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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeDec 4th 2011
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   Author: TobyBartels
   Format: MarkdownItexI 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorColin Tan
   • CommentTimeDec 24th 2013
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   Author: Colin Tan
   Format: MarkdownItexWhat 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!

   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!

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeDec 24th 2013
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   Author: TobyBartels
   Format: MarkdownItexRight, it\'s $a_k$ *applied to* $x - c$, rather than $a_k$ *multiplied by* $(x-c)^k$.

   Right, it's $a_k$ applied to $x - c$, rather than $a_k$ multiplied by $(x-c)^k$.

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeDec 24th 2013
   • (edited Dec 24th 2013)
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   Author: TobyBartels
   Format: MarkdownItexWhy did you remove the claim that (for functions on a subset of the complex plane) analytic functions are differentiable?

   Why did you remove the claim that (for functions on a subset of the complex plane) analytic functions are differentiable?

5. • CommentRowNumber5.
   • CommentAuthorColin Tan
   • CommentTimeDec 25th 2013
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   Author: Colin Tan
   Format: TextI write X is Y to mean X if and only if Y. Generally, I write X is Y asymmetrically to indicate the hard direction is X implies Y. The "is" matches our use of the word in the English language. Another example of such usage is "an bounded entire function is constant." Do revert if you would.

   I write X is Y to mean X if and only if Y. Generally, I write X is Y asymmetrically to indicate the hard direction is X implies Y. The "is" matches our use of the word in the English language.
   
   Another example of such usage is "an bounded entire function is constant."
   
   Do revert if you would.
6. • CommentRowNumber6.
   • CommentAuthorColin Tan
   • CommentTimeDec 25th 2013
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   Author: Colin Tan
   Format: TextI take back that portion of my comment about English language usage. I dislike the predicate "if and only if". When formulating a theorem, I wish that the resulting verbal articulation be easily remembered. Do you have a means to preserve the symmetric technical meaning yet have a graceful articulation?

   I take back that portion of my comment about English language usage.
   
   I dislike the predicate "if and only if". When formulating a theorem, I wish that the resulting verbal articulation be easily remembered. Do you have a means to preserve the symmetric technical meaning yet have a graceful articulation?
7. • CommentRowNumber7.
   • CommentAuthorColin Tan
   • CommentTimeDec 25th 2013
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   Author: Colin Tan
   Format: TextI reverted the theorem back to the original phrasing until a better alternative can be found.

   I reverted the theorem back to the original phrasing until a better alternative can be found.
8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeDec 25th 2013
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   Author: Todd_Trimble
   Format: MarkdownItexColin, 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".

   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”.

9. • CommentRowNumber9.
   • CommentAuthorColin Tan
   • CommentTimeDec 25th 2013
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   Author: Colin Tan
   Format: TextTodd, I realize that the suggestion you gave is what we commonly write for definitions. In the case of this entry, following this syntax gives "each function in one complex variable is analytic if differentiable." Would we want to write that? Yes, the part on usage of "is" is what I wish to take back.

   Todd, I realize that the suggestion you gave is what we commonly write for definitions. In the case of this entry, following this syntax gives "each function in one complex variable is analytic if differentiable." Would we want to write that?
   
   Yes, the part on usage of "is" is what I wish to take back.
10. • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 25th 2013
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    Author: Todd_Trimble
    Format: MarkdownItexIf 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.

    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.

11. • CommentRowNumber11.
    • CommentAuthorTobyBartels
    • CommentTimeDec 28th 2013
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    Author: TobyBartels
    Format: MarkdownItex@ 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.

    @ 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.

12. • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 28th 2013
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    Author: Todd_Trimble
    Format: MarkdownItex@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'.

    @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’.