Fundamentals of Plasma Physics

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

. (8.94)