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Questions tagged [catalans-constant]

For questions about special identities and problems involving Catalan's constant as well as general questions about the constant itself.

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Show that $\int_0^{\frac\pi 2}\frac{\theta-\cos\theta\sin\theta}{2\sin^3\theta}d\theta=\frac{2C+1}4$

While trying to evaluate $\int_0^1 k^2K(k)dk$ related to elliptic integral of the first kind, by integral switching method, I reached the trigonometric integral $$\int_0^{\frac\pi 2}\frac{\theta-\cos\...
Bob Dobbs's user avatar
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1 vote
1 answer
102 views

$\int_0^1E(k)dk$ without switching integrals

Let $E=E(k), K=K(k)$ be the complete elliptic integrals of the second and the first kind respectively. It is well-known that $\frac12\int_0^1K\,dk=G$ is the Catalan constant. We can also find, by ...
Bob Dobbs's user avatar
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3 votes
1 answer
143 views

closed form for $\psi^{(-2)}\left(\frac{1}{4} \right)$

the polygamma function of negative order exist in a lot of complicated integrals like this integral: $$\int_0^{\frac{1}{4}} x \psi(x) dx=\left(x \psi^{(-1)}(x) \right)^{\frac{1}{4}}_0-\int_0^{\frac{1}{...
Faoler's user avatar
  • 1,379
6 votes
2 answers
273 views

$I = \int_0^{\pi/2} \left( \int_0^{\pi/2} \frac{\log(\cos(x/2)) - \log(\cos(y/2))}{\cos(x) - \cos(y)} dx\right) \ dy $

\begin{align*} I = \int_0^{\pi/2} \left( \int_0^{\pi/2} \frac{\log(\cos(x/2)) - \log(\cos(y/2))}{\cos(x) - \cos(y)} dx\right) \ dy \end{align*} What I do so far Let $u = \cos(x/2)$ and $v = \cos(y/2)$....
Martin.s's user avatar
11 votes
0 answers
247 views

Solve the integral $\int_0^1 \frac{\ln^2(x+1)-\ln\left(\frac{2x}{x^2+1}\right)\ln x+\ln^2\left(\frac{x}{x+1}\right)}{x^2+1} dx$

I tried to solve this integral and got it, I showed firstly $$\int_0^1 \frac{\ln^2(x+1)+\ln^2\left(\frac{x}{x+1}\right)}{x^2+1} dx=2\Im\left[\text{Li}_3(1+i) \right] $$ and for other integral $$\int_0^...
Faoler's user avatar
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26 votes
4 answers
806 views

Is there an expression for $I(n)=\int_{0}^{\frac{\pi}{4}}x\tan^{n}x dx$?

I've played around a little with this integral, and I can straightforwardly evaluate it with a substitution $\tan x\mapsto x$ in terms of the Beta function if the bounds were $(0,\pi /2)$. But for the ...
Kisaragi Ayami's user avatar
8 votes
3 answers
246 views

Struggling to figure out why $\int\limits_{0}^{1}\frac{\ln \left(x^{2}\right)}{\left(1+x^{2}\right)^{2}}dx = -G-\frac{\pi}{4}$.

Struggling to figure out why $$\int\limits_{0}^{1}\frac{\ln \left(x^{2}\right)}{\left(1+x^{2}\right)^{2}}dx = -G-\frac{\pi}{4}$$ I've taken note of the exponent in the logarithm, tried substituting $...
Kisaragi Ayami's user avatar
8 votes
2 answers
767 views

Another integral representation of Catalan's Constant?

The integral in question: $$I=\int_{0}^{1} {\frac{\arctan\left(\frac{x-1}{\sqrt{x^2-1}}\right)}{\sqrt{x^2-1}}}dx.$$ So I have a solution and it's rather straightforward, but I don't much like it. I'm ...
Alejandro Jimenez Tellado's user avatar
0 votes
2 answers
104 views

Ways to solve $\int_0^{\frac{\pi}{2}} \frac{x}{\sin{x}}dx$ [duplicate]

$$\int_0^{\frac{\pi}{2}} \frac{x}{\sin{x}}dx$$ I tried using $$\int_0^{\frac{\pi}{2}} \frac{2ix}{e^{ix}-e^{-ix}}dx$$ And then factoring and making use of the Geometric Series $$\int_0^{\frac{\pi}{2}} \...
Anik Patel's user avatar
4 votes
2 answers
204 views

