DOC.
18
REPLY TO LAUE 91
validity
of
(6)
the conditions
(7a) are
sufficient
to
guarantee
the statistical
indepen-
dence of the Fourier coefficients.
In this
manner
we
reach the
following preliminary
result: Since
we
have
to
assume,
for natural
radiation,
that its statistical
properties are
not
changed by
superposition
of incoherent
partial
radiations,
one can say
the
equations
(7a) are
sufficient conditions for natural radiation to
assure
the statistical
independence
of
its
Fourier coefficients.
§2.
Proof
of the
Statistical
Independence
of the Fourier
Coefficients
of Natural Radiation
Let
F(t)
be
a component
of
the radiation
vector
of
stationary
natural
radiation,
given
for
an
infinite time
period.
T is
a large
time
span
when
compared
to the oscillation
[p.
883]
period
of
the
longest wavelength
in the radiation.
F(t)
is to
be
represented
between
t0
and
t0
+
T
by
the Fourier series
E
A
cos
2irn
n
n
+
B"
sin
2rrn
T
T.
(4a)
The Fourier coefficients
An,
Bn,
of
F(t) will
obviously depend upon
the choice
of
time
epoch
t0.
By
imagining
the
expansion
to be carried
out
for
very
many
and
arbitrarily
selected
t0,
we
obtain statistical data for the derivation
of
the statistical
properties
of
the coefficients
An,
Bn
that
we
must
necessarily
demand in natural
radiation.
In order
to
derive these
properties, we
first
expand
F(t)
into
a
Fourier series
between
the times 0 and
6,
where
d
is
very large compared
to
T.
For this
time
interval
let
there be
F(f)
=
E
av
cos
(2tv
j
+
(f").
(8)
The coefficients
An
and
Bn
can
be
expressed
with
t0
and the coefficients
av
and
/v
from the
expansion (8)
if
t0
is
chosen
between
t
=
0 and
t
=
6
-
T.
One
next
gets
2
o+
T
J
av
cos
|2irv
^
+
4
cos
2vn
t
~
T
at
2
f'o +
T
*"
=
^E
!a
cos
r
27TV
4
+
0
0
sin
\2vn
t
-
to
[o
T
dt
(9)
{2}