Fundamentals of Plasma Physics

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)