304 DOC. 30
QUASIPERIODIC
PROCESSES
With this
we
have obtained the
general
relation between the
intensity
function
I and the characteristic
X,
where this relation
is in
a
form that is
independent
of the
choice of
T.
The task that
we
originally
set
ourselves
requires finally
that
we
solve
(3a)
for
I.
If
we
by
W(A),
i.e.,
denote the characteristic of the additive
constant,
if
we
set
X(A)
-
x(°°)
=
WA),
•••(4)
then
we
have,
because the
integral
in
(3a)
vanishes
in
any
case
for
infinitely large
A,
oo
[p. 3]
2t|/(A)
=
JI(x)cos(xA)dx.
...(3b)
o
We
multiply
this
equation by
cos
yA
and then
integrate
between
A
=
0
and
A
=
G,
where
we
conceive of the
quantity
G
as a
large
number that
we
later let
grow
to
infinity.
We obtain
G
x-
o°
A= G
J
V (A)
cos
(M)
dA
= J
I(x)dx
J
cos (xA) cos (yA)
c
^=0
X=0
A=0
X=
oo
=
J
/ (x)dx -Z(x,y),
x-
0
where
A= G
Z
(x,
y)
2
_
j
sin(x
+
j)A
sin
(x-y)
A
+
x
+
y x-y
A=0
For the lower
integration
limit
A
=
0,
z
always vanishes,
even
when
x
-
y
=
0.
For
the
upper integration
limit
A
=
G,
the first
term
of
Z is
a
rapidly alternating
function
of
x,
which,
when
multipled by
Ix
and
integrated,
does
not
make
any
finite
contribution
to
the double
integral
that has
to
be built.
Thus,
the latter reduces
to
1
J,(I) sin(x
-
yJG&
*=o
* y
The factor sin
(x
-
y)G,
which alternates
infinitely rapidly
for
a
variable
x,
would
bring
about the
vanishing
of this
integral too,
if
the
function
I(x)/x-y
were
not to
become infinite for
x
=
y.
Thus,
the
parts
of the
integration region
that
are
not
very
close
to
the
point
x
=
y
cannot,
in
any
case,
contribute
anything
to
the
integral.
In
the
light
of
this,
one
easily
obtains