Evaluation of tricky Gamma infinite sum

I want to prove that: $$\sum_{n=0}^\infty\frac{n}{(n+1)^2}\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n+1}{2}\right)}=\frac{4G}{\sqrt{\pi}}+\sqrt{\pi}\log2$$ and $$\sum_{n=0}^\infty(-1)^n\...
Zima's user avatar
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5 votes
0 answers
179 views

Is there a simple way to evaluate $\int_0^1 \arctan x \ln \left(\frac{1-x}{1+x}\right) d x?$

After trying many methods but fail, I try the following substitution. Letting $t=\frac{1-x}{1+x}$ preserves the interval and transforms the integral into $$I=\int_0^1 \arctan x \ln \left(\frac{1-x}{1+...
Lai's user avatar
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5 votes
2 answers
126 views

Calculating a limit of a sequence of integrals that approaches the catalan constant

Let $f\colon[0, 1] \to \mathbb{R}$, $f(x)=\frac{\arctan(x)}{x}, x \in (0, 1], f(0)=1$. Prove that $$\lim_{n \to \infty} n \left(\frac{\pi}{4}-n\int_0^1\frac{x^n}{1+x^{2n}}dx \right)=\int_0^1f(x)dx$$ ...
Shthephathord23's user avatar
3 votes
2 answers
275 views

For $n\ge m\ge 1$, how far can we walk with $ \int_0^{\frac{\pi}{2}} \frac{x^n}{\sin^m x} d x$?

In the post, I tackled the integral by power series and integration by parts and obtained that $$ \int_0^{\frac{\pi}{2}} \frac{x^2}{\sin x} d x=2\pi G-\frac{7}{2}\zeta(3) $$ where $G$ is the Catalan’s ...
Lai's user avatar
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8 votes
3 answers
415 views

Are there any other decent methods to evaluate $\int_0^1 \frac{\ln \left(1-x^4\right)}{1+x^2} d x?$

We first split the integrand into 3 parts as \begin{aligned} \int_0^1 \frac{\ln \left(1-x^4\right)}{1+x^2} d x &= \underbrace{\int_0^1 \frac{\ln \left(1+x^2\right)}{1+x^2} d x}_J+\underbrace{\...
Lai's user avatar
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3 votes
1 answer
145 views

Explicit value of $\operatorname{Li}_2(1/2-{\rm i}/2)$

When you ask Wolfram Alpha about the value of $\operatorname{Li}_{2}\left(1/2-{\rm i}/2\right)$ it gives you $$ \frac{5\pi^{2}}{96} - \frac{\ln^{2}\left(2\right)}{8} + {\rm i}\left[\frac{\pi\ln\left(2\...
Òscar Pérez Massanet's user avatar
0 votes
1 answer
66 views

Contour integral for $f(z)=\frac{1}{2}\frac{e^{2z}}{\cosh(e^z)}$

By substituting $u=e^{e^x}$ in the Catalan's constant C's integral $\int_1^{\infty}\frac{\ln u}{u^2+1}du=C$, I obtained the integral $\int_{-\infty}^{\infty}\frac{1}{2}\frac{e^{2x}}{\cosh(e^x)}$. Let $...
Bob Dobbs's user avatar
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4 votes
3 answers
299 views

Computing $\int_{0}^{\pi} \ln (\sin x+2) d x$ and $\int_{0}^{\pi} \ln (2-\sin x) d x$

I first encountered this integral $$ I=\int_{0}^{\pi} \ln (\sin x+2) d x $$ several months ago without any idea and had tried many methods such as integration by parts, substitution and Fourier series ...
Lai's user avatar
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5 votes
3 answers
532 views

Is it hard to evaluate $\int_{0}^{\infty} \frac{\ln \left(1-x+x^{2}\right)}{1+x^{2}} d x$?

Latest Edit As suggested by @Quanto, $I(a)$ can be utilised to give more examples as below. $$ \boxed{\begin{aligned} I(a)&= \int_{0}^{\infty} \frac{\ln \left(1+2 x \sin a+x^{2}\right)}{1+x^{2}} d ...
Lai's user avatar
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4 votes
2 answers
287 views

Hard integral $\int_{0}^{\frac{\pi}{4}} \ln ^{n}(\tan \theta) d \theta$?

