Gary
GaryVasco
Posts: 3,352 
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Post by Gary on Aug 12, 2017 16:39:10 GMT
Vasco,
In reading VIII The Fundamental Theorem, 1 Introduction, p. 403, I found this regarding path-independence: “Recall that we were only able to apply Cauchy’s Theorem after having explicitly produced an example (a circle) of a loop integral that vanishes in spite of enclosing the singularity.” Do you think Needham was referring to functions such as $\frac{1}{z^m}$, where $m < -1$ (p. 396)? If so, it is surprising that he did not highlight it as an example of vanishing that contradicts Cauchy’s Theorem.
Gary
P.S. On reading subsection 3, pp. 404-406, we find that Needham did highlight the example $\frac{1}{z^2}$ and pointed out on p. 406, para. 1 that "since B = A for a closed loop, the [integral] vanishes." But he did not make the point that it contradicts Cauchy's Theorem, only that path independence permits use of the Fundamental Theorem "thus avoiding the attendant anxiety over the singularity at the origin." What is the distinction between something that causes "attendant anxiety" and a contradiction?
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Post by Admin on Aug 14, 2017 9:45:50 GMT
Vasco, In reading VIII The Fundamental Theorem, 1 Introduction, p. 403, I found this regarding path-independence: “Recall that we were only able to apply Cauchy’s Theorem after having explicitly produced an example (a circle) of a loop integral that vanishes in spite of enclosing the singularity.” Do you think Needham was referring to functions such as $\frac{1}{z^m}$, where $m < -1$ (p. 396)? If so, it is surprising that he did not highlight it as an example of vanishing that contradicts Cauchy’s Theorem. Gary P.S. On reading subsection 3, pp. 404-406, we find that Needham did highlight the example $\frac{1}{z^2}$ and pointed out on p. 406, para. 1 that "since B = A for a closed loop, the [integral] vanishes." But he did not make the point that it contradicts Cauchy's Theorem, only that path independence permits use of the Fundamental Theorem "thus avoiding the attendant anxiety over the singularity at the origin." What is the distinction between something that causes "attendant anxiety" and a contradiction? Gary Just as a first comment on this, (more to follow), this result for $\frac{1}{z^2}$ does not contradict Cauchy's Theorem. The theorem says that if the loop does not contain any singularities then the integrals of the inverse power functions around it must vanish. It does not say the converse, that if the integral vanishes then the loop does not contain any singularities. Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Aug 14, 2017 14:28:20 GMT
Vasco,
Thank you. That clears up some things, but leaves me a little puzzled about others. It helps in understanding the last two paragraphs of Section I, Introduction, beginning p. 377, but it might have helped to add a qualification to the last paragraph there to warn the reader that two integrals from a to b can agree in some cases where there is a singularity between the two contours. In section V he discusses the case of 1/z, a function with a region which is analytic at every point but zero with an integral that does not vanish. And then in section VIII, ss 4, we learn that integrals on a closed loop can (generally do?) vanish in spite of containing a singularity. I had not sorted all this out. But it still seems to me that the sentence regarding "avoiding the attendant anxiety" is not quite accurate. Isn't it the case that Cauchy's Theorem just does not apply to the integration of $z^2$ on all contours because it does not apply to the case when the contour contains 0? But I now see why in the last paragraph of VIII, 3, p. 406, he can not use Cauchy's Theorem to extrapolate from the vanishing of an integral on a circle to the vanishing for more general loops. Previously, I just did not get the point. Then it seems that Cauchy's theorem is an interesting, no doubt important result, that has no application to closed curves containing singularities (20). Those contours must be integrated by other means. This also helps me to understand XI and the significance of the General Cauchy Theorem.
Gary
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Post by Admin on Aug 15, 2017 10:41:32 GMT
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Aug 16, 2017 4:57:44 GMT
Vasco, I agree with your comments. I went back through the chapter to try to write a synopsis focusing on Cauchy's Theorem. That effort is provided in the attachment, the text of which I will be adding to my notes on Ch. 8. nh.ch8.notes.synopsis.pdf (51.87 KB) Gary |
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Post by Admin on Aug 16, 2017 5:25:16 GMT
Gary
I have just read your document and I agree with your comments.
Vasco
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