Intereting Posts

Every open set in $\mathbb{R}$ is the union of an at most countable collection of disjoint segments
Some three consecutive numbers sum to at least $32$
Showing that $\displaystyle\lim_{s \to{1+}}{(s-1)\zeta(s)}=1$
Relationship Between Sine as a Series and Sine in Triangles
Show that a proper continuous map from $X$ to locally compact $Y$ is closed
Element of, subset of and empty sets
Proving by induction that $ \sum_{k=0}^n{n \choose k} = 2^n$
Is there a name for the group of complex matrices with unimodular determinant?
Trace of the $n$-th symmetric power of a linear map
Guessing one root of a cubic equation for Hit and Trial
Solution of $A^\top M A=M$ for all $M$ positive-definite
Find length of $CD$ where $\measuredangle BCA=120^\circ$ and $CD$ is the bisector of $\measuredangle BCA$ meeting $AB$ at $D$
category of linear maps
Analyzing limits problem Calculus (tell me where I'm wrong).
suppose $n$ people are in a party and every two of them have exactly one common friends,prove that there is one who is friend to all.

I’m interested in this following definite integral:

$$

\int_0^\infty du \frac{\sin(\beta u)}{1+u^\alpha},

$$

where $\beta>0$ and $\alpha\geq1$. Is there any closed form for this integral? I would be fine if the answer involves special functions, it would just be nice to have the answer in closed form.

- Definite integral, quotient of logarithm and polynomial: $I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$
- Integral identity related with cubic analogue of arithmetic-geometric mean
- Integral $\int_0^{\pi/4}\log \tan x \frac{\cos 2x}{1+\alpha^2\sin^2 2x}dx=-\frac{\pi}{4\alpha}\text{arcsinh}\alpha$
- The value of $\int^{\pi/2}_0 \frac{\log(1+x\sin^2\theta)}{\sin^2\theta}d\theta$
- Evaluating $\int_0^{\pi/4} \ln(\tan x)\ln(\cos x-\sin x)dx=\frac{G\ln 2}{2}$
- Transforming an integral equation using taylor series

I’m also interested in the behavior of this integral in the limit $\beta\gg1$. In particular, I’d like to know if it decays exponentially at large values of $\beta$.

- Find $\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$
- Integral ${\large\int}_0^1\ln(1-x)\ln(1+x)\ln^2x\,dx$
- On Galois groups and $\int_{-\infty}^{\infty} \frac{x^2}{x^{10} - x^9 + 5x^8 - 2x^7 + 16x^6 - 7x^5 + 20x^4 + x^3 + 12x^2 - 3x + 1}\,dx$
- Integral $\int_0^1 \frac{\ln|1-x^a|}{1+x}\, dx$
- Finding $\pi$ factorial
- Let $f:\to\mathbb{R}$ be a continuous function. Calculate $\lim\limits_{c\to 0^+} \int_{ca}^{cb}\frac{f(x)}{x}\,dx$
- Prove that $ \int \limits_a^b f(x) dx$ = $ \int \limits_a^b f(a+b-x) dx$
- Evaluating $\int_0^{\large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $
- Integrating around a dog bone contour
- Change of limits in definite integration

Assuming $\beta >0$, considering $$I_a=\int_0^\infty \frac{\sin(b u)}{1+u^a}\,du$$ it seems that $a=1$ corresponds to a very particular case $$I_1=\text{Ci}(b) \sin (b)-\text{Si}(b) \cos (b)+\frac{1}{2} \pi \cos (b)$$ where appear the sine and cosine integrals. $I_1$ seems to decrease as an hyperbola. Expanding $I_1$ for large values of $b$ up to $O\left(\frac{1}{b^9}\right)$ gives $$I_1\approx \frac{1}{b}-\frac{2}{b^3}+\frac{24}{b^5}-\frac{720}{b^7}+\cdots$$

Fo the other cases, using a CAS, it seems that $I_a$ systematically express in terms of the Meijer G function. For example

$$I_2=\frac{1}{4} \sqrt{\pi } b G_{1,3}^{2,1}\left(\frac{b^2}{4}|

\begin{array}{c}

0 \\

0,0,-\frac{1}{2}

\end{array}

\right)$$

$$I_3=\frac{G_{1,7}^{5,1}\left(\frac{b^6}{46656}|

\begin{array}{c}

\frac{5}{6} \\

\frac{1}{6},\frac{1}{3},\frac{1}{2},\frac{5}{6},\frac{5}{6},0,\frac{2}{3}

\end{array}

\right)}{2 \sqrt{3 \pi }}$$ $$I_4=\frac{1}{2} \sqrt{\frac{\pi }{2}} G_{1,5}^{3,1}\left(\frac{b^4}{256}|

\begin{array}{c}

\frac{3}{4} \\

\frac{1}{4},\frac{3}{4},\frac{3}{4},0,\frac{1}{2}

\end{array}

\right)$$ I have not been able to see any clear pattern.

For $b >1$, these functions start at $0$, go through a maximum value and seem to also decrease as hyperbolas at large values of $b$ (just as Taozi commented).

- Rational Numbers – LCM and HCF
- Show that $\gcd\left(\frac{a^n-b^n}{a-b},a-b\right)=\gcd(n d^{n-1},a-b)$
- Half order derivative of $ {1 \over 1-x }$
- Proof by contradiction: $ S \subseteq \emptyset \rightarrow S= \emptyset $
- Is $\{ \sin n^m \mid n \in \mathbb{N} \}$ dense in $$ for every natural number $m$?
- Image of ring homomorphism $\phi : \mathbb{Z} \to \mathbb{Q}$?
- Minimum set of US coins to count each prime number less than 100
- How do I know when to use “let” and “suppose” in a proof?
- What is the 0-norm?
- Find $f:C\to\mathbb{R}^2$ continuous and bijective but not open, $C\subset\mathbb{R}^2$ is closed
- $S\subseteq V \Rightarrow \text{span}(S)\cong S^{00}$
- Normal space- equivalent condition: $\forall U, V$ – open $: U \cup V = X \ \ \ \exists F \subset U \ , G \subset V$ – closed $: F \cup G = X$
- How does the determinant change with respect to a base change?
- Find $\sum_{k=1}^{\infty}\frac{1}{2^{k+1}-1}$
- The additive groups $\mathbb{R}^n$ for $n\geq 1$ are all isomorphic.