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        general >> complex analysis >> Minimum Real Part theorem issue on RHP (unbounded 
        
(Message started by: aleberto69 on Oct 11^th, 2013, 4:08am)

Title: Minimum Real Part theorem issue on RHP (unbounded
Post by aleberto69 on Oct 11^th, 2013, 4:08am

Hello to everybody.. this is my first login...
I need some help with the minimum real part theorem ( similar to the max-min modulus theorem) of a complex function analytic in a (bounded) D domain...
The proof of the theorem in the case of a limited (bounded) domain is similar to that of max modulus and so it is clear.
Furthermore some authors use the theorem dealing with not bounded domain ( eg the half right plane).
E.A. Guillemin "TheMathematicsOfCircuitAnalysis" pag.412
write that "the real part of an analytic function on the real half plane ( unbounded domain) reach the min ( and the max) on the immaginary axis being the imaginary axis the contour..."
In previous chapter he demonstrate the theorem but just for bounded domain and not for the half R. plane..
Why the theorem is still valid for the unbounded RHP?
Why the contour, to be considered, is just the imaginary axis?
Does anybody know a proof of that or has a reference to that proof?

I tried to work out but te only consideration was...
"If I consider the same function in a huge half a circle ( diameter on the imaginary axis and circumference on the RHP...)so that this is a bounded domain, the theorem is sure valid, but the minimum could be yield on the circumference and not on the imaginary axis.."

thank you for your replies..

Title: Re: Minimum Real Part theorem issue on RHP (unboun
Post by aleberto69 on Oct 12^th, 2013, 2:19am

I find out that the theorem applied to the Right Half Plane is not valid in the case of the maximum of the real part.
I meant that the maximum could not lie on the imaginary axis that some authors consider the contour of the RHP..

The function
F(z)=z+1
is analytic but
Re(ix)=1
while
Re(z)> 1 for any z with Re(z)>0 (belonging to RHP)

The theorem is instead valid if we consider a limited domain and its whole contour..

Anyway my interest is not on the max but on the minimum of the real part of f(z)...
May be the theorem of the minimum is valid in the RHP with the imaginary axis as contour?

The discussion is not just a mathematical curiosity and arise from some theorems on Positive Real Functions that are useful in linear circuit theory.
A PRF is defined as an f(z) that Re(f(z))>0 for Re(z)>0.
In many text books you can find the statment that
a function f(z)
analitycal in the RHP
and having
Re(f(z))>=0 in the imaginary axis
is a Real Positive function

this is an interesting property since you move the problem of the positive realeness test from the hole RHP to just the imaginary axis.

The authors demonstrated the theorem referring to the "minimum real part theorem" appling it to the RHP(unbounded) and considering the imaginary axis as its contour.

Unfortunately I think this is not a rigorous proof...

some ideas?

Title: Re: Minimum Real Part theorem issue on RHP (unboun
Post by aleberto69 on Oct 12^th, 2013, 2:26am

sorry ... some imperfections about the example.

the correct example is

The function
F(z)=z+1
is analytic but
Re(F(ix))=1 ( on the immaginary axis)
while
Re(F(z))> 1 for any z with Re(z)>0 (belonging to RHP)