By substituting t + τ = t1, t = t2 into (2.18) we have
(3.1)
Thus
(3.2)
where, in the first equality the variable changes t1 = t + τ and t2 = t are made, and in the last equality the result
(3.3)
is used. Thus, accounting for Assumption 2.2.16, the sufficient condition of Theorem 2.2.15 turns out to be satisfied.
where, in the second equality, the variable change t′ = t − τ is made. (3.4) is equivalent to (2.52) from which (2.54) follows.
From (2.54) we have
(3.5)
(3.6)
Therefore, (2.55) follows. In addition, one has
(3.7)
which is equivalent to (2.56).
(3.8)
from which we obtain (2.57). From (2.57) it follows that
(3.9)
Therefore
(3.10)
from which we have (2.58) In addition, one has
(3.11)
which leads to (2.59).
(3.12)
from which we obtain (2.60). From (2.60) it follows that
(3.13)
Therefore
(3.14)
from which we have (2.61). In addition, one has
(3.15)
which leads to (2.62).
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