Steve
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 Nov 29 awarded Scholar Nov 29 accepted Fundamental motivation for several complex variables Nov 28 comment Fundamental motivation for several complex variables Thank you for this answer. I will have to take some time to study and digest it. Maybe it would have been better to put a focus on domains of holomorphy in my question. Much of the research going on in SCV seems to deal with characterizing these domains. Nov 28 comment Fundamental motivation for several complex variables @Daniel - this "multiple single variable complex analysis" is mostly what I have seen applied - not the scv phenomena which are unique to that field. I think the proof you have offered is awesome though! Nov 27 awarded Nice Question Nov 27 comment Fundamental motivation for several complex variables @fedja - I, and I suspect a lot of other mathematicians, like to have a general feeling for where and why the different fields of mathematics are useful. I may not have much use for cohomology in my day to day mathematical life, but I know that if I am confronted with a local to global problem, searching for a cohomological interpretation will be useful. Or if my problem is invariant under the action of some group, the representation theory of that group will probably be useful. I know SCV is an important field, I just don't know why yet. Maybe someone here can provide some insight. Nov 27 awarded Student Nov 27 comment Fundamental motivation for several complex variables I have not yet seen a real application of SCV in complex algebraic geometry - mostly it is just using the "multivariable calculus" of complex numbers. I don't see where the unique features of SCV come into the picture (namely: domains of holomorphy) Nov 27 asked Fundamental motivation for several complex variables Nov 11 comment A question that arises in trying to make mathematically precise a well known informal statement about analytic functions It is sort of sad that such basic observations are not usually given to our calculus students, or complex analysis students for that matter. It was many years after learning these subjects, and feeling somewhat dissatisfied with my state of knowledge, that I finally figured these things out for myself. This plus analyticity is, in my opinion, the real reason for the identity theorem. Nov 11 comment A question that arises in trying to make mathematically precise a well known informal statement about analytic functions Here is the recipe for f''(0) which you should be able to generalize. Take 3 points within a distance of $\delta$ from 0. Find the unique quadratic passing through all 3 points. Half the lead term of this quadratic is an approx of the second derivative. Form a sequence of such approximations as $/delta$ goes to 0. The limit of this sequence is the second derivative. Nov 9 awarded Teacher Nov 9 answered A question that arises in trying to make mathematically precise a well known informal statement about analytic functions