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I wrote analytic function, mostly just a definition. I found a reference that treated the infinite-dimensional case in pretty fair generality (slightly more than I actually did) without making the definition any more complicated (well, except one place where one must insert the word ‘continuous’), so I did that.
What is a homogeneous operator? Apparently you do mean $\sum a_k (x-c)$ rather than $\sum a_k (x-c)^k$, as I would suppose. Merry Christmas!
Right, it's $a_k$ applied to $x - c$, rather than $a_k$ multiplied by $(x-c)^k$.
Why did you remove the claim that (for functions on a subset of the complex plane) analytic functions are differentiable?
Colin, in the case of definitions, an ’if’ as in the sentence “a topological space is Hausdorff if any two distinct points have disjoint neighborhoods” invariably translates to ’if and only if’. Otherwise, for the sake of clarity, one should use the full phrase ’if and only if’ (if and only if it applies, of course). There are substitutes such as ’exactly when’ which might sound more graceful to some ears.
Perhaps you mean to take back the assertion that “X is Y” means (or should mean) “X if and only if Y”. This is patently false. For example, we say “a compact Hausdorff space is normal”.
If we were speaking exclusively about functions of a single complex variable, $f: U \to \mathbb{C}$ where $U \subseteq \mathbb{C}$ is an open domain, then yes, it would be perfectly legitimate to write “we say $f$ is analytic if it is complex differentiable” and use that as a definition. In fact, this is the definition given in many accounts, e.g., the text by Ahlfors.
For a definition, I would not write (and I’ve never seen anyone write) “each function in one complex variable is analytic if differentiable” because that ’each’ would be confusing, making it would look much more like an assertion or proposition than a definition.
@ Todd: There is no definition in the material whose phrasing Colin and I were discussing; the definition of ‘analytic’ is given earlier in a more general context, the definition of ‘differentiable’ is given on its own page, and their equivalence (in a certain context) is being given as a theorem.
@ Colin: As a matter of mathematical English grammar, the phrase ‘A [adjective-1] [noun] is [adjective-2].’ doesn't imply that every [adjective-2] [noun] is [adjective-1]. Some phrasing other than ‘if and only if’ might be possible, but not that one.
@Toby: that’s neither here nor there, because I wasn’t discussing the article; I was discussing English language usage (e.g., meaning of ’is’ and of ’if and only if’). I mentioned definitions as an instance where one could get away with saying ’if’ in place of ’if and only if’.
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