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Post by Admin on Apr 8, 2017 5:40:17 GMT
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Gary
GaryVasco
Posts: 3,352 
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Post by Gary on Apr 18, 2017 23:45:57 GMT
Vasco,
Did you find Ex. 10 unproblematic? It strikes me as reversing $f(z)$ and $g(z)$. With the positive coefficient, $e^a$, $|z|^n$ must always be smaller than $e^z$. Hence, $|z|^n$ must be the short leash and we can't use it to find the zeros, but the long leash $e^z$ has no zeros.
Gary
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Post by Admin on Apr 19, 2017 16:23:09 GMT
Gary
I find that $|z^ne^a|>|e^z|$. If you want any more detail, let me know.
Vasco
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Apr 20, 2017 16:43:50 GMT
Vasco, Here is my answer. nh.ch7.ex10.pdf (95.03 KB) I didn't see how I could write an inequality that seemed to contradict the equation, so I went another direction. I'll look at your answer now. Gary |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Apr 20, 2017 17:05:24 GMT
Vasco,
I looked at your answer and even though I could follow all the steps, I still could not reconcile $z^ne^a = e^z$ with $|z^ne^a| > |e^z|$ until I realized that the equality has solutions (z's) for which it is true and that it doesn't have to be true for z's on the unit circle.
Gary
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Post by Admin on Apr 20, 2017 18:19:56 GMT
Gary
No wonder you were perplexed! Yes, the inequality is on $\Gamma$, but the solutions are inside $\Gamma$.
I don't understand how you find 5 zeroes inside the unit circle. The function $z^ne^a$ has $n$ zeroes inside the unit circle and so you have proved that $z^ne^a=e^z$ also has $n$ zeroes inside the unit circle.
Vasco
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Apr 20, 2017 21:19:10 GMT
Gary No wonder you were perplexed! Yes, the inequality is on $\Gamma$, but the solutions are inside $\Gamma$. I don't understand how you find 5 zeroes inside the unit circle. The function $z^ne^a$ has $n$ zeroes inside the unit circle and so you have proved that $z^ne^a=e^z$ also has $n$ zeroes inside the unit circle. Vasco Vasco, Agreed. I think I was thinking of Exercise 11 when I wrote 5 zeros. I have replaced my answer. Gary |
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