Note that, since Fk is real valued, we can also represent Fk by
∞
(A.35)
Fk(x) α α0,k + Σ (αj,kcos(2 π jχ) + βj,ksin(2 π jχ)),
where
co,k = αo,k ; for j ≥ 1: cj
α ., - i β ,
J,k J,k
-------ɔ-------, cj, k
α ., + i β ,
J, k ,i, k
2
(A.36)
Since
∫exp(2iπjχ)dχ III(■=0),
(A.37)
it follows that
F(x)dx O 0 implies c0,k - α0,k - 0,
(A.38)
hence
Fk(x ) = Σ cj, keχp(2 i π jx )∙
Next, observe that
(A∙39)
x
0
i π jy ) dy =
eχp(2 iπjχ) - 11(j.≠0)
2 i ∏ j
÷ xI(j'=0)
(A∙40)
and
x
Jy exp(2 i π jy ) dy =
0
λx exp(2 i π jx)
ч 2iπj
exp(2 i π jx) - 1
(2 i π j )2
ʌ
I(j'≠0)
√
÷ -2x21(j' 0),
(A∙41)
hence, for j1 ≠ 0, j2 ≠ 0,
44