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

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.

 Quote by nigels 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.
 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

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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 $e^x$ would no longer be "functions". So we drop that requirement.