14.2 Vlasov non-linearity and quasi-linear velocity space diffusion 403
This expression can be further evaluated by invoking the parity properties ofE ̃ 1 (k)and
ω(k).These parity properties are established by writing
E 1 (x,t)=
1
2 π
∫
dkE ̃ 1 (k)eikx−iω(k)t (14.20)
and noting that the left hand side is real because it describes a physical quantity. Taking the
complex conjugate of Eq.(14.20) gives
E 1 (x,t)=
1
2 π
∫
dkE ̃ 1 ∗(k)e−ikx+iω
∗(k)t
. (14.21)
The parity properties are determined by defining a temporary new integration variablek′=
−kand noting that
∫∞
−∞dkcorresponds to
∫∞
−∞dk
′sincedk′=−dkand the klimits
(−∞,∞)correspond to thek′limits(∞,−∞). Thus Eq.(14.21) can be recast as
E 1 (x,t) =
1
2 π
∫
dk′E ̃ 1 ∗(−k′)eik
′x+iω∗(−k′)t
=
1
2 π
∫
dkE ̃ 1 ∗(−k)eikx+iω
∗(−k)t
(14.22)
where the primes have now been removed in the second line. Since Eq.(14.20)and (14.22)
have the same left hand sides, the right hand sides must also be the same. BecauseE ̃ 1 (k)
andω(k)are arbitrary, they must individually satisfy the respective parity conditions
E ̃ 1 (k) = E ̃∗ 1 (−k)
ω(k) = −ω∗(−k) (14.23)
or equivalently
E ̃ 1 (−k) = E ̃ 1 ∗(k)
ω(−k) = −ω∗(k). (14.24)
If the complex frequency is written in terms of explicit real and imaginary partsω(k) =
ωr(k)+iωi(k)then the frequency parity condition becomes
ωr(−k)+iωi(−k) = −[ωr(k)+iωi(k)]∗
= −ωr(k)+iωi(k) (14.25)
from which it can be concluded that
ωr(−k) = −ωr(k)
ωi(−k) = ωi(k). (14.26)
The real part ofωis therefore an odd function ofkwhereas the imaginary part is an even
function ofk.
Application of the parity conditions gives
E ̃ 1 (−k)E ̃ 1 (k)=E ̃ 1 ∗(k)E ̃ 1 (k)=
∣
∣
∣E ̃ 1 (k)
∣
∣
∣
2
(14.27)