• 5

PLASMA DISPERSION FUNCTION

Definition16 (first form valid only for Im £ > 0):

, dt exp (-t2)                                                                                        . rc                         , x

Z(C) = 7T-1/2 / ---------------------- y =2iexp(-C2) / dt exp (-t2) .

J — OO                                                                                             J — oo

Physically, £ = x + 22/ is the ratio of wave phase velocity to thermal velocity. Differential equation:

dZ                                 ,                        / x 1/2 d2Z dZ

------------------  = -2 (1 + £Z) , Z(0) = ivr7 ; ------------------------  + 2<--------- h 2Z = 0.

d< d<2 d<

Real argument (2/ = 0):

Z(x) = exp ^ — x2^j ^itv1/2 — 2 J^ dt exp

Imaginary argument (x = 0):

Z(iy) = i7r1/2 exp (y2) [1 - erf(y)] . Power series (small argument):

Z(C) = i7t1/2 exp (-C2) - 2C (l - 2<2/3 + 4<4/15 - 8<6/105 + •••). Asymptotic series,           1 (Ref. 17):

Z(0 = i7r1/2<rexp (-<2) - C"1 (l + l/2<2 + 3/4C4 + 15/8C6 + •••), where

r 0 y > |x|_1 cr = < 1 < |x|-1

U i/ < -i^r1

Symmetry properties (the asterisk denotes complex conjugation):

Z(C*) = -[Z(-C)]*; Z(C) = [Z(0] * + 2i7T1/2 exp[-(C*)2] (y > 0).

Two-pole approximations18 (good for £ in upper half plane except when y < 7t1//2x2 exp( — x2), x 1):

z x 0.50 + 0.81i 0.50 - 0.81i z(C)« ------------------- T            „ , . , a = 0.51-0.8H;

a - C                                                                    a* + C

. x 0.50 + 0.962 0.50 - 0.962 z' (C) « —;---- ^----- h —             —, b = 0.48 - 0.912.

Авторы: 1379 А Б В Г Д Е З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

Книги: 1908 А Б В Г Д Е З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я