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I am wondering if the field of "Complex Analysis in One Variable" is different from the field called "Complex Function Theory". I hope this is not a ridiculous question, because I have tried finding out the difference on Wikipedia first, but "Complex Function Theory" had nothing come up.

Or is Complex Function Theory a subset of Operator Theory or is it different from it as well?

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AFAIK they mean the same thing. – Pete L. Clark Jun 29 '11 at 7:27
I agree that these terms should mean the same thing. However, if I came across a book titled "Complex Analysis" I would expect to read about the theory of holomorphic functions of one variable. If I came across a book titled "Complex Function Theory" I would expect a more advanced text, maybe with material on several complex variables or more general (not necessarily holomorphic/meromorphic) functions. A quick Amazon search shows that my expectations would be wrong, though. – Corey Jun 29 '11 at 8:05
AFAIK they are the same, the confusion may come from the fact that "complex analysis" or "complex calculus" is in German simply "Funktionentheorie", although taken at face values this means "theory of functions" (which results in misunderstands of what the topic is about among German freshmen sometimes). – Tim van Beek Jun 29 '11 at 8:07
@Corey : I would not even expect that. From the books I've seen, a title of "Complex Function Theory" can go anywhere from introductory complex analysis to things like Remmert's Classical Topics in Complex Function Theory, so it is in this case especially unwise to judge a book by its cover. – Willie Wong Jun 29 '11 at 9:16
@Willie: As Tim says, Funktionentheorie without further specification refers to complex analysis in one variable. I've never heard it used in another way except by uninformed people. The specification "Funktionentheorie in einer komplexen Veränderlichen" would be a bit nicer, I think, but "Variablen" is perfectly fine. – t.b. Jun 29 '11 at 9:38
up vote 3 down vote accepted

For the question as stated exactly, I would say that "complex analysis in one variable" is a subset of "complex function theory", which in principle can allow more than one variables. (If the question were about "complex analysis" without the qualifier "in one variable", I would completely agree with Pete and Tim's comments that they are two names for the same thing.)

In the AMS MSC, one complex variable is in 30-XX, while several complex variables is in 32-XX, reflecting the fact that there are important differences between the two sub-disciplines. So one may arguably say that the name "complex analysis" or "complex function theory" encompass two fields under the same heading.

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+1. Nice to know the AMS MSC. – Jack Jul 17 '11 at 22:24

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