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Post by Admin on May 1, 2017 12:21:42 GMT
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Gary
GaryVasco
Posts: 3,352 
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Post by Gary on May 3, 2017 21:29:17 GMT
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on May 3, 2017 21:52:15 GMT
Vasco,
I have had a look at your solution. I would venture that the first sentence of the problem should be rewritten as "Let f(z) be an odd power of z, and consider its effect when applied to points on the unit circle." My reasoning is that f(z) really has no effect on the circle itself other than to expand the radius by $|z|^n$. I understand that if f(z) is applied to every point on the circle, it will also rotate points as well as expand their values, but the focus appears to be on $p$ and $-p$, the amount of winding induced by the power, and the added semicircle.
Gary
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Post by Admin on May 3, 2017 23:07:57 GMT
Vasco, I have had a look at your solution. I would venture that the first sentence of the problem should be rewritten as "Let f(z) be an odd power of z, and consider its effect when applied to points on the unit circle." My reasoning is that f(z) really has no effect on the circle itself other than to expand the radius by $|z|^n$. I understand that if f(z) is applied to every point on the circle, it will also rotate points as well as expand their values, but the focus appears to be on $p$ and $-p$, the amount of winding induced by the power, and the added semicircle. Gary Gary I have always taken "applying a mapping to the unit circle" to mean "applying a mapping to the points on the unit circle". However the exercise is not asking us to consider the mapping $z$ raised to an odd power, but the more general mapping when $f$ is a continuous function with the same properties ie $f(-p)=-f(p). Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on May 4, 2017 1:32:05 GMT
Vasco, I have had a look at your solution. I would venture that the first sentence of the problem should be rewritten as "Let f(z) be an odd power of z, and consider its effect when applied to points on the unit circle." My reasoning is that f(z) really has no effect on the circle itself other than to expand the radius by $|z|^n$. I understand that if f(z) is applied to every point on the circle, it will also rotate points as well as expand their values, but the focus appears to be on $p$ and $-p$, the amount of winding induced by the power, and the added semicircle. Gary Gary I have always taken "applying a mapping to the unit circle" to mean "applying a mapping to the points on the unit circle". However the exercise is not asking us to consider the mapping $z$ raised to an odd power, but the more general mapping when $f$ is a continuous function with the same properties ie $f(-p)=-f(p). Vasco Vasco, I see that I missed the challenge to generalize. Sometimes you have to hit me over the head to impart some sense. I will be reposting very shortly with an added paragraph. Gary |
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Post by Admin on May 6, 2017 8:28:18 GMT
Gary
I interpret net rotation in a different way to you I think. As $z$ moves round the semicircle the net rotation of $f$ still counts the number of times $f$ goes round the origin. The way I see it is that the word net is used because as $z$ moves round the semicircle, the function $f$ may not always rotate in the same direction (sometimes anticlockwise, sometimes clockwise). All we know is that it finally points in the opposite direction to the way it was pointing initially, and the net rotation is the net rotation anticlockwise, which can be more than $\pi$.
I understand the hint as meaning "How is the rotation from $f(-p)$ back to $f(p)$ related to the first rotation from $f(p)$ to $f(-p)$?" Is it twice the 1st rotation, is it the negative of the first rotation, is it the first rotation plus $2\pi$, or what?
Vasco
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on May 8, 2017 23:54:14 GMT
Vasco,
I just needed a bit more time (and study of your answer). I see it now and I reposted.
Gary
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Post by Admin on May 9, 2017 13:53:54 GMT
Vasco, I just needed a bit more time (and study of your answer). I see it now and I reposted. Gary Gary Looks good to me! Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on May 9, 2017 16:17:39 GMT
Vasco, I just needed a bit more time (and study of your answer). I see it now and I reposted. Gary Gary Looks good to me! Vasco Vasco, Thank you. I added a figure and will repost. Gary |
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