'''	\"  eqn file | troff -mm [-Tdevname] | ...
.EQ
delim $$
.EN
.PH "'EQN:  File(\n(.F)''Device(\*(.T)'"
.P
.HU Proof
.DS 3
.EQ
-i~cos~omega t~=~i~cos~omega t~=~-i~
{e sup it~-~e sup -it} over 2~=~
{e sup +it~-~e sup -it} over 2i
.EN
.DE
.P
Let us consider $delta (t)$, the unit
impulse (or Dirac) function.
We shall define $delta (t)$ conceptually
as a limit of time functions
$delta sub n (t)$, which: (i) vanish
outside $(- epsilon sub n ,+ epsilon sub n )$,
(ii) are nonnegative in $(- epsilon sub n sup n ,+ epsilon sub n )$,
(iii) have $int from {- epsilon sub n} to {epsilon sub n}~
delta (t)dt~=~1$
(iv) $lim from s~int from {- epsilon sub n} to {epsilon sub n}~
delta (t)dt~=~0$
(other alternatives for the approximating
functions would also have been reasonable).
.sp 2
\l'6i'
.sp 2
.P
We are now in a position to define and describe
the important concept of the transfer function
of an IS box.
We have already shown that if
$cos~omega t~->~B over A~cos ( omega t + delta )$
and $K$ and $n$ are integers, then
$K over n~cos~omega t~->~K over n~B over A~cos ( omega t + delta )$.
If there is even a shred of continuity, we
may extend the result for rationals to the reals.
Then for $lambda$ real,
.DS C
$lambda~cos~omega t~->~lambda~B over A~cos ( omega t + delta )~$.
.DE
.sp 2
\l'6i'
.sp 2
.P
Now the expectations of the second and third
series vanish term by term, while the others yield
$-2~left ( sqrt {1+r over 2}~+~sqrt {1-r over 2} right )~
e sup {-~{2 pi sup 2} over h sup 2}~cos~{2 pi} over h~phi sub 0$,
which we have already seen to be small.
Here is another line of text so that it will continue
underneath the in-line equation, and we can see how
it looks.
Beware:  The square root symbol can only be as large as the largest
size available on the target device.
