Find f(z) when |z|<=1 and |f(z)|=|e^z| for |z|=1

In summary: I'm glad I didn't took you tire...In summary, there is another possibility for f(z) that I can think of.
  • #1
Cosmossos
100
0
f(z) is analytic and not equal to zero in the unit circle (|z|<=1) . we also know that |f(z)|=|e^z| for |z|=1. find f(z)

How should i approch to this question? I know that on inside the boundry it can't get the maximum nor the minimum .. but it doesn't help at all. i have no idea what to do.
 
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  • #2
Does the question imply that there is a unique answer? It is obvious that the function [itex]f(z)= e^z[/itex] itself satisfies all of those properties,
 
  • #3
no, but I'm asked to find it, so I think i should do more than that
 
  • #4
Cosmossos said:
no, but I'm asked to find it, so I think i should do more than that

Jesus dude, you kinda' antagonizing huh? I mean you got a flat tire, I say, "hey, here's one that's not flat that'll fit, use it," and you say, "oh no, that one won't work." So I say, "dude, it's the same diameter, same lugs, and best of all, it's ain't flat." Then you say, . . . ok, I'll stop being anoying.
 
  • #5
The question is written in a way that implies that the answer is not just ez. But maybe the question is just worded badly.
 
  • #6
jackmell said:
Jesus dude, you kinda' antagonizing huh? I mean you got a flat tire, I say, "hey, here's one that's not flat that'll fit, use it," and you say, "oh no, that one won't work." So I say, "dude, it's the same diameter, same lugs, and best of all, it's ain't flat." Then you say, . . . ok, I'll stop being anoying.

Huh? the simple answer isn't going to work here and it is like giving the trivial solution for a set of equations . This question isn't a stupid college question but some levels above that and needed a little bit more then just stating the answer . jackmell , next time, think before you shoot...

By the way, this isn't the complete answer ... so my friend, I'm glad I didn't took you tire...

And for the one who actually try to answer my question, thank you very much and I didn't meant to insult you in any way.
 
  • #7
BruceW said:
The question is written in a way that implies that the answer is not just ez. But maybe the question is just worded badly.

You are correct
 
  • #8
It's a teachable moment. Sometimes in math, the obvious solution is in fact the answer. Don't try and make it more complicated than it is. Final exam, you ask me that question, I'm going with e^x. Bingo-bango, wait a while so I'm not the first one to leave the class then I'm outta there.
 
  • #9
jackmell said:
It's a teachable moment. Sometimes in math, the obvious solution is in fact the answer. Don't try and make it more complicated than it is. Final exam, you ask me that question, I'm going with e^x. Bingo-bango, wait a while so I'm not the first one to leave the class then I'm outta there.

In that case you got 5 out of 20. enjoy
 
  • #10
I don't think there is a single teacher teaching Complex Analysis in the entire world that would disagree with me. Well, maybe not the smart-elic stuf. Mostly just playing around.
 
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  • #11
Cosmossos said:
By the way, this isn't the complete answer ...

You sound like you have some idea of what the actual answer is..?
If I had to guess, I would say ez was a correct answer, despite the strangely-worded question.
 
  • #12
Cosmossos said:
Huh? the simple answer isn't going to work here and it is like giving the trivial solution for a set of equations .
Why not? That was why I asked "Does the question imply that there is a unique answer?" The question said " find f(z)" which seems to me to imply that there is a unique solution- in which case it would have to be [itex]e^z[/itex].

This question isn't a stupid college question but some levels above that and needed a little bit more then just stating the answer . jackmell , next time, think before you shoot...

By the way, this isn't the complete answer ... so my friend, I'm glad I didn't took you tire...

And for the one who actually try to answer my question, thank you very much and I didn't meant to insult you in any way.
 
  • #13
There is another possibility for f(z) that I can think of.
 
  • #14
BruceW said:
There is another possibility for f(z) that I can think of.

Do you mind sharing?
 
  • #15
Does it matter? Its another obvious one. And you said "but I'm asked to find it, so I think i should do more than that ... the simple answer isn't going to work here and it is like giving the trivial solution for a set of equations ... By the way, this isn't the complete answer " So I'm guessing you've been told that its not just an easy answer?
 
  • #16
BruceW said:
Does it matter? Its another obvious one. And you said "but I'm asked to find it, so I think i should do more than that ... the simple answer isn't going to work here and it is like giving the trivial solution for a set of equations ... By the way, this isn't the complete answer " So I'm guessing you've been told that its not just an easy answer?

come on ... The level of the course is more than that. believe me , I KNOW.
No one would give us this question if that was the answer
 
  • #17
If it is not an easy answer, then I can't for the life of me think what the answer is. Sorry I haven't really been able to help
 
  • #18
Thank you everyone for try to help me!
 

FAQ: Find f(z) when |z|<=1 and |f(z)|=|e^z| for |z|=1

What does the expression "f(z)" refer to in this context?

In this context, "f(z)" refers to a function that takes a complex number "z" as an input and outputs a complex number "f(z)".

What does the notation "|z|" mean?

The notation "|z|" represents the absolute value or modulus of the complex number "z". This is calculated by taking the square root of the sum of the squares of the real and imaginary parts of "z".

How can I find the value of "f(z)" when |z| is less than or equal to 1?

To find the value of "f(z)" for |z| <= 1, you can substitute the value of "z" into the function and evaluate it. In this case, you can use the given information that |f(z)| = |e^z| and the fact that |z| = 1 for |z| = 1, and solve for "f(z)".

Is there a specific method to find "f(z)" for this problem?

Yes, there are several methods that can be used to find "f(z)" for this problem. One method is to use the given information that |f(z)| = |e^z| and apply the properties of absolute value to solve for "f(z)". Another method is to use the definition of a complex exponential function and solve for "f(z)" using algebraic techniques.

Can "f(z)" have multiple solutions for |z| <= 1?

No, in this context, "f(z)" can only have one solution for |z| <= 1. This is because the given information |f(z)| = |e^z| restricts the possible values of "f(z)" for any given value of "z". Therefore, there can only be one function "f(z)" that satisfies both conditions for |z| = 1.

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