JJexp(2 i π j 1 x )exp(2 i π j 2 y )min(x, y ) dxdy
x
= ∣exp(2 i π j 1 x)ʃy exp(2 i π j2y)dydx - ∣x exp(2 i πj 1 x)∣exp(2 i π j2y)dydx
x exp(2 iπ (j'i +j2)x) ʃ exp(2 iπ (j'i +j2)x) ʃ exp(2 iπ ji x)
______________dx ʃ I__________LJ___dx + I_________1__dx
(A.42)
2 i π j 2 2 (2 i ∏ j 2)2 2 (2 i π j 2)2
ʃx exp(2 iπ (j' 1 +j2) x) ʃxexp(2 iπ j 1 x)
- I____________LJ___dx + I___________1__dx
2 2 i π j 2 2 2 i π j 2
= _ 1 + 1(j' 1+j 2=0)
4 π 2j 12 4 π2j 2
and
x
∣exp(2 i π j 1 x)∣exp(2 i π j2y)dydx ʃ ʃ
0
exp(2 iπ (j' 1 +j2) x)
2 i π j 2
dx -
exp(2 i π j 1 x )
2 i π j 2
dx
I(j-1÷j 2=0)
2 i π.j,
(A.43)
It follows now from (A.38) and (A.42) that
ʃʃ':(x)Fm(y)min(x,у)dxdy = —2 Σ
jj 4π2 j≠0
( V
j2
Λj
4π2
Ij-1
cc
∞
j-i
A
Ï
c....
Ï
c_..„
(A.44)
ʌ Vλ β j,kβ j,
2^t 2 2
j=1 j
V
∞
λ.1
A
. A
βjm
and it follows from (A.38) and (A.43) that
ʃFk(x )fFm (У ) dydx = ɪ Σ cjk-j
0 ∙0 2 i π j≠0 j
« jkβ j
1 τL « ∙t∣
= ⅛∑ —
4π
Moreover,
45
A
j « β t
Γλ j, mtJ,k
j=1
(A.45)