Propagation of MHD-Beams - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 253

where

s m,g = ix ( k 0 √ ε m,g − k (1) S ) .

Here and below the upper sign refers to ' m ', t he lowe r to ' g '. For s = s m,g

we have k m,g ( s m,g ) = k (1) S . The bra nchof √ s √ s m,g ) bei ng determined by

the c ondi tion Re √ s> 0( Re √ s m,g > 0). If Re k 0 √ ε m,g > Re k (1) S , then
Im √ s m,g > 0, and if Re k 0 √ ε m,g < Re k (1) S , then Im √ s m,g < 0 . functions
∆ T m,g
√ s ±

C − 1
√ s m,g k m,g ( s )

k (1) S

−

are regular in the vicinity of s =0and s = s m,g
Consider the integral

√ s exp (

∞

C − 1
√ s m,g

− s )

I m,g = ±

k (1) S ds. k m,g ( s ) −

0

Use new variable s = t 2

and, with (8.47),

t 2 exp − t 2

∞

2 ix C − 1
√ s m,g

I m,g = ∓ dt.

t 2 − s m,g

0

Let us rewrite it in the form

⎛ ⎞

exp − t 2

s m,g −

√ π
2

∞

2 ix C − 1
√ s m,g

2 s m,g
√ π

⎝ 1 ⎠ .

I m,g = ∓ −

t 2 dt

(8.50)

0

The integral can be expressed in terms of the Kramp function w ( V ) ([1], [8])
which for Im V> 0 is defined by the integral

exp t 2

∞

−

w ( V )= 2 iV
π

dt, (8.51)

V 2 − t 2

0

and for Im V< 0 it is found by the analytic continuation of (8.51) into the
lower half-plane. As a result,
∞

exp − t 2

2 V w ( V )+ 0

iπ at Im V> 0 ,

iπ exp − V 2 /V

dt = − (8.52)

at Im V< 0 .

V 2 − t 2

0

From (8.50) and (8.52) we get

i √ π

√ s m,g C − 1 1+ i √ πs m,g w ( √ s m,g )

x

I m,g = ∓

0

Im √ s m,g > 0 , at

∓

s m,g ) tIm √ s m,g < 0 . 2 πx C − 1 exp ( −