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I was reading a article on proving the relationship between gamma and beta functions but there are some things I don't understand.

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Then the variables are changed as
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Then they have obtained
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this.But I don't understand $dv$is obtained.(when changing variables v = s+t has to be differentiated) .Since v include both s and t how is it differentiated and with respect to what.Also how does v change from 0 to v

clarkson
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  • Hello clarkson, have you had experience with double integrals and changes of coordinates before? What you are looking for is called the Jacobian and information can be found here and here. The first link should teach you how to use it and the second link should explain what it is. – Brad Mar 10 '14 at 16:13
  • I don't know Jacobian.I'll read those links.Is there any other method to prove this gamma beta relationship without having to use Jacobian – clarkson Mar 10 '14 at 16:41
  • I understand the Jacobian part.But still I don't understand how v vary from 0 to v – clarkson Mar 10 '14 at 17:28
  • Be sure to note that it's $u$ that varies from $0$ to $v$. In my mind I see that $u = s$ and that, because $v$ goes from $0$ to $\infty$ and $u$ goes to $v$, you will cover all values of $s$ from $0$ to $\infty$. Likewise, $t$ must also go to $\infty$ because $v$ does. – Brad Mar 10 '14 at 18:59

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