Fundamentals of Plasma Physics

8.5 Analysis of linear mode conversion 243

To see this, it should first be recalled that the theory of complex variables shows that
any function of a complex variable must satisfy the Cauchy-Riemann conditions, which
means thatfrandfisatisfy

∂fr

∂u

=

∂fi

∂v

,

∂fr

∂v

=−

∂fi

∂u

. (8.63)

Also, on defining the gradient operator in theu,vplane∇= ˆu∂/∂u+ ˆv∂/∂vit is seen
that

∇fr·∇fi=

∂fr

∂u

∂fi

∂u

+

∂fr

∂v

∂fi

∂v

= 0. (8.64)

This means that contours of constantfrare everywhere orthogonal to contours of constant
fi.
Sinceficorresponds to the phase ofexp(fr(p) + ifi(p))andfrcorresponds to the
amplitude, Eq.(8.64) shows that a path in the complexpplane which is arranged to follow
the gradient offrwill automatically be a contour of constantfi,i.e., a contour of constant
phase. Thus, the optimum path is to follow∇frbecause on this pathfiwill be constant
and so there will not be any alternating positive and negative contributions as on any other
path. In fact, because it is constant, the phase can be factored from the integral when using
this optimum path, giving

y(x) = eifi

∫

C‖∇fr

efr(p)dp. (8.65)

Clearly, the maximum contribution to this integral comes from the vicinity of where
frassumes its maximum value. Since the maximum occurs where∇fr= 0,most of the
contribution to the integral comes from the vicinity of where∇fr= 0.The extrema of
fare always saddle points because the Cauchy-Reimann conditions imply∂^2 fr,i/∂u^2 +
∂^2 fr,i/∂v^2 = 0.Thus, the vicinity of∇fr= 0must be a saddle point.
The discussion in the previous paragraph implies that once the endpoints of the contour
have been chosen, for purposes of evaluation it is advantageous to deform the contour so
that it follows the gradient offr;this called the steepest ascent/descent path (usually just
called steepest descent). This optimum choice of path ensures that (i) there is a localized
region wherefrassumes a maximum value and (ii) the phase does not vary along the path.
The contribution to the integral is concentrated into a small region of the complexp−plane
in the vicinity of wherefrhas its maximum value. Simple integration techniques may be
used to evaluate the integral in this vicinity, and the contribution from this region dominates
other contributions becauseexp(fr(p))at the maximum offris exponentially larger than
at other places. Furthermore if the contour is chosen so thatfr→−∞at the endpoints
then Eq.(8.58) will be satisfied and the chosen contour will be a solutionof the original
differential equation.
The saddle points are located wheref′(p) = 0.For the Airy function,

f(p) = p^3 /3 +px

f′(p) = p^2 +x

f′′(p) = 2p (8.66)