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Post by Admin on Jul 4, 2017 7:42:22 GMT
Gary
The second sentence of the last paragraph of subsection 1 on page 412 doesn't make sense in my printing of the book (year 2000).
I have come to the conclusion that the word "same" is missing and that the sentence should read:
"This leads us to believe that each term must die away with the same reciprocal dependence on $\epsilon$ as governs the growth of the number of terms in the series".
This is the only way I can make sense of it, and it fits in perfectly with the discussion in the paragraph as a whole.
Vasco
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Gary
GaryVasco
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Post by Gary on Jul 4, 2017 14:18:47 GMT
Gary The second sentence of the last paragraph of subsection 1 on page 412 doesn't make sense in my printing of the book (year 2000). I have come to the conclusion that the word " same" is missing and that the sentence should read: "This leads us to believe that each term must die away with the same reciprocal dependence on $\epsilon$ as governs the growth of the number of terms in the series". This is the only way I can make sense of it, and it fits in perfectly with the discussion in the paragraph as a whole. Vasco Vasco, I am not bothered by the sentence construction as it stands, as it seems to me that the construction "as governs ...." refers to something in the immediately preceding exposition. But I agree that "same" would be a bit more clear. This is a nice problem in anaphora. My problem is with "reciprocal", as I am at a loss for a referent. My best guess at this point is that the dying away of the area of terms is "reciprocal" to the growth of terms (increasing number of squares), so reciprocal refers to the inverse relationship, the division by number of terms. I am more used to thinking of "reciprocal" in terms of reciprocity, a mutual influence. Gary |
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Post by Admin on Jul 4, 2017 15:06:35 GMT
Gary
The way I see it is as follows:
The sum of the series in (18) is approximately equal to $N\times\displaystyle\oint_{\square}f(z)dz$, where $N$ is the number of terms in the series and $\displaystyle\oint_{\square}f(z)dz$ is a typical term in the series. Since we know that this sum will be nonzero and finite then the two parts of this product must be the reciprocal of each other in terms of their dependence on $\epsilon$,
($1/N\approx\displaystyle\oint_{\square}f(z)dz$),
and so because $N$ grows like $(1/\epsilon^2)$ then each term must die away as the reciprocal, that is as $\epsilon^2$, otherwise the sum of the series is either infinite or zero as $\epsilon$ approaches zero.
Vasco
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Jul 4, 2017 15:35:40 GMT
Gary The way I see it is as follows: The sum of the series in (18) is approximately equal to $N\times\displaystyle\oint_{\square}f(z)dz$, where $N$ is the number of terms in the series and $\displaystyle\oint_{\square}f(z)dz$ is a typical term in the series. Since we know that this sum will be nonzero and finite then the two parts of this product must be the reciprocal of each other in terms of their dependence on $\epsilon$, ($1/N\approx\displaystyle\oint_{\square}f(z)dz$), and so because $N$ grows like $(1/\epsilon^2)$ then each term must die away as the reciprocal, that is as $\epsilon^2$, otherwise the sum of the series is either infinite or zero as $\epsilon$ approaches zero. Vasco Vasco, That is what I guessed, though you have it worked out more explicitly. Gary |
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