5.2 The Landau problem 163dropping pole,
v∥
ip
k(a)
(b)
(c)
integration contour
integration contour
integration contour
pole,v∥
ip
kdropping pole ,
v∥
ip
kIm v∥
Im v∥
Im v∥
Re v∥
Re v∥
Re v∥
complex v∥plane
complex v∥plane
complex v∥plane
Figure 5.4: ComplexvplaneD(p)involves a similar integration along the realv‖axis. It also has a pole that is
initially in the upper half plane whenRep> 0 , but then drops to being below the axis as
Repis allowed to become negative. Thus analytic continuation ofD(p)is also constructed
by deforming the path of thev‖integration so that the contour always lies below the pole.
Equipped with these suitably constructed analytic continuations ofN(p)andD(p)
into the left-handpplane, evaluation of Eq.(5.52) can now be undertaken. As shown in
the simple example, it is computationally advantageous to deform the Bromwich contour
into the left handp-plane. The deformed contour evaluates to the same result as the orig-
inal Bromwich contour (provided the deformation does not jump over any poles) and this
evaluation may be accomplished via the method of residues. In the general case where
N(p)/D(p)has several poles in the left handpplane, then as shown in Fig.5.2(c), the