Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [catalans-constant]

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

In mathematics, Catalan's constant $G$, is defined by $$G = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^2} = \frac{1}{1^2} - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2} + \frac{1}{9^2} - \cdots.$$ Its numerical value is approximately $0.915965594177219015054603514932384110774$.

It is not known whether $G$ is irrational, let alone transcendental. Catalan's constant was named after Eugène Charles Catalan. It has many relations to special series, integrals and well-known functions.

The constant also appears in combinatorics and low-dimensional topology.

105 questions
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A double series yielding Riemann's $\zeta$

Can you give me some hints to prove equality: $$\sum_{m,n=1}^{\infty} \frac1{(m^2+n^2)^2} =\zeta (2)\ G-\zeta(4)=\frac{\pi^2}{6}\ G-\frac{\pi^4}{90}$$ where $\zeta (t):= \sum\limits_{n=1}^{+\infty} \frac{1}{n^t}$ is the Riemann zeta function and $G…
gugo82
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26
votes
4 answers

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 bounds $(0,\pi /4)$ the substitution takes the…
Kisaragi Ayami
  • 783
26
votes
2 answers

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 said post was by Quanto, who posted this…
Robert Lee
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23
votes
2 answers

Prove the integral evaluates to $\frac{K}{\pi}$

Yesterday I received the following integral that might require some tedious steps to do $$\int_0^{\infty}{\small\left[ \frac{x}{\log^2\left(e^{\large x^2}-1\right)}- \frac{x}{\sqrt{e^{\large x^2}-1}\log^2\left(e^{\large…
user 1591719
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23
votes
10 answers

Relationship between Catalan's constant and $\pi$

How related are $G$ (Catalan's constant) and $\pi$? I seem to encounter $G$ a lot when computing definite integrals involving logarithms and trig functions. Example: It is well known that $$G=\int_0^{\pi/4}\log\cot x\,\mathrm{d}x$$ So we see that…
clathratus
  • 18,002
17
votes
3 answers

Evaluate $\int_{0}^{\frac{\pi}{4}}\ln(\cos t)dt$

$$\int_{0}^{\frac{\pi}{4}}\ln(\cos t)dt=-\frac{{\pi}}{4}\ln2+\frac{1}{2}K$$ I ran across this integral while investigating the Catalan constant. I am wondering how it is evaluated. I know of this famous integral when the limits of integration are…
Cody
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4 answers

Integral $I=\int_0^\infty \frac{\ln(1+x)\ln(1+x^{-2})}{x} dx$

Hi I am stuck on showing that $$ \int_0^\infty \frac{\ln(1+x)\ln(1+x^{-2})}{x} dx=\pi G-\frac{3\zeta(3)}{8} $$ where G is the Catalan constant and $\zeta(3)$ is the Riemann zeta function. Explictly they are given by $$ G=\beta(2)=\sum_{n=0}^\infty…
Jeff Faraci
  • 10,393
14
votes
1 answer

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}{2}\right)+\frac{1}{n}\right)\sin(2nx)\,\mathrm{d}x…
vitamin d
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14
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3 answers

Catalan's constant and $\int_{0}^{2 \pi} \int_{0}^{2 \pi} \ln(\cos^{2} \theta + \cos^{2} \phi) ~d \theta~ d \phi$

According to my book (The Nature of computation, page 691): $$\int_{0}^{2 \pi} \int_{0}^{2 \pi} \ln(\cos^{2} \theta + \cos^{2} \phi) ~d \theta ~d \phi= 16 \pi^2 \left(\frac{C}{\pi}- \frac{\ln2}{2}\right),$$ where $C$ is Catalan's constant. I have…
Mencia
  • 273
13
votes
1 answer

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_0^1 \frac{\ln\left(\frac{2x}{x^2+1}\right)\ln…
Faoler
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  • 6
  • 18
13
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5 answers

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 Catalan constant, following some heuristic effort. But, I…
Quanto
  • 120,125
13
votes
4 answers

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}…
No-one Important
  • 1,246
13
votes
6 answers

Evaluate $\int_{-\pi/4}^{\pi/4}\frac{x}{\sin x}\mathrm{d}x$

I am working on the integral $$I=\int_{-\pi/4}^{\pi/4}\frac{x}{\sin x}\mathrm{d}x=2\int_0^{\pi/4}\frac{x}{\sin x}\mathrm{d}x$$ Which I am fairly confident has a closed form, as $$\int_{0}^{\pi/2}\frac{x}{\sin x}\mathrm{d}x=2G$$ Where $G$ is…
clathratus
  • 18,002
12
votes
2 answers

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}\zeta(3)\ln 2+\frac{1}{24}\ln^4 2+\pi…
FDP
  • 15,643
12
votes
1 answer

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 constant. These identities should be nontrivial (that…
clathratus
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