Basic Complex Analysis?

Question

How do you determine where a function is analytic? And what exactly does analytic mean?

I've got a list of functions such as

(3z-1) / (3-z)

Where z is a complex number and I'm being asked to find where its analytic and to compute its derivative.

Answer 1

A function is analytic at a point if it is differentiable everywhere in a neighbourhood of that point.
An equivalent definition is that the function can be expanded into a power series that converges in a neighbourhood of that point.

In your example, (3z - 1)/(3 - z) is differentiable at every point other than 3, and therefore analytic at every point except z = 3.
You can find its derivative using the quotient rule.

Answer 2

See the link for what analytic means. Can't help you more than that.