e^(-i * x) not well-defined. Why??


by nigels
Tags: complex analysis, holomorphic, well-defined
nigels
nigels is offline
#1
Nov4-12, 06:18 PM
P: 25
Hi, Just saw this as a step in an example that demonstrates the differentiability of holomorphic function. But I can't for the life of me figure out why e^(-2iθ) is ill-defined.
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Hetware
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#2
Nov4-12, 06:27 PM
P: 125
Quote Quote by nigels View Post
Hi, Just saw this as a step in an example that demonstrates the differentiability of holomorphic function. But I can't for the life of me figure out why e^(-2iθ) is ill-defined.
What do you mean it is ill-defined. Why do you say that? It's well defined in the opinions of many smart people. I can't go beyond that until I know what objection there is to the conventional definition there is.
Vargo
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#3
Nov6-12, 03:50 PM
P: 350
It is well defined as a function of a real variable theta. But as a function on the plane, considering theta=theta(z), it has a problem at z=0. Does this answer your question? It is hard to tell without more context.

As a function of z, this function is the complex conjugate of (z/|z|)^2

HallsofIvy
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#4
Nov8-12, 07:23 AM
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e^(-i * x) not well-defined. Why??


The standard definition of "function" for real numbers requires that "if x= y, then f(x)= f(y)"- i.e. that f is "well-defined". For functions of complex variables, that is simply too restrictive. "Functions" that we would like to be able to use, such as [itex]e^x[/itex] would no longer be "functions". So we drop that requirement.
Klungo
Klungo is offline
#5
Nov8-12, 08:13 AM
P: 136
Some functions are given the requirement of Principal Value to make some functions of a complex variables become actual functions.

I.e , let z=Re^(ix), and restrict x to be in
(-pi,pi].

This restriction works since e^(ix)=e^(ix+i*2n*pi) for all integers n.

On the other hand, it makes functions behave less like we would wabt them to.

That is, Log(xy)=/=Log(x)+Log(y) generally for principal value logarithm. On the other hand, log(xy)=log(x)+log(y).

(correct me if I'm wrong as I am just typing from memory)

Edit: x,y is a complex number for the logarithm examples.


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