In my post, I had found an elegant integral $$\int_{0}^{\frac{\pi}{4}} \ln (\tan x)~ d x=-G. $$ I then try to generalise the integral $$ I_{n}=\int_{0}^{\frac{\pi}{4}} \ln ^{n}(\tan \theta) d \theta, ~...
Lai's user avatar
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2 votes
3 answers
138 views

Evaluating $\int_{0}^{\infty} \frac{x}{\cosh (a x)} d x$ for a positive constant $a$

We first simplify the integral by a substitution and then convert the integrand into a power series as below: $$ \begin{aligned} \int_{0}^{\infty} \frac{x}{\cosh (ax)} d x & \stackrel{ax\mapsto x}{...
Lai's user avatar
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5 votes
1 answer
111 views

Showing $\lim_{x\rightarrow 1^-} \sum_{k=0}^\infty (-x)^k (2k+1) \ln(2k+1) = \frac{-2C}{\pi}$

While I was playing around with divergent summation, I noticed that the following appears to be true: $$\lim_{x\rightarrow 1^-} \sum_{k=0}^\infty (-x)^k (2k+1) \ln(2k+1) = \frac{-2C}{\pi}$$ where $C = ...
user196574's user avatar
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3 votes
4 answers
224 views

The integral $\int_{0}^{1} \frac{ \log (1-x)}{1+x^2}dx$

Recently a very interesting result $\int_0^1\frac{\ln(1-x)}{1+x^2}\mathrm{d}x+\int_0^1\frac{\arctan x}{x(1+x)}\mathrm{d}x=0$ has been proved in a more than elegant way. See Show that $\int_0^1\frac{\...
Z Ahmed's user avatar
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2 votes
2 answers
127 views

How to tackle $\int_{0}^{\frac{\pi}{2}}y \ln (1+\cos y)\,d y$?

I recently encounter an integral problem consisting the integral $$ I:=\int_{0}^{\frac{\pi}{2}} y \ln (1+\cos y) d y, $$ I tried to tackle $I$ using the double angle formula and the result of $$ \int_{...
Lai's user avatar
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4 votes
1 answer
203 views

Is there any other method to compute $\int_{0}^{\frac{\pi}{4}} y \ln (\cos y) d y$?

After investigating the integral $$ \int_{0}^{\frac{\pi}{2}} y \ (\cos y) d y $$ in the post. I keep on finding the integral with smaller limit $$ I:=\int_{0}^{\frac{\pi}{4}} y \ln (\cos y) d y. $$ As ...
Lai's user avatar
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1 vote
2 answers
128 views

Narrower the limits harder the integral!

When I met the integral $\int_{0}^{\frac{\pi}{2}} \ln (1+\sin x-\cos x) d x$, I evaluated it by squaring $ 1+\sin x -\cos x. $ $ \begin{aligned}\int_{0}^{\frac{\pi}{2}} \ln (1+\sin x-\cos x) d x&=\...
Lai's user avatar
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0 votes
3 answers
163 views

How to find the closed form for the integral $I_n:=\int_{0}^{1} \frac{x^{n} \ln x}{1+x^{2}} d x, {} $ where $ n\in N$?

Inspired by the post, I started to investigate the integral in general, $$ I_n:=\int_{0}^{1} \frac{x^{n} \ln x}{1+x^{2}} d x, $$ where $ n\in N.$ Using the infinite geometric series, $$\frac{1}{1+t}=\...
Lai's user avatar
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3 votes
0 answers
201 views

Estimate: $\pi \to e \to \log(2) \to G$ by sampling uniform distribution

Successively: $\pi \to e \to \log(2) \to G$ were calculated/estimated by sampling uniform distributions. Method: With a normal distribution $\pi$ can be calculated with help of the PDF (probability ...
Vincent Preemen's user avatar
25 votes
2 answers
1k views

How to prove $\int_0^1 \frac{\arctan^2(x)\ln\left(\frac{x}{(1-x)^2}\right)}x \, \mathrm{d}x=G^2$?

A while back I made a post asking for examples of integrals which evaluated to famous irrational constants (or constants that were very likely irrational but yet unproven to be). The top answer in ...
Robert Lee's user avatar
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2 votes
1 answer
217 views

Can this formula for $\zeta(3)$ be proven or simplified further?

