248 Chapter 8. Vlasov theory of warm electrostatic waves in a magnetized plasma
for largepas
f(p) ≃
p^3
3
+pξ−lnp
= p
(
p^2
3
+ξ
)
−lnp
=
2
3
pξ−lnp (8.87)
while for smallpit can be approximated as
f(p) ≃ pξ−
μ
p
−lnp
= p
(
ξ−
μ
p^2
)
−lnp
= p
(
ξ−
μ
p^2
)
−lnp
= 2pξ−lnp. (8.88)
Forξ> 0 and large roots, the quantitiespl,f(pl)andf′′(pl)are respectively
pl=±(−ξ)^1 /^2 , f(pl) =∓
2
3
(−ξ)^3 /^2 −lnpl, f′′(pl) =±2(−ξ)^1 /^2. (8.89)
Forξ> 0 the corresponding quantities for the small roots are
ps=±(−μ/ξ)^1 /^2 , f(ps) =±2(−ξμ)^1 /^2 −lnps, f′′(ps) =∓ 2
(−ξ)^3 /^2
μ^1 /^2
. (8.90)
The Gaussian integrals corresponding to steepest descent paths over theseξ> 0 sad-
dle points are
large roots:
∫
vicinity
ofsaddle
ef(p)dp=
√
∓π
(−ξ)^3 /^2
e∓
(^23) (−ξ)^3 /^2
(8.91)
small roots :
∫
vicinity
ofsaddle
ef(p)dp=
√
±π
(−μξ)
1 / 2 e
±2(−ξμ)^1 /^2 (8.92)
where the logarithmic term inf(p)has been taken into account. Forξ< 0 ,the large
root quantities are
pl=±|ξ|^1 /^2 , f(pl) =∓
2
3
|ξ|^3 /^2 −lnpl, f′′(pl) =± 2 |ξ|^1 /^2 (8.93)
and the small root quantities are
ps=±(μ/|ξ|)^1 /^2 , f(ps) =∓2(|ξ|μ)^1 /^2 −lnps, f′′(ps) =∓ 2
|ξ|^3 /^2
μ^1 /^2