It is currently Fri Apr 20, 2018 8:06 am


All times are UTC [ DST ]




Post new topic Reply to topic  [ 2 posts ] 
Author Message
 Post subject: Root integral
PostPosted: Wed Mar 01, 2017 3:19 pm 

Joined: Sat Nov 14, 2015 6:32 am
Posts: 137
Location: Melbourne, Australia
Do not think dirty about the "root" here.

Let $k \in \mathbb{N} \cup \{0\}$. Evaluate the parametric integral:

$$\mathcal{J}(k)= \int_1^2 \frac{x^{2k+1}}{\sqrt{(x^2-1)(4-x^2)}} \, {\rm d}x$$

_________________
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$


Top
Offline Profile  
Reply with quote  

 Post subject: Re: Root integral
PostPosted: Mon May 29, 2017 7:28 am 

Joined: Thu Dec 10, 2015 1:58 pm
Posts: 59
Location: India
Making the change of variable $x \mapsto x^2$, the integral changes to: $\displaystyle \mathcal{J}(k) = \frac{1}{2}\int_1^4 \frac{x^k}{\sqrt{(x-1)(4-x)}}\,dx$.

Let, $\displaystyle f(z) = \frac{z^k}{\sqrt{(z-1)(4-z)}}$ and consider the integral of $f(z)$ about a positively oriented dumbell contour ($\gamma_{\epsilon}$) around the points $1$ and $4$ and the branch-cut along the line $[1,4]$. Let, $\gamma_{\epsilon}$ shrink 'nicely' to the line $[1,4]$ as $\epsilon \to 0^{+}$ we have:

$$\displaystyle \lim\limits_{\epsilon \to 0^{+}} \int_{\gamma_{\epsilon}} f(z)\,dz = 2\mathcal{J}(k)(-1 + e^{\pi i}) = -2\pi i \text{Res}_{z = \infty} \left(f(z)\right)$$
Hence, $$\begin{align*} \mathcal{J}(k) = \frac{1}{2}\pi i \text{Res}_{z = \infty} \left(e^{-i\pi/2}z^{k-1}\left(1 - \frac{1}{z}\right)^{-1/2}\left(1 - \frac{4}{z}\right)^{-1/2}\right) &= \frac{\pi}{2} \text{Res}_{z = 0} \left(z^{-(k+1)}\left(1 - z\right)^{-1/2}\left(1 - 4z\right)^{-1/2}\right) \\&= \frac{\pi}{2} \sum_{j=0}^{k} \binom{2j}{j}\binom{2(k-j)}{k-j}\frac{1}{4^{j}} \end{align*}$$


Top
Offline Profile  
Reply with quote  

Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 2 posts ] 

All times are UTC [ DST ]


Mathimatikoi Online

Users browsing this forum: No registered users and 1 guest


You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot post attachments in this forum

Search for:
Jump to:  
cron
Powered by phpBB® Forum Software © phpBB Group Color scheme created with Colorize It.
Theme created StylerBB.net