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Results tagged with integration
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user 82588
Questions related to various forms of integration including the Riemann integral, Lebesgue integral, Riemann–Stieltjes integral, double integrals, line integrals, contour integrals, surface integrals, integrals of differential forms, ...
17
votes
Interesting integral
This integral is due to Lobachevskii. He gave it in more general form as follows
which can be found in his work "Application of imaginary geometry to certain integrals" (1836). Also see equation 3.84 …
- 5,624
7
votes
Accepted
Number theoretic interpretation of the integral $\int_{-\infty}^\infty\frac{dx}{\left(e^x+e^...
The following formula gives a parametric extension of $(1)$ for $|a|$ sufficiently small
\begin{align}
\small\int_{-\infty}^\infty\frac{dx}{\left(e^x+e^{-x}+e^{a+ix\sqrt{3}}\right)^2}+e^a\int_{-\infty …
- 5,624
23
votes
1
answer
1k
views
Number theoretic interpretation of the integral $\int_{-\infty}^\infty\frac{dx}{\left(e^x+e^...
Is there any explanation based on algebraic number theory that the integral
$$
\int_{-\infty}^\infty\frac{dx}{\left(e^x+e^{-x}+e^{ix\sqrt{3}}\right)^2}=\frac{1}{3}\tag{1}
$$
has a closed form? Analyt …
- 5,624
4
votes
Accepted
Approximating a finite sum with an integral
First, we rewrite the sum as a sum over the full period
$$
S(a,N)=\frac{2}{N+1}\sum_{j=1}^{N+1} \sin^2\left( \frac{2\pi j}{N+1} \right)\sin \left( 2 a \cos \left( \frac{2\pi j}{N+1} \right) \right).
…
- 5,624
11
votes
"sinc-ing" integral
A more general result is due to C. Störmer (Acta Mathematica December 1895, Volume 19, Issue 1, pp 341–350)
- 5,624
7
votes
Accepted
Integral of power of binomials equal to sum of power of binomials?
The generalization looks like this
$$
\int_{-\infty}^{\infty} \binom{n}{\alpha x}^l dx =\sum_{k=-\infty}^\infty\binom{n}{\alpha k}^l,\quad 0<\alpha\le 2/l,~l\in\mathbb{N}\tag{1}
$$
where $n$ need not …
- 5,624
7
votes
Accepted
Identity involving an improper integral (with geometric application)
Since the main contribution to the integral comes from $t<<1$, analytically one has
\begin{align}
\lim_{c\to 0^+}\int_c^{\pi/2}\frac{c}{t}\sqrt\frac{1+t^2}{t^2-c^2}dt&=\lim_{c\to 0^+}\int_c^{\pi/2}\fr …
- 5,624
24
votes
Is there a transformation or a proof for these integrals?
UPDATE
The integral in T. Amdeberhan's question was taken from his own paper that was written 10 years earlier before he posted his question: https://arxiv.org/abs/0808.2692 A dozen integrals: Russell …
- 5,624