Looking At The Landau Contour As It Applies To Plasma Physics

Sketch of Landau contour when   w _i =  0.  The straight line portions of the contour are exactly on the real (Re) axis.

The Landau contour represents a special case used in plasma physics.  This is illustrated in the sketch above, where the pole can be anywhere in the upper right quadrant.

This is for the complex variable, v. What Lev Landau did was to extend his di-electric function, ε  to the entire   w- plane. In the graphic the contour follows the bold line along the Re(v_x) abscissa and then goes round the residue at x.  One must ensure the path of integration does not meet a pole, so must go around it, i.e  based on a deformed contour.  

 Hence, to make ε  an entire function one needs to follow an analytic continuation process  from the upper half plane to the lower half plane.  One must then ensure the path of integration does not meet the pole but goes around it by way of a deformed contour.  Then if u = w/ k  we will have (for  w _i  > 0):

I =  ò ^¥_-¥   du  du f _o / (u - w/ k)           (w _i  > 0)

And thence: 

I =  ò ^¥_-¥   du  du f _o / (u - w/ k)  +  2 πi (Residues)

When the pole is on the Re-axis:

I =  P ò ^¥_-¥   du  du f _o / (u - w/ k)  +  πi (Residues)

How do we know this analytic continuation is correct?

We can take the integral I⁺  (i.e. from just above the pole):

 I⁺   =  ò ^¥_-¥   du  du f _o / (u - w/ k -   i/ d)

Also:   I^-   =  ò ^¥_-¥  du  du f _o / (u - w/ k +  i/ d) + 2πi (Res)

Taking the average of the integrations:

I*  =  ½ [I⁺  + I^-  ]  

I*  =     lim _{d ® 0}    [I⁺  + I ^-  ]   =

 lim _{d ® 0} ½ [ ò ^¥_-¥   du  du f _o / (u - w/ k -   i d)

+  ò   du  du f _o / (u - w/ k -   id)] + πi(Res)

Where the bracketed quantity is the Cauchy Principal Value.

See, e.g. 

• Complex Analysis – continued: The Principal Value

For advanced plasma physics, say applied to solar phenomena (e.g. corona, auroral substorms) , with normal mode (steady state) behavior assumed so we only need Landau's basic prescription.   There is no need to introduce or go back to the Laplace transform to apply it- since that contains more information than we actually need.   

What we can do for insurance is to look at the di-electric function to see if any damping is associated with the normal mode (as has been done in various numerical models of  auroral substorms, as developed at the Geophysical Institute in Fairbanks, AK). Thus we look at:

 ε   =  1 -   w _pe ²  /k ²  [ò  du  {du f _o  d(u) /  (u - w/ k )]

Suggested Problem:

For  a particular numerical model a space physicist wants to show the di-electric function can be expressed in real and imaginary parts such that: ε  _r    +  i ε _i  = 0  

Show this using integration by parts of the equation for ε   and then do a Taylor expansion.