Register to reply

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

Share this thread:
nigels
#1
Nov4-12, 06:18 PM
P: 27
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.
Phys.Org News Partner Science news on Phys.org
Wildfires and other burns play bigger role in climate change, professor finds
SR Labs research to expose BadUSB next week in Vegas
New study advances 'DNA revolution,' tells butterflies' evolutionary history
Hetware
#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
#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
#4
Nov8-12, 07:23 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,353
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
#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.


Register to reply

Related Discussions
Show that matrices of defined form have inverse of the same same defined form Calculus & Beyond Homework 2
Well-defined map Calculus & Beyond Homework 2
Well defined Calculus & Beyond Homework 12
How is 0^0 defined? General Math 15