An In Depth Treatment Of The Landau Contour - And The Role Of the Inverse Laplace Transform. (Part 1)

In this post we take a deep dive beyond the basic presentation of the Landau contour to examine the role not only of the Laplace transform in its  creation but the inverse Laplace transform as well. To that end, given the increased math detail it may be useful to review residues:

Brane Space: The Residue Theorem and Complex Integrals

And contour integration:

Brane Space: Reviewing Contour Integrals

We saw in previous posts on the Landau contour that an initial value equation is required, based on the Poisson equation:

ik (E₁ ) =  -  4pe ò  d ₃ v f ₁ 

And:

 ¶ _t f ₁ / ¶ t  + ik v x  f ₁ =  (e/ m)  E ₁  ¶ _vx f_o )   

Where  ¶ _t f ₁   denotes a total derivative.  For temporal variation we must do a Laplace transform leading to:

-i w f ₁   -  f ₁   (t = 0) + ik v x  f ₁ =  (e/ m)  E ₁  ¶ _vx f_o 

  The relevant Laplace transform pair being:

A)  E ₁ ( w )  =   ò^¥_o  dt  e ^iwt  E ₁  (t)   And:

B) E ₁ (t)  =  ò_L     dw/2p    e ^-iwt  E ₁  (w)  

Where 'L' for the integral denotes the deformed Laplace contour used in obtaining the inverse Laplace transform.

 The contour L  must pass over all poles of  E ( w ) which from (B) includes all zeros of e (k,w).

Note that E ₁ ( w )  is defined only in the upper half plane of the complex  w- plane.

The preceding pair are based on 2 generic Laplace transforms:

    i) f( w )  =   ò^¥_o  dt  e ^iwt  f(t) 

And:

   ii)  f(t)  =  ò_L     dw/2p    e ^-iwt  f(w)  

In the specific case shown,  E (w ) is defined only for sufficiently large as well as the inverse Laplace transform, defined above by E ₁ (t).  The inverse Laplace transform (B) is accomplished by analytically continuing E (w ) to the entire complex w- plane. By eqn. (A) when one analytically continues E (w ) one must also analytically continue e (k,w). 

e (k,w)  =1 - we ²/ k^{2   ò  du  d} u g(u))/  u   -  w/ k

This  means when w crosses the real w  - axis (from above to below) we cannot allow the pole to cross the integration contour, so we get:

A. Non-deformed contour

If it did cross then the value of  e (k,w)  would jump by -2pi times the residue at u =  w/ k.  Hence the integration contour must be deformed when w < 0    in the complex u-plane.

B. Deformed integration contour

The two sets of contours (A and B) shown in the above graphs are known (in concert)  as the Landau contour. An analogous contour is needed to evaluate the integral :

E (w ) =  4pe/ k² e (k,w)   ò  dv f₁  (k, v, t= 0)/  v x    -  w/ k

which  includes all zeros of  e (k,w). 

With the Laplace contour shown above we have  w _i > 0 everywhere on the contour and hence  wi  > 0 in the evaluation of e (k,w) from the previous equation.

If :

 e (k,w)  = 0  we get a normal mode dispersion relation.

Landau's contribution:

What Landau did was to extend e (k,w) to the entire w- plane. In velocity space the pole can be anywhere in the upper right quadrant (since w _i >  0). To make e (k,w) an entire function one needs to follow an analytic continuation process, i.e. from the upper half plane to the lower half plane.  

For the contour with w _i <  0

Thus we need:

i) I =   ò ^¥_-¥  ^du f_o /  (u   -  w/ k)                              (w >0)

ii) I =   ò ^¥_-¥  ^du f_o /  (u   -  w/ k)  + 2pi (Res)          (w _i > 0)

In part 2 I examine the multiple poles associated with the inverse Laplace transform given by:

E ₁ (t)  =  ò_L     dw/2p    e ^-iwt  E (w)  

And the normal modes associated with them