Dear ArialAlok Srivastava
Thank you for your email. I think exercise 7.4 is OK as stated.
(a) You should use the inequalities the way they are stated. Note that we
are looking
for a sufficient condition. (For example, a sufficient condition for x <<
3 is to have
x << 2 (since 3>2).)
(b) The inequalities given in the hint follow from the triangle
inequality for complex numbers.
Let a, b complex numbers. Then
|a+b| <<= |a| + |b| (*)
Selecting a=1 and b=L gives:
|1+L| <<= 1 + |L| (which is the one you are referring to, but NOT the one
to use here - it
would give a necessary condition for RP)
But (*) can be rearranged to:
|a| >= |a+b| - |b| (**)
Selecting a=1+L and b=-L gives a+b=1 and (**) yields:
|1+L| >= 1 - |L|
Similarly, selecting a=1+L and b=-1 gives a+b=L and (**) yields:
|1+L| >= |L| - 1
which are the two desired conditions.
Good luck with your course.
Best regards,
Sigurd Skogestad
At 10:11 AM 10/29/00 +0800, you wrote:
>>>>
ArialProf. Sigurd,
I am studying your book in my M.Sc course and find it very good. I have
one small query to ask you. In Exercise 7.4, hint is provided in form of
an inequality i.e
Arialmagnitude of (1+L)>=1-magnitude
of L and
magnitude of (1+L)>=magnitude of L -1
Arial(a) Shoudn't these inequalities
have less than or equal to sign instead of more than or equal to. I
could get the upper and lower bounds of L as given in equation 7.65 and
7.66 only if the inequalities have less than or equal to sign.
Arial(b) Another question I would
like to ask you from where do we get these inequalities.
ArialI am sorry if these are
elementary questions and take up some of your invaluable time.
ArialAppreciate your help
ArialBest regards
ArialAlok Srivastava
<<<<<<<<
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Sigurd Skogestad, Professor Fax: +47-7359-4080
Dept. of Chemical Engineering Phone: +47-7359-4154
Norwegian Univ. of Science and Techn.(NTNU) Home: +47-7393-6855
N-7491 Trondheim email: sigurd.skogestad@chembio.ntnu.no
Norway http://www.chembio.ntnu.no/users/skoge
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