Earth-Ionosphere Cavity Resonator - Ultra and Extremely Low Frequency Electromagnetic Fields - page 133

In the text we need to develop the boundary condition ( 4.12 ) as a series in Legendre
polynomials. For that one should use the following representation of the delta-
function

X

ı./
sin D

C n P n .cos /: (4.59)

nD0

In order to obtain the undetermined coefficients C n , one should multiply Eq. ( 4.59 )
by sin P m .cos / and then integrate both sides of this equation between zero and .
On account of Eqs. ( 4.57 ) and ( 4.58 ) one can find

n C P n .1/ D n C

1
2

1
2 :

C n D (4.60)

In the course of this text, we also need the following sum

n . n C 1 / P n .cos / D ln sin 2

X

.2n C 1/

1: (4.61)

2

nD1

Rearrangement of Solution

Before rearranging Eq. ( 4.33 ), it should be noted that this equation contains the
delta-function of because the source function, that is, the current density j s
includes the same factor, i.e., ı./. In order to eliminate the delta-function, one
should rearrange the sum in Eq. ( 4.33 ). Taking the notice of

! n
! n ! 2 D

! 2
! n ! 2 C 1

(4.62)

and accounting of Eqs. ( 4.59 ) and ( 4.60 ), we obtain

X X

! n .2n C 1/ ! 2 .2n C 1/

! n ! 2

2ı./
sin : (4.63)

! n ! 2 P n .cos / D P n .cos / C

n D 0 n D 0

If the delta-function is omitted and ! 2 in the sums is replaced by the ! 2 C

ic!Z.!/=d,Eq.( 4.33 ) is reduced to ( 4.38 ).