This question is related to the equivalence of formulas (1) and (2) below where formula (1) is from a post on the Harmonic Series Facebook group and formula (2) is based on evaluation of the integral ...
Steven Clark's user avatar
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8 votes
2 answers
492 views

Finding $\int_{1}^{\infty} \frac{1}{1+x^2} \frac{\operatorname{Li}_2\left ( \frac{1-x}{2} \right ) }{\pi^2+\ln^2\left(\frac{x-1}{2}\right)}\text{d}x$

Prove the integral $$\int_{1}^{\infty} \frac{1}{1+x^2} \frac{\operatorname{Li}_2\left ( \frac{1-x}{2} \right ) }{ \pi^2+\ln^2\left ( \frac{x-1}{2} \right ) }\text{d}x =\frac{96C\ln2+7\pi^3}{12(\pi^2+...
Setness Ramesory's user avatar
2 votes
1 answer
109 views

How many ways are there to prove that $ \int_{0}^{\frac{\pi}{2}} \frac{x}{\sin x} d x=2 G?$

In my post, I had found the values of the couple of integrals, $$ \int_{0}^{\frac{\pi}{4}} x \tan x d x=-\frac{\pi}{8} \ln 2+\frac{G}{2}\tag*{(1)} $$ and $$ \int_{0}^{\frac{\pi}{4}} x \cot x d x=\...
Lai's user avatar
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1 vote
1 answer
119 views

How do we prove that $\int_{0}^{\frac{\pi}{4}} x\cot xd x= \frac{\pi}{8} \ln 2+\frac{G}{2},$ where G is the Catalan’s constant?

After evaluating the integral in my post,$$ \int_{0}^{\frac{\pi}{4}} x \tan x d x =-\frac{\pi}{8} \ln 2+\frac{G}{2}, $$ then I want to know more about the value of its “partner” integral $$\...
Lai's user avatar
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6 votes
3 answers
313 views

Integral $\int_0^{\pi/3}\frac{x}{\cos(x)}\,dx$

I am trying to compute the integral $$\int_0^{\pi/3}\frac{x}{\cos(x)}\,dx \tag{1}$$ Context: Originally I was trying to prove the following result: $$\sum_{n=0}^\infty\frac{1}{(2n+1)^2\binom{2n}{n}}=\...
Ricardo770's user avatar
  • 2,801
4 votes
2 answers
246 views

How many ways to deal with the integral $\int_{0}^{\frac{\pi}{4}} \ln (\tan x) d x$ and $\int_{0}^{\frac{\pi}{4}} \ln (\sin x) d x$?

After finding that $\displaystyle \int_{0}^{\frac{\pi}{2}} \ln (\tan x) d x =0$ in my post, I was curious about the value of the integral with different upper limit $\dfrac{\pi}{4} $. The answer is ...
Lai's user avatar
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4 votes
2 answers
224 views

How do I evaluate $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}H_n}{2n+1}?$

So, I am coming from this question. I managed to bring the answer to this series: $$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}H_n}{2n+1}$$ where $H_n$ is the $n$th harmonic number. However, as seen in the ...
MandelBroccoli's user avatar
11 votes
3 answers
843 views

Finding $\int_0^{\frac{\pi}{2}} \frac{x}{\sin x} dx $

Is there a way to show $$\int_0^{\frac{\pi}{2}} \frac{x}{\sin x} dx = 2C$$ where $C=\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^2} $ is Catalan’s constant, preferably without using complex analysis? The ...
Vishu's user avatar
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3 votes
0 answers
126 views

Question related to potential closed-form representation of Catalan's constant

The motivation for this question is to find a closed-form representation for Catalan's constant. Formula (1) below for the Dirichlet beta function $\beta(s)$ (which I believe is globally convergent) ...
Steven Clark's user avatar
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6 votes
2 answers
827 views

Catalan constant's integral representation

Catalan constant is known to have a rich source of integral identities, here is the formula I found: $$ \int_0^\infty \frac{\sin^{-1}(\sin(x))}{x} \,dx \ =2G.$$ This can be proved by analyzing the ...
Quý Nhân Đặng Hoàng's user avatar
0 votes
0 answers
93 views

Question on Difference Root representations of $\eta(2 n+1)$ and $\beta(2n)$ where $n\in\mathbb{N}$

I've noticed the Dirichlet eta function $\eta(s)$ and the Dirichlet beta function $\beta(s)$ can be represented by difference roots at odd and even positive integers respectfully. For example, ...
Steven Clark's user avatar
  • 7,621
12 votes
2 answers
552 views

Computation of $\int_0^1 \frac{\arctan^2 x\ln x}{1+x}dx$

I'm searching for a "simple" proof of: \begin{align}\int_0^1 \frac{\arctan^2 x\ln x}{1+x}dx=-\frac{233}{5760}\pi^4-\frac{5}{48}\pi^2\ln ^2 2+\text{Li}_4\left(\frac{1}{2}\right)+\frac{7}{16}\...
FDP's user avatar
  • 13.9k
4 votes
1 answer
270 views

How do I solve this ominous integral?

