244 Chapter 8. Vlasov theory of warm electrostatic waves in a magnetized plasma
so the saddle points (denoted by subscripts) are located atps=±(−x)^1 /^2. In the vicinity
of the saddle point,fcan be Taylor expanded as
f(p)≃f(ps) +
(p−ps)^2
2
f′′(ps). (8.67)
It is now convenient to define the origin of the coordinate system to be at the saddle point
and also use cylindrical coordinates,r,θso thatp−ps=rexp(iθ)and for fixedθ,dp=
dreiθ.Also, phasor notation is used forf′′sof′′=|f′′|exp(iψ)whereψis the phase of
f′′.Thus, in the vicinity of the saddle point we write
f(p)≃f(ps) +r^2
|f′′(ps)|
2
exp(2iθ+ iψ). (8.68)
Choosing the contour to follow the path of steepest descent corresponds to choosingθsuch
that 2 θ+ψ=±πin which case
f(p)≃f(ps)−r^2
|f′′(ps)|
2
(8.69)
and so Eq.(8.61) becomes
y(x)≃
∫∞
−∞
ef(ps)−
(^12) r (^2) |f′′(ps)|
dreiθ (8.70)
where the approximation has been made that nearly all the contribution to the integral
comes from smallr;it is not necessary to worry about errors in the Taylor expansion
at larger,since the contributions from largerare negligible because of the exponential
behavior. Thusfris maximum at the saddle pointr= 0and the main contribution to
the integral comes from the contour going over the ridge of the saddle. Therintegral is a
Gaussian integral
∫
exp(−ar^2 )dr=
√
π/aso
y(x) ≃
∫∞
−∞
ef(ps)−r
(^2) |f′′(ps)|/ 2
dreiθ
= ef(ps)+iθ
∫∞
−∞
e−r
(^2) |f′′(ps)|/ 2
dr
= ef(ps)+iθ
√
2 π
|f′′(ps)|
= ef(ps)
√
2 πei2θ
|f′′(ps)|
= ef(ps)
√
πei(±π−ψ)
|f′′(ps)|
= ef(ps)
√
−
2 π
f′′(ps)
. (8.71)
Sincef(ps) =p
(
p^2 /3 +x
)
=±
√
−x(2x/3) =∓2(−x)^3 /^2 / 3 andf′′(ps) =± 2
√
−xit
is seen that ∫
ofsaddlevicinity
ef(p)dp≃
√
∓π
(−x)^1 /^2
e∓2(−x)
3 / (^2) / 3
. (8.72)