5.2 The Landau problem 167The ‘P’ has been dropped from the right hand side of Eq.(5.69) because there is no
longer any problem with a singularity. These Gaussian-type integrals may be evalu-
ated by taking successive derivatives with respect toaof the Gaussian
1
π^1 /^2∫
dξexp(−aξ^2 )=1
a^1 /^2(5.70)
and then settinga=1.Thus,
1
π^1 /^2∫
dξξ^2 exp(−ξ^2 )=1
2
,
1
π^1 /^2∫
dξξ^4 exp(−ξ^2 )=3
4
(5.71)
so Eq.(5.69) becomesP
1
π^1 /^2∫∞
−∞dξ
exp(−ξ^2 )
(ξ−α)=−
1
α[
1+
1
2 α^2+
3
4 α^4+...
]
. (5.72)
In summary, for|α|>> 1 ,the plasma dispersion function has the asymptotic formZ(α)=−1
α[
1+
1
2 α^2+
3
4 α^4+...
]
+iπ^1 /^2 exp(−α^2 ). (5.73)- |α|<< 1 case.
In order to evaluate the principle part integral in this regime the variableη=ξ−αis
introduced so thatdη=dξ.The integral may be evaluated as follows:P
1
π^1 /^2∫∞
−∞dξ
exp(−ξ^2 )
(ξ−α)=
1
π^1 /^2∫∞
−∞dη
e−η(^2) − 2 αη−α 2
η
=
e−α2π^1 /^2∫∞
−∞dηe−η2η
1 − 2 αη+
(− 2 α)^2
2!
+
(− 2 α)^3
3!+...
= − 2 αe−α
2π^1 /^2∫∞
−∞dηe−η2[
1+
2 η^2 α^2
3+...
]
= − 2 α(
1 −α^2 +...)
(
1+
α^2
3+...
)
= − 2 α(
1 −
2 α^2
3+...
)
(5.74)
where in the third line all odd terms from the second line integrated to zero due to
their symmetry. Thus, forα<< 1 ,the plasma dispersion function has the asymptotic
limit
Z(α)=− 2 α(
1 −
2 α^2
3+...
)
+iπ^1 /^2 exp(−α^2 ). (5.75)