Integration Challenge!

Find the general form for the following integral -

\[\displaystyle \large \int\limits_{0}^{\infty}{\frac{{x}^{s}}{a{x}^{2} + bx + c} dx}\]

for constants $a, b, c, s$ such that the above integral converges. Find the condition for convergence as well.

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Master is here as many of you might have guessed.

First we will notice some facts about Chebyshev Polynomials of 2nd kind.

$\displaystyle {U}_{n}(a) = \frac{sin((n+1){cos}^{-1}(a))}{sin({cos}^{-1}(a))}$

And hence it can be seen that

$\displaystyle \sum_{n=0}^{\infty}{{U}_{n}(a) {(-x)}^{n}} = \frac{1}{1 + 2ax + {x}^{2}}$

$\displaystyle \sum_{n=0}^{\infty}{{U}_{n}(a) \Gamma(n+1) \frac{{(-x)}^{n}}{n!}} = \frac{1}{1 + 2ax + {x}^{2}} ------\boxed{1}$

Well, that may be all of the non-English we require, I guess. Back to the integral,

$\displaystyle \int_{0}^{\infty}{\frac{{x}^{s}}{a{x}^{2} + bx + c} dx} = \frac{1}{c} \int_{0}^{\infty}{\frac{{x}^{s}}{\frac{a}{c}{x}^{2} + \frac{b}{c}x + 1} dx}$

$\displaystyle x \rightarrow \sqrt{\frac{a}{c}} x$

$\displaystyle \frac{1}{c} {\left(\frac{c}{a}\right)}^{\frac{s+1}{2}} \int_{0}^{\infty}{\frac{{x}^{s}}{{x}^{2} + \frac{b}{\sqrt{ac}}x + 1} dx}$

Now, we can easily use Ramanujan Master Theorem(considering $\boxed{1}$ ),

$\displaystyle = \frac{1}{c} {\left(\frac{c}{a}\right)}^{\frac{s+1}{2}} \Gamma(s+1)\Gamma(1-s-1) {U}_{-s-1}\left(\frac{b}{2\sqrt{ac}}\right)$

$\displaystyle = - \frac{1}{c} {\left(\frac{c}{a}\right)}^{\frac{s+1}{2}} \frac{\pi}{sin((s+1)\pi)} \frac{sin\left(s {cos}^{-1}\left(\frac{b}{2\sqrt{ac}}\right)\right)}{\sqrt{1 - \frac{{b}^{2}}{4ac}}}$

Kartik Sharma - 5 years, 10 months ago

I see that your last formula implies the divergence if the integral at $s=0$ . How is that?

Hasan Kassim - 5 years, 9 months ago

Oh sorrry, I didn't see the $s$ behind the $\cos ^{-1}$

Hasan Kassim - 5 years, 9 months ago

I will post the answer on $17$ th August.

Kartik Sharma - 5 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$