Timeline for Is there a solution to $\int_{\theta-1}^{x} \left(\frac{y+1-\theta}{\theta} \right)^{-\frac{1}{\epsilon+3}} y^{-\frac{\epsilon+4}{\epsilon+3}}dy$?
Current License: CC BY-SA 4.0
23 events
| when toggle format | what | by | license | comment | |
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| Dec 31, 2021 at 12:02 | vote | accept | Felipe Augusto de Figueiredo | ||
| Dec 29, 2021 at 20:28 | comment | added | Felipe Augusto de Figueiredo | Dear @Zachary, could you elaborate on your solution to the integral, please? What are the steps involved in finding it? | |
| Dec 29, 2021 at 19:24 | comment | added | Dispersion | The expansion does work, although perhaps not if $x/\alpha$ is too large (remember that $\alpha=\theta-1$). As I stated, the expansion is valid for $|x-\alpha|<\alpha$, i.e for $x\in(0, 2\alpha)$, so even if $x>\alpha$, the series converges as long as $x\in(\alpha, 2\alpha)$. This result is of course derived within a certain limit, so it won't be valid for all $x$ such that $x\ge \alpha$; asking to know the exact behavior of your integral not in any limit but for any values of your parameters is not very reasonable given your integral. | |
| Dec 29, 2021 at 13:45 | answer | added | Carlo Beenakker | timeline score: 5 | |
| Dec 29, 2021 at 11:42 | history | edited | Felipe Augusto de Figueiredo | CC BY-SA 4.0 |
added 12 characters in body
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| Dec 29, 2021 at 11:41 | comment | added | Felipe Augusto de Figueiredo | @Zachary, OK, thanks. The only problem is that $x > \theta -1$. So I don't think the expansion will work. | |
| Dec 29, 2021 at 2:56 | comment | added | Dispersion | This is a solution to the integral (I didn't include the factor of $\theta^{f(\epsilon)}$ that can be taken out of the integral in your post). | |
| Dec 29, 2021 at 1:34 | comment | added | Felipe Augusto de Figueiredo | Dear @Zachary, is that a solution to the integral or just an expansion to its integrand? | |
| S Dec 28, 2021 at 19:18 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
made the exponents larger to be readable + pushed the integration limits above the top and below the bottom of the integral sign.
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| S Dec 28, 2021 at 19:18 | history | suggested | username | CC BY-SA 4.0 |
made the exponents larger to be readable
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| Dec 28, 2021 at 19:15 | comment | added | Dispersion | Letting $\alpha=\theta-1$, if $|x-\alpha|<\alpha$ (i.e. if $x/\alpha$ is within the unit-ball centered at $x=1$), then you have a binomial series expansion: $$\alpha^{-1-f(\epsilon)}\sum_{k\ge 0}(-1)^k \alpha^{-k} {k+f(\epsilon)\choose k} \frac{(x-\alpha)^{k-f(\epsilon)+1}}{k-f(\epsilon)+1},$$ where $f(\epsilon)=\frac{1}{3+\epsilon}$. | |
| Dec 28, 2021 at 18:58 | comment | added | Carlo Beenakker | the "solution" can be written in terms of a hypergeometric function, in view of the indefinite integral $$\int y^a(y+b)^c\,dy=(a+1)^{-1}y^{a+1} (b+y)^c \left(y/b+1\right)^{-c} \, _2F_1\left(a+1,-c;a+2;-y/b\right)$$ no idea if this useful for you... | |
| Dec 28, 2021 at 18:51 | comment | added | username | Is $x>\theta -1$ ? A closed form solution means not very much in general: it is much more useful to know how to compute it numerically for practical purposes, or study its dependance on various elements. So what do you really want to know : how to evaluate it efficiently (for which range of parameters) or to know its behavior for $\theta$ close to $x+1$, or near $1$, or for $x$ large...? | |
| Dec 28, 2021 at 18:45 | review | Suggested edits | |||
| S Dec 28, 2021 at 19:18 | |||||
| S Dec 28, 2021 at 18:30 | history | suggested | Dispersion | CC BY-SA 4.0 |
Fixed LaTeX.
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| Dec 28, 2021 at 18:29 | review | Suggested edits | |||
| S Dec 28, 2021 at 18:30 | |||||
| Dec 28, 2021 at 18:29 | comment | added | Felipe Augusto de Figueiredo | @AlessandroDellaCorte, thanks for pointing that out, that was wrong. | |
| Dec 28, 2021 at 18:28 | history | edited | Felipe Augusto de Figueiredo | CC BY-SA 4.0 |
added 3 characters in body
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| Dec 28, 2021 at 18:23 | comment | added | Alessandro Della Corte | $\epsilon=1$? You really mean that? So why you write like that? | |
| Dec 28, 2021 at 18:15 | comment | added | Felipe Augusto de Figueiredo | @Zachary, thanks for your comment. I've just added some more information. | |
| Dec 28, 2021 at 18:14 | history | edited | Felipe Augusto de Figueiredo | CC BY-SA 4.0 |
added 217 characters in body
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| Dec 28, 2021 at 18:05 | comment | added | Dispersion | What is the range of $x, \theta, \epsilon$? It would also be helpful to know the context in which this integral came up, and if estimates are useful if a closed-form solution isn't possible. | |
| Dec 28, 2021 at 17:24 | history | asked | Felipe Augusto de Figueiredo | CC BY-SA 4.0 |