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 wildrabbitt 2022-07-12 17:52

a contour integral device

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

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

/ \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?

 paulunderwood 2022-07-12 18:37

[QUOTE=wildrabbitt;609373]Hi, I'm reading a book and I need to know how to evaluate this integral :

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

/ \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?[/QUOTE]

Encapsulate the $$\LaTeX$$ in backslash left braket and backslash right bracket. For inline use parentheses.

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

 wildrabbitt 2022-07-12 19:50

Thanks.

 charybdis 2022-07-12 22:55

[QUOTE=wildrabbitt;609373]Perhaps someone recognises it and knows a page online or a book where I could find it?[/QUOTE]

This (without the easier y=1 case) is a lemma that appears in the proof of [URL="https://en.wikipedia.org/wiki/Perron%27s_formula"]Perron's formula[/URL]. [URL="https://abel.math.harvard.edu/archive/213b_spring_05/perron_formula_without_error_estimate.pdf"]Here[/URL] is a reference I found.

 wildrabbitt 2022-07-13 06:13

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

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