New answers tagged integration
3
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Accepted
Volume of 3-dimensional region
$\newcommand\si\sigma$The expression $\Gamma(\Pi)$ makes no sense, and hence the definition of $F$ in your post makes no sense. Replacing $\Gamma$ by $\phi$, we get a definition which does makes sense:...
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Integration in the surreal numbers
In a recent article in the Notices of the AMS, Philip Ehrlich briefly describes some progress in this area. Below is a relevant excerpt from the article.
Conway originally expressed doubt that “...
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Is there a formula or algorithm to remove infinitesimal and oscillating parts from an expression while keeping finite and infinite ones?
This isn't an answer, just a long comment.
If this is from an established field, and I'd guess it is, that needs to be part of the question. Not knowing one, I will blindly sally forth because I am ...
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Class of Riemann integrable functions with antiderivative
Just a modest proposal so a comment (but I am not entitled). Every Riemann (even Lebesgue) integrable function, say on the line, can be regarded in a natural way as a distribution (the distributional ...
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Class of Riemann integrable functions with antiderivative
A classical and simple example is given by the function $x \mapsto \sin\frac{1}{x}$ extended by $0\mapsto 0$, on (say) $[-1,1]$:
It is discontinuous at $0$.
It is a derivative (= has an ...
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Definite integral of Bessel function of the first kind times $x^{3/2}$
The integral requires $\nu>-5/2$ for convergence, and then becomes a hypergeometric function:
$$\int_0^a x^{3/2} J_\nu (bx) dx=\frac{2^{1-\nu} a^{\nu+\frac{5}{2}} b^{\nu}}{(2 \nu+5) \Gamma (\nu+1)}\...
- 151k
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References for $\int_{-\infty}^\infty\binom{1}{t}^3\,\mathrm dt=\frac{3}{2}+\frac{6}{\pi^2}$ and related integrals?
comment
$$
\binom{t}{1} = \frac{1}{\Gamma(t+1)\Gamma(2-t)} = \frac{\sin((t+1)\pi)}{t(t-1)\pi}
$$
Using this, Maple does the indefinite integrals in terms of the functions ${\rm Si}$ and ${\rm Ci}$. ...
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Could variable be still function in x and y after performing Reynolds averaging over area
Based on the comment by @CarloBeenakker, Coarse-grain averaging could be performed on a subset of the domain (spatial or temporal) so that the averaged value still varies with the independent ...
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