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Our next goal is to examine so-called “max/min” problems for coplex-valued functions of complex variables.Since order makes no sense for complex numbers, we will investigate max/min problems for the absolute value of a complex-valued function.For the corresponding question for real-valued functions of real variables, we have as our basic resultthe First Derivative Test. Indeed, when searching for the poinhts where a differentiable real-valued function f on an interval [ a , b ] attains its extreme values, we consider first the poinhts where it attains a local max or min.Of course, to find the absolute minimum and maximum, we must also check the values of the function at the endpoints.

Our next goal is to examine so-called “max/min” problems for coplex-valued functions of complex variables.Since order makes no sense for complex numbers, we will investigate max/min problems for the absolute value of a complex-valued function.For the corresponding question for real-valued functions of real variables, we have as our basic resultthe First Derivative Test ( [link] ). Indeed, when searching for the poinhts where a differentiable real-valued function f on an interval [ a , b ] attains its extreme values, we consider first the poinhts where it attains a local max or min, to which purpose end [link] is useful. Of course, to find the absolute minimum and maximum, we must also check the values of the function at the endpoints.

An analog of [link] holds in the complex case, but in fact a much different result is really valid. Indeed, it is nearly impossible for the absolute value ofa differentiable function of a complex variable to attain a local maximum or minimum.

Let f be a continuous function on a piecewise smooth geometric set S , and assume that f is differentiable on the interior S 0 of S . Suppose c is a point in S 0 at which the real-valued function | f | attains a local maximum. That is, there exists an ϵ > 0 such that | f ( c ) | | f ( z ) | for all z satisfying | z - c | < ϵ . Then f is a constant function on S ; i.e., f ( z ) = f ( c ) for all z S . In other words, the only differentiable functions of a complex variable, whose absolute value attains a local maximum on the interior of a geometric set, are constant functions on that set.

If f ( c ) = 0 , then f ( z ) = 0 for all z B ϵ ( c ) . Hence, by the Identity Theorem ( [link] ), f ( z ) would equal 0 for all z S . so, we may as well assume that f ( c ) 0 . Let r be any positive number for which the closed disk B ¯ r ( c ) is contained in B ϵ ( c ) . We claim first that there exists a point z on the boundary C r of the disk B ¯ r ( c ) for which | f ( z ) | = | f ( c ) | . Of course, | f ( z | | f ( c ) | for all z on this boundary by assumption. By way of contradiction, suppose that | f ( ζ ) | < | f ( c ) | for all ζ on the boundary C r of the disk. Write M for the maximum value of the function | f | on the compact set C r . Then, by our assumption, M < | f ( c ) | . Now, we use the Cauchy Integral Formula:

| f ( c ) | = | 1 2 π i C r f ( ζ ) ζ - c d ζ | = 1 2 π | 0 2 π f ( c + r e i t ) r e i t i r e i t d t | 1 2 π 0 2 π | f ( c + r e i t ) | d t 1 2 π 0 2 π M d t = M < | f ( c ) | ,

and this is a contradiction.

Now for each natural number n for which 1 / n < ϵ , let z n be a point for which | z n - c | = 1 / n and | f ( z n ) | = | f ( c ) | . We claim that the derivative f ' ( z n ) of f at z n = 0 for all n . What we know is that the real-valued function F ( x , y ) = | f ( x + i y ) | 2 = ( u ( x , y ) 2 + ( v ( x , y ) ) 2 attains a local maximum value at z n = ( x n , y n ) . Hence, by [link] , both partial derivatives of F must be 0 at ( x n , y n ) . That is

2 u ( x n , y n ) t i a l u t i a l x ( x n , y n ) + 2 v ( x n , y n ) t i a l v t i a l x ( x n , y n ) = 0

and

2 u ( x n , y n ) t i a l u t i a l y ( x n , y n ) + 2 v ( x n , y n ) t i a l v t i a l y ( x n , y n ) = 0 .

Hence the two vectors

V 1 = ( t i a l u t i a l x ( x n , y n ) , t i a l v t i a l x ( x n , y n ) )

and

V 2 = ( t i a l u t i a l y ( x n , y n ) , t i a l v t i a l y ( x n , y n ) )

are both perpendicular to the vector V 3 = ( u ( x n , y n ) , v ( x n , y n ) ) . But V 3 0 , because V 3 = | f ( z n ) | = | f ( c ) | > 0 , and hence V 1 and V 2 are linearly dependent. But this implies that f ' ( z n ) = 0 , according to [link] .

Since c = lim z n , and f ' is analytic on S 0 , it follows from the Identity Theorem that there exists an r > 0 such that f ' ( z ) = 0 for all z B r ( c ) . But this implies that f is a constant f ( z ) = f ( c ) for all z B r ( c ) . And thenm, again using the Identity Theorem, this implies that f ( z ) = f ( c ) for all z S , which completes the proof.

REMARK Of course, the preceding proof contains in it the verification that if | f | attains a maximum at a point c where it is differentiable, then f ' ( c ) = 0 . This is the analog for functions of a complex variable of [link] . But, [link] certainly asserts a lot more than that. In fact, it says that it is impossible for the absolute value of a nonconstant differentiable function of a complexvariable to attain a local maximum. Here is the coup d'grâs:

Maximum modulus principle

Let f be a continuous, nonconstant, complex-valued function on a piecewise smooth geometric set S , and suppose that f is differentiable on the interior S 0 of S . Let M be the maximum value of the continuous, real-valued function | f | on S , and let z be a point in S for which | f ( z ) | = M . Then, z does not belong to the interior S 0 of S ; it belongs to the boundary of S . In other words, | f | attains its maximum value only on the boundary of S .

  1. Prove the preceding corollary.
  2. Let f be an analytic function on an open set U , and let c U be a point at which | f | achieves a local minimum; i.e., there exists an ϵ > 0 such that | f ( c ) | | f ( z ) | for all z B ϵ ( c ) . Show that, if f ( c ) 0 , then f is constant on B ϵ ( c ) . Show by example that, if f ( c ) = 0 , then f need not be a constant on B ϵ ( c ) .
  3. Prove the “Minimum Modulus Principle:” Let f be a nonzero, continuous, nonconstant, function on a piecewise smooth geometric set S , and let m be the minimum value of the function | f | on S . If z is a point of S at which this minimum value is atgtained, then z belongs to the boundary C S of S .

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
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it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
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