How to calculate the integral
$$ \int_b^c \! (c-s)^{-3/4} (s-b)^{-3/4}\,\mathrm{d}s $$
Thanks.
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2You can call $t-a$ as your new $a$ to reduce the number of symbols in your problem. – Nov 27 '11 at 19:20
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Thanks, I have just changed. – Richard Nov 27 '11 at 19:31
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5Are you familiar with the Beta function? – Did Nov 27 '11 at 19:34
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2After that, you might have a look at this page of particular values of the Gamma function. – Did Nov 27 '11 at 19:37
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1Thanks for help. Now I understand. By substitution $t=\frac{s-b}{c-b}$ we obtain that the integral is equal $(c-b)^{-1/2} B(\frac{1}{4}, \frac{1}{4})$. – Richard Nov 27 '11 at 19:50
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2Yep. You could post this as an answer to your question (yes, this is allowed on the site, even recommended). // Thanks to you, I refreshed my memories about which values of Gamma are known... Not so many! – Did Nov 27 '11 at 20:06
1 Answers
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Let $I=\displaystyle\int_b^c (c-s)^{-3/4}(s-b)^{-3/4}ds$. By substitution $t=\dfrac{s-b}{c-b}$ we have $s-b=(c-b)t$, $c-s=(c-b)(1-t)$, $ds=(c-b)dt$. Hence $$ I=\int_0^1 (c-b)^{-3/4}(1-t)^{-3/4}(c-b)^{-3/4}t^{-3/4}(c-b)dt=(c-b)^{-1/2}B\left(1/4,1/4\right), $$ where $B$ is Beta function, defined as $B(u,v)=\displaystyle\int_0^1 (1-t)^{v-1}t^{u-1}dt$ for every positive $u$ and $v$.
Zev Chonoles
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Richard
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1+1. Cleaned up a little bit your LaTeX and added a link. If you mind, cancel these. – Did Nov 28 '11 at 08:53
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2@Asaf, nightowl vandalized my editing and you did not clean up the mess. Consider that at present $I$ is not the integral the OP is interested in, that the arguments of Beta are wrong, and that the link to the WP page on the Beta function that I added is gone. Great. I suggest to come back to my version but I will not fight for this. In fact, at this point, I wish to (1) apologize to Richard and (2) disassociate myself from the editing process of this post. – Did Nov 28 '11 at 10:24
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@Didier: I am terribly sorry. It's hard to see through latex edits sometimes. Richard, I apologize for the mess I caused. – Asaf Karagila Nov 28 '11 at 10:35
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@Richard, did I mention that if the answer suits you (wink, wink), you may accept it? (Good job, should you ask.) – Did Nov 29 '11 at 08:01