophysics and QuantumElectronics, 19(2), 212217, doi:10.1007/BF01038529.

Fadeeva, V. N., and N. M. Terentev (1954), Tables of Values of the Probabil-ity Integral for Complex Arguments, State Publishing House for TechnicalTheoretical Literature, Moscow.

Fried, B. D., and S. D. Conte (1961), The Plasma Dispersion Function, Aca-demic Press, New York.

Gordeyev, G. V. (1952), Plasma oscillations in a magnetic field,Sov. Phys. JETP, 6, 660669.

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Jackson, E. A. (1960), Drift instabilites in a maxwellian plasma, Phys. Fluids,

3(5), 786, doi:10.1063/1.1706125.Mikhailovskiy, A. B. (1975), Theory of Plasma Instabilities, Atomizdat, in

Russian.

Paris, R. B. (1998), The asymptotic expansion of gordeyevs integral,Zeitschrift fur Angewandte Mathematik und Physik (ZAMP), 49(2), 322338, doi:10.1007/PL00001486.

Poppe, G. P. M., and C. M. J. Wijers (1990), More efficient computation ofthe complex error function, ACM Transactions on Mathematical Software,16(1), 3846, doi:10.1145/77626.77629.

Sitenko, A. G. (1982), Fluctuations and Non-linear Wave Interactions inPlasmas, Pergamon Press, New York.

Stix, T. H. (1962), The Theory of Plasma Waves, McGraw-Hill Book Com-pany, New York.

Weideman, J. A. C. (1994), Computation of the complex error func-tion, SIAM J. Numerical Analysis, 31 (5), 14971518, available online athttp://www.jstor.org/stable/2158232.

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