Fundamentals of Plasma Physics

8.1 Uniform plasma 233

whereτ=t′−t.Certain Bessel function relations are now of use. The first of these is the
integral representation of theJnBessel function, namely

Jn(z)=

1

2 π

∫ 2 π

0

eizsinθ−inθdθ. (8.21)

The inverse of this relation is

eizsinθ=

∑∞

n=−∞

Jn(z)einθ (8.22)

which may be validated by taking theθFourier transform of both sides over the interval
from 0 to 2 π.
The phase integral can be evaluated using Eq.(8.22) in Eq.(8.20) to obtain

Iphase(x,t) = eik·x−iωt

∑∞

n=−∞

Jn

(

k⊥v⊥

ωcσ

)

e

−ik⊥vω⊥cσsinφ

∫^0

−∞

dτei(k‖v‖−ω)τ+in(ωcστ+φ)

= eik·x−iωte−

ik⊥v⊥sinφ

ωcσ

∑∞

n=−∞

Jn

(

k⊥v⊥

ωcσ

)

[

ei(k‖v‖−ω)τ+in(ωcστ+φ)

] 0

−∞

−i

(

ω−k‖v‖−nωcσ

).

(8.23)

In Eqs. (8.16) and (8.23) there is a lower limit att=−∞;this limit corresponds to the
phase in the distant past and is essentially the information regarding theinitial condition of
the system in the distant past. We saw in our previous discussion of the Landau problem
that initial value problems are properly discussed using a Laplace transform approach and
that Fourier transforms do not work properly when there is an initial valueproblem. How-
ever, if we ignore the initial conditions and use the Plemelj formula with Fourier transforms
the same result as the Laplace method is obtained. This short-cut procedurewill now be fol-
lowed and so any reference to the initial conditions will be dropped andFourier transforms
will be used with invocation of the Plemelj formula whenever it is necessary to resolve any
singularities in the integrations. Hence, terms evaluated att=−∞in Eqs. (8.16) and
(8.23) are dropped since these are initial values. After making these simplifications, the
perturbed distribution function becomes

fσ 1 (x,v,t)=−

qσ ̃φ 1 fσ 0 eik·x−iωt

κTσ











1 −e−

ik⊥v⊥sinφ

ωcσ

∑

n

Jn(

k⊥v⊥

ωcσ

)ωeinφ

(

ω−k‖v‖−nωcσ

)











.

(8.24)

The next step is to evaluate the density perturbation Eq.(8.5), an operation which involves
integrating the perturbed distribution function over velocity;the velocity integrals are eval-
uated using the Bessel identity (Watson 1922)

∫∞

0

zJn^2 (βz)e−α

(^2) z 2
dz=

1

2 α^2

e−β

(^2) / 2 α 2
In(β
2
2 α^2 ). (8.25)