Let $ n\ge 1 $ be a positive integer. How do I prove the generalization: $$ \int_0^1\frac{\arctan(x)\log^{2n}(x)}{1+x} \, dx =\frac{\pi}{4}\left(1-2^{-2 n}\right) \zeta(2 n+1)(2 n)!+\frac{1}{2} \beta(...
QLimbo's user avatar
  • 2,344
3 votes
4 answers
443 views

Compute $\int_{0}^{1}\frac{\ln(1+x)\ln(1+x^2)}{x}\,dx=\frac{\pi}{2}G-\frac{33}{32}\zeta(3)$

I saw the following result and I am trying to prove it. $G$ is Catalan´s constant. $$\boxed{\int_{0}^{1}\frac{\ln(1+x)\ln(1+x^2)}{x}\,dx=\frac{\pi}{2}G-\frac{33}{32}\zeta(3)}$$ I could not figure out ...
Ricardo770's user avatar
  • 2,801
13 votes
1 answer
379 views

Is there an integral for $\frac{\pi}{\mathrm{G}}$?

I would like to find an integral of the form $$\int_a^bf(x)dx=\frac{\pi}{\mathrm G},$$ or at least an infinite series $$\sum_{n\ge k}a_n=\frac{\pi}{\mathrm G},$$ where $\mathrm G$ is Catalan's ...
clathratus's user avatar
  • 17.3k
14 votes
1 answer
591 views

Beautiful monster: Catalan's constant and the Digamma function

The problem I have been trying for a while now to show that this monster $$\begin{align} &\int_0^{\pi/4}\tan(x)\sum_{n=1}^{\infty}(-1)^{n-1}\left(\psi\left(\frac{n}{2}\right)-\psi\left(\frac{n+1}{...
vitamin d's user avatar
  • 5,793
11 votes
5 answers
498 views

Prove $ \int_{5\pi/36}^{7\pi/36} \ln (\cot t )dt +\int_{\pi/36}^{3\pi/36} \ln (\cot t )dt = \frac49G $

I have the conjecture for the integral $$ \int_{\frac{5\pi}{36}}^{\frac{7\pi}{36}} \ln (\cot t )\>dt +\int_{\frac{\pi}{36}}^{\frac{3\pi}{36}} \ln (\cot t )\>dt = \frac49G $$ where $G$ is the ...
Quanto's user avatar
  • 99.4k
13 votes
4 answers
478 views

How to Evaluate $ \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \sum_{k=1}^{n}\frac{1}{4k-1} $

How can I evaluate $$ \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \sum_{k=1}^{n}\frac{1}{4k-1} \approx - 0.198909 $$ The Sum can be given also as $$ \frac{1}{2} \int_{0}^{1} \frac{1}{(x+1)\sqrt[4]{(-x)^{3}}}\...
No-one Important's user avatar
2 votes
1 answer
267 views

Why does this proof fail? Catalan Constant's Exact Value.

A failed attempt at finding an exact value for Catalan's Constant "C" Definition : $$ C+ \int_{\frac{\pi}{2}}^{\pi} \frac{x \sin{(x)}}{\sqrt{1+\cos{(x)}^2}} = \pi \ln(1+\sqrt{2}) $$ Let $$I(...
No-one Important's user avatar
1 vote
1 answer
153 views

Evaluate $\int_0^1 \frac{\ln{\left(ax^2+2bx+a\right)}}{x^2+1} \; \mathrm{d}x, \;a>b>0$

Evaluate $$\int_0^1 \frac{\ln{\left(ax^2+2bx+a\right)}}{x^2+1} \; \mathrm{d}x, \;a>b>0$$ I tried- $$=\frac{\pi \ln{a}}{4}+\int_0^1 \frac{\ln{\left(x^2+2tx+1\right)}}{x^2+1} \; \mathrm{d}x$$ ...
user avatar
3 votes
2 answers
225 views

integral arising in statistical mechanics

The following integral arises in statistical mechanics of lattice models: $ \displaystyle I = \int_{0}^{\pi/2} \ln\left(\, 1 + \sqrt{\, 1 - a^{2}\sin^{2}\left(\phi\right)\,}\,\right)\, \mathrm{d} \phi\...
sangara's user avatar
  • 69
8 votes
1 answer
208 views

A double sum identity for Catalan constant

While playing around with the problems here 1 and here 2, I came to found the double sum identity for Catalan constant, G by making few change in those problems. ie $$ G= \sum_{m=1}^{\infty}\sum_{n=...
Naren's user avatar
  • 3,442