Propagation of MHD-Beams - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 252

∆ T m,g is the difference between the values of the transmission matrix taken
from the corresponding sheets on contours Γ 2 m and Γ 2 g :

∆ T m = T 1 − T 2 , ∆ T m = T 4 − T 3 ,

or

∆ T g = T 1 − T 4 . (8.46)

In (8.46) the subscripts attached to T indicate the index of the sheet from
which the value of T is taken (see Table 7.2). With the chosen integration
contours, the residues (see (8.C.2), (8.C.3)) are taken from the 2-nd sheet of
the Riemannian surface T ( k ) . The poles k = k ( n ) S are determined by zeroes
of ∆ SK (see (8.36)) covered by the deformation of the integration contours at
the transition from Γ 1 and Γ 2 = Γ 2 m + Γ 2 s .
The computation of integral terms in (8.43) can be complicated by the
presence of a pole near the integration contour. For example, in the case
shown in Fig. 8.6, for the ground conductivities 10 5 s − 1 10 6 s − 1 the

pole k (1) S ( σ g ) is located on the 1-st sheet (the physical sheet, Im κ g > 0
and Im κ s > 0). For σ g ≈

σ g

10 6 s − 1 the pole k (1) S ( σ g ) intersects the right
bank of cut Im( κ g ) = 0 and for 10 6 s − 1 10 6 s − 1 is located on

the 4-th sheet (Im κ s > 0, Im κ g < 0). Thus, the pole can occur close to
the contour of integration affecting the accuracy of the numerical compu-
tation. When the pole is located between the contours Γ 2 m and Γ 2 g ,itis
necessary in computing G ( g ) to take into account the residue in the pole.
However, one can select a singular part T ( k ) and calculate explicitly the
contribution caused by the singularities in the integrand with the help of the
residue.
Let us change the variable of integration in (8.45). Substitution
k = k 0 √ ε m,g + i s

σ g 2 ×

x = k m,g ( s )

(8.47)

reduces (8.45) to

i exp ik 0 √ ε m,g x
x

∞

1
2 π

∆ T m,g ( k m,g ( s )) exp ( G m,g = − s ) ds. (8.48)

0

In the neighborhood of k = k (1) S , the corresponding branch of the matrix
function T ( k ) can be presented in the form of the Laurent series:

C − 1

T ( k )= + C 0 + C 1 ( k − k (1) S )+ ··· , (8.49)

k − k (1) S

where C − 1 is the residue matrix T at k = k (1) S . Taking into account that

∝ √ s , we write near s = 0 the functions ∆ T m,g ( s )

∆ T m,g
√ s = ∆ T m,g

C − 1
√ s m,g k m,g ( s )

C − 1
√ s m,g k m,g ( s )

√ s ±

k (1) S ∓ k (1) S

,

− −