Fundamentals of Plasma Physics

242 Chapter 8. Vlasov theory of warm electrostatic waves in a magnetized plasma

and

xy = a

∫

Ce

p^3 / 3

(

d

dp

epx

)

dp

= a

[

ep

(^3) /3+px]p^2
p 1
−a

∫

C

d

dp

(

ep

(^3) / 3 )
epxdp
= −a

∫

Cp

(^2) ep^3 /3+pxdp.

(8.60)

Addition of these results corresponds to Eq.(8.54).

2. Suppose the integrand is analytic in some region and consider two different contours
   in this region having thesamepair of endpoints. We can take one of these contours and
   deform it into the other. Thus, the two contours give the same result and so correspond
   to the same solution.


3. Independence of contours: Since Eq. (8.54) is a second order ordinary differential
   equation it must have two linearly independent solutions. This means theremust be
   two linearly independent contours since a choice of contour corresponds to a solution.
   Linear independence of the contours means that one cannot be deformed into the other.
   One possible way for the two contours to be linearly independent is for them tohave
   different pairs of end points. In this case, one contour can only be deformed into
   the other if the end points can be moved without changing the integral. If moving
   the end points changes the integral, then the two contours are independent. Another
   possibility is a situation where the two contours have the same pairs of endpoints, but
   the integrand is not analytic in the region between the two contours. For example
   there could be a pole or branch cut between the contours. Then, one contour could
   not be deformed into the other because of the non-analytic region separating the two
   contours. The two contours would then be linearly independent.

8.5.2 Steepest descent contour

The solution to the Airy equation is thus

y(x) =

∫

C

ef(p,x)dp (8.61)

whereCsatisfies the conditions listed above and the solution can be multiplied by an
arbitrary constant. In order to evaluate the integral it is useful tofirst separate the complex
functionf(p,x)into its real and imaginary parts,

f(p,x) =fr(p,x) + ifi(p,x). (8.62)

Also, it should be remembered that for the purposes of thepintegration,xcan be consid-
ered as a fixed parameter. Thus, when performing thepintegration,xneed not be written
explicitly and so we may simply writef(p) =fr(p) + ifi(p)wherep=u+ ivandu,v
are the coordinates in the complexpplane. For an arbitrary contour in thepplane, bothfr
andfiwill vary. The variation of phase means that there will be alternatingpositive and
negative contributions to the integral in Eq.(8.61), making evaluation of this integral very
delicate. However, evaluation of the integral can be made almost trivialif the contour is
deformed to follow a certain optimum path.