38
The FT of function Yιjn(r)∕ri can be easily found using following equality
^-2π∕f.r = 4π^(-0∕7∙(2π∕r)yz,m(∕)^m(r),
l,m
(A.5)
which yields
⅛(r)
r3
= 4πY(-i)‰4∕) f dχj^-
, , ⅜J ∙^∙
l,m'
Γ 4π ʌ
J drY2,m(rWlm^ = --Y2,m(f), (A.6)
writing down explicitly
1 - 3cos20 4ττ ɔ , zι .
--Я3-- = -y(l-3∞*4)∙
The derivative terms can also be calculated using FT
V2<Mr) = F’1 [nv4(r)]] = -4π2<F^1 [∕2⅛(f)]. (A.7)
For practical numerical calculation, FT is implemented via discrete Fourier transform
(DFT). That is, for an array of N uniformly sampled data hn = h(tn) (n = 0,1,2,..., N - 1),
if the sampling time-step is Δ, the frequency interval is ʌʃ = ɪ, i.e.,
I N N ∖ 1
f’-' (-2..... l∙ "∙' ∙-∙T.V∆' (A'8)
The FT of ∕ι(r) is approximately calculated as
ʃv-l N-I
dth(t)e-2π'f"' ≈ Y1 hke^2πif"tt A = Δ У hke-2nikn/N, (A.9)
fc=0 k=0
the DFT is then defined as
N-I
h^Yhke-2”ikn/N, (A. 10)
Ic=O