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Old 2022-07-12, 17:52   #1
wildrabbitt
 
Jul 2014

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Default a contour integral device

Hi, I'm reading a book and I need to know how to evaluate this integral :



\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{y^s}{s}ds


/ \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{y^s}{s}ds forgotten how to get latex in a post again

I know it equals 0 on (0,1), 1/2 for y = 1 and 1 for y > 1 but I can't find a proof anywhere.


Perhaps someone recognises it and knows a page online or a book where I could find it?

Last fiddled with by wildrabbitt on 2022-07-12 at 17:59
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Old 2022-07-12, 18:37   #2
paulunderwood
 
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Sep 2002
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Quote:
Originally Posted by wildrabbitt View Post
Hi, I'm reading a book and I need to know how to evaluate this integral :



\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{y^s}{s}ds


/ \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{y^s}{s}ds forgotten how to get latex in a post again

I know it equals 0 on (0,1), 1/2 for y = 1 and 1 for y > 1 but I can't find a proof anywhere.


Perhaps someone recognises it and knows a page online or a book where I could find it?
Encapsulate the \(\LaTeX\) in backslash left braket and backslash right bracket. For inline use parentheses.

\[\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{y^s}{s}ds\]

Last fiddled with by paulunderwood on 2022-07-12 at 18:43
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Old 2022-07-12, 19:50   #3
wildrabbitt
 
Jul 2014

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Thanks.
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Old 2022-07-12, 22:55   #4
charybdis
 
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Apr 2020

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Quote:
Originally Posted by wildrabbitt View Post
Perhaps someone recognises it and knows a page online or a book where I could find it?
This (without the easier y=1 case) is a lemma that appears in the proof of Perron's formula. Here is a reference I found.
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Old 2022-07-13, 06:13   #5
wildrabbitt
 
Jul 2014

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Thanks a lot. That's just the sort of thing I was looking for.
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