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#1 2009-08-22 02:54:11

dannyv
Member
Registered: 2007-09-20
Posts: 34

On the maximum of real-valued functions with complex image

Hi, I have the following problem

given a function f(k) defined on the reals and a complex constant z0, what is the maximum of following function?

The maximum of the module is clearly the value k such that

right? because when you take the module, the squares of the real and imaginary parts are maximum and hence the module is maximum.

But what happens when you cannot factorize the complex constants? e.g., given the following fuction

where k is only real, and z1 and z2 are complex constants. Can we still derivate g, make it equal to 0 and still say we can find a critical point? i.e., does solving


gives you a critical point?

thanks in advance for the help  up

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#2 2009-08-22 06:12:00

juriguen
Member
Registered: 2009-07-05
Posts: 59

Re: On the maximum of real-valued functions with complex image

Hi!


If you want the maximum of the modulus, first of all you should calculate the absolute value of g:


After that, you can differentiate that rather ugly expression smile, solve it equal to 0 and find for what k is maximum


Jose


“Make everything as simple as possible, but not simpler.” -- Albert Einstein

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#3 2009-08-22 12:58:11

dannyv
Member
Registered: 2007-09-20
Posts: 34

Re: On the maximum of real-valued functions with complex image

thanks for the reply. I understand that solves the problem, but actually what I want to know if this:

does solving |g(k)|'=0 and g'(k)=0 give the same critical points?

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#4 2009-08-22 13:10:24

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: On the maximum of real-valued functions with complex image

Hi dannyv;

dannyv wrote:

does solving |g(k)|'=0 and g'(k)=0 give the same critical points?

I'm not sure, but I worked a few simpler g(k) that had complex constants. I found the g'(k)=0 and it wasn't a maxima or a minima.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#5 2009-08-23 03:46:14

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: On the maximum of real-valued functions with complex image

given a function f(k) defined on the reals and a complex constant z0, what is the maximum of following function?

You first need to phrase your question in a way that it is possible to answer.  Since z_0f(k) is of a complex value, there is no concept of maximum or minimum.  In fact, you can't even say one complex number is greater than another.  For example, which is greater, 1 or i?  You can talk about which modulus is greater (in which case, |1| = |i| = 1), and you can talk about when z_0f(k) achieves its maximum modulus.  Is this what you want?


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#6 2009-08-23 16:08:49

dannyv
Member
Registered: 2007-09-20
Posts: 34

Re: On the maximum of real-valued functions with complex image

yes, I meant maximum of the module. I the question, as I wrote above

does solving |g(k)|'=0 and g'(k)=0 give the same critical points?

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#7 2009-08-24 10:10:59

Avon
Member
Registered: 2007-06-28
Posts: 80

Re: On the maximum of real-valued functions with complex image


Suppose
is a continuous function and let

If

is differentiable at
and
then
on some neighbourhood of
and so

If
is differentiable at
and
then
on some neighbourhood of
and so

Hence if

is differentiable at
and
then
is also differentiable at
and
if and only if

It follows that a critical point of

must be either a critical point of
or a zero of

If the maximum of

is attained at a zero of
then we must have that
is identically zero, and so every point is a critical point of

Therefore if

has a maximum value, it is attained at a critical point of

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#8 2009-08-24 10:37:06

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: On the maximum of real-valued functions with complex image

Avon, we're are talking about functions that take on complex values.  Saying f(x) > 0 makes no sense.

does solving |g(k)|'=0 and g'(k)=0 give the same critical points?

Sorry, but once again I have to answer you with a question.

given a function f(k) defined on the reals

Does it map into the reals?  Or into the complex numbers?


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#9 2009-08-25 01:45:01

dannyv
Member
Registered: 2007-09-20
Posts: 34

Re: On the maximum of real-valued functions with complex image

k  takes values on the reals and maps into the complex numbers

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