All Questions
16 questions
7
votes
2
answers
661
views
For a manual evaluation of a definite integral
I note that Mathematica could yield the identity
$$\int_0^1\frac{\log(1+x^2(x-1)/2)}{x^2(x-1)}dx=\frac{\pi(\pi-4)-12\log^22+24\log2}{16}.\tag{1}\label{1}$$
But I don't know how Mathematica got this.
...
- 1,440
10
votes
1
answer
572
views
Are “most” bounded derivatives not Riemann integrable?
Given $a,b\in\mathbb R$ with $a<b$. Let
$$X=\{f\in C([a,b]): f \text{ is differentiable on } [a,b] \text{ with }f' \text{ bounded }\},$$
and
$$A=\{f\in X: f' \text{ is Riemann integrable}\}. $$
It ...
- 3,312
7
votes
5
answers
1k
views
Generalizations of the Euler–Maclaurin Summation Formula
I'm using the Euler–Maclaurin formula in a research project I'm working on. While brilliant, the elementary proof found in Apostol - An Elementary View of Euler's Summation Formula does not give me ...
- 2,689
4
votes
1
answer
162
views
Definite integral of power of sine ratio
I stumbled on the following rather appealing trigonometric definite integral,
\begin{equation}
\int_0^y \left(\frac{\sin x}{\sin (y-x)}\right)^a \mathrm{d}x = \pi \frac{\sin(ya)}{\sin(\pi a)}
\end{...
- 109k
3
votes
1
answer
268
views
References for $\int_{-\infty}^\infty\binom{1}{t}^3\,\mathrm dt=\frac{3}{2}+\frac{6}{\pi^2}$ and related integrals?
In user dxdydz's answer to the question "Unexpected appearances of $\pi^{2}/6$", the identity $$\int_{-\infty}^\infty\binom{1}{t}^3\,\mathrm dt=\frac{3}{2}+\frac{6}{\pi^2}$$ is mentioned.
...
- 41.1k
6
votes
1
answer
584
views
Integral representation of $\frac{355}{113}-\pi$? [duplicate]
It is well known that
$$
\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2},
$$
Given that $\frac{355}{113}$ is an excellent approximation of $\pi$, is there any known integral representation of ...
- 82.7k
1
vote
1
answer
164
views
Two trigonometric integrals: looking for a transformation
I have two integrals of trigonometric functions and I would like to ask:
QUESTION. Is there a transformation rule (or general principle) to show this equality?
$$\int_0^{\frac{\pi}2}\frac{d\theta}{\...
- 105k
1
vote
0
answers
210
views
Questions about iterating the Euler-Maclaurin summation formula
Introduction
The Euler–Maclaurin summation formula is as follows for a positive integer $p$ and a continuous function $f(\cdot)$ that is $p$ times continuously differentiable on the interval $[m,n]$ : ...
- 4,829
7
votes
2
answers
2k
views
The source of the Integral
Wolfram alpha calculates the integral
$$\int\limits_0^\infty \frac{x^2\ln{x}}{e^x-1}dx=2\zeta^\prime(3)+3\zeta(3)-2\gamma\zeta(3).$$
However, I need to cite the source of this identity (the table of ...
- 1,986
11
votes
1
answer
566
views
Integral representation of product of two Whittaker functions
Does anyone know anything about the following formula involving special functions:
$$\begin{multline*}
W_{\kappa,\mu}(z)W_{\lambda,\mu}(w)=\frac{e^{-(z+w)/2}(zw)^{\mu+1/2}}{\Gamma(1-\kappa-\lambda)}\...
- 47.3k
3
votes
1
answer
183
views
On integral representation of Whittaker $W$ functions
According to NIST, the integral representation of Whittaker $W$ functions
$$
W_{\kappa,\mu}\left(z\right)=\frac{z^{\mu+\frac{1}{2}}2^{-2\mu}}{\Gamma\left(%
\frac{1}{2}+\mu-\kappa\right)}\int_{1}^{\...
- 188k
9
votes
2
answers
1k
views
A tricky integral to evaluate
I came across this integral in some work. So, I would like to ask:
QUESTION. Can you evaluate this integral with proofs?
$$\int_0^1\frac{\log x\cdot\log(x+2)}{x+1}\,dx.$$
- 105k
5
votes
1
answer
4k
views
Rigorous multivariate differentiation of integral with moving boundaries (Leibniz integral rule)
The Leibniz integral rule, in its multivariate form, deals with differentiation of the following sort:
$$ \frac{\partial}{\partial t} \int_{D(t)} F({\bf x}, t) \, d{\bf x} \, , \qquad D(t)\in \mathbb{...
- 3,574
4
votes
0
answers
673
views
Proofs of the second fundamental theorem of calculus
I am referring to the following version of the theorem, in the setting of the Lebesgue integral.
Theorem Let $f: [a,b] \rightarrow \bf R$ be an everywhere differentiable function whose derivative is ...
- 105k
10
votes
1
answer
833
views
This is not a dyadic cosine-product
The double-angle formula, $\sin2x=2\sin x\cos x$, turns the scary-looking integral
$$\int_0^{\infty}dz\prod_{k=1}^{\infty}\cos\frac{z}{2^k}$$
into fun once you realize $\prod_k\cos\frac{z}{2^k}=\frac{\...
- 105k
5
votes
2
answers
454
views
Reference for the fact that the images of the narrow and wide Denjoy integrals are respectively $ACG_\ast$ and $ACG$?
The narrow Denjoy integral (which also goes by the names Henstock-Kurzweil integral, Perron integral, and Lusin integral) is a transfinite integration process defined by Denjoy in the early 20th ...
- 3,312