Robert Lee's user avatar
Robert Lee's user avatar
Robert Lee's user avatar
Robert Lee
  • Member for 4 years, 8 months
  • Last seen this week
25 votes
2 answers
1k views

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

16 votes
7 answers
1k views

How to prove $\int_{0}^{\infty} \frac{(1-x^2) \, \text{sech}^2\left(\frac{\pi x}{2} \right)}{(1+x^2)^2}\, dx = \frac{\zeta(3)}{\pi}$?

13 votes
5 answers
382 views

How to show $ \int_{-\infty}^{\infty} \frac{e^{-(x+1)^2}}{1+e^{-x}}\mathrm{d}x = \frac{\left(2\sqrt[4]{e} -1 \right)\sqrt{\pi}}{2e}$?

12 votes
4 answers
370 views

Find matrix $A\in \mathcal{M}_n (\mathbb{N})$ such that $A^k =\left( \sum_{i=1}^{k}10^{i-1} \right)A$.

10 votes
2 answers
1k views

How can I prove that these definitions of curl are equivalent?

7 votes
5 answers
463 views

If $f''(x) \le 0, \, \forall x \neq 0$ and $f$ is minimum at $0$, prove $f'(0)$ doesn't exist.

7 votes
4 answers
738 views

Examples of non-elementary integrals, but whose definite integral IS solvable with power series.

7 votes
1 answer
170 views

Ways to show $\int_0^{\infty}\frac{\sin^2(\pi x)}{x^2}\Big\lvert x-\Big\lfloor x +\frac12 \Big\rfloor \Big\rvert \, \mathrm{d}x = \frac{\pi^2}{8}$?

7 votes
0 answers
103 views

Evaluating $\int_{0}^{\infty} \frac{\mathrm{d}x}{x(I_n(x)^2 + K_n(x)^2)}$ and similar integrals.

6 votes
4 answers
635 views

Examples of non-trivial exclusively irrational integrals?

5 votes
0 answers
136 views

What property is being used to relate $\arctan(x)$ and $\ln(1 + x^2)$ in these complex integrals?

5 votes
2 answers
368 views

How can I solve the differential equation $f'(x) + f\left( x^2\right) =0$?

5 votes
1 answer
293 views

Given $H\subseteq G$ with $G$ a group, then $HH^{-1} =H \implies aH = H \ \forall a \in H$.

5 votes
4 answers
1k views

Prove $\lim_{z \to 0} \frac{z}{\overline{z}}$ doesn't exist using $\varepsilon-\delta$.

5 votes
1 answer
463 views

Prove complex numbers $a$ and $b$ are antipodal under stereographic projection $\iff a \overline{b} = -1$

5 votes
1 answer
9k views

Does there exist a gradient chain rule for this case?

4 votes
2 answers
209 views

When does equality hold in $\Bigr\lvert\sum_{k=1}^n a_kb_k\Bigr\rvert^2 \le \left(\sum_{k=1}^n |a_k|^2\right)\left(\sum_{k=1}^n |b_k|^2\right)$?

4 votes
2 answers
72 views

Is there a closed form for this sequence $a_n = m, \ \binom{m}{2}\le n < \binom{m+1}{2}$?

4 votes
3 answers
196 views

What is the probability of picking a full set from multiset after $m$ draws?

3 votes
3 answers
180 views

Show that every two consecutive rationals in any row of this infinite tree of Calkin-Wilf trees are the endpoints of a "kite".

3 votes
2 answers
214 views

How to evaluate $\int_0^{\pi} e^{i \zeta e^{ ix}} \ dx$?

3 votes
1 answer
294 views

What is the value of $1 -\omega^h + \omega^{2h} -...+(-1)^{n-1} \omega^{(n-1)h}$ when $\omega$ is a root of unity?

3 votes
2 answers
118 views

How to show $\sum_{k=0}^{\infty} \frac{1}{k!} \left( \int_{1}^{x} \frac{1}{t} \ dt \right)^k =x$?

3 votes
2 answers
1k views

Is the Jordan normal form uniquely determined by the characteristic and minimal polynomial?

3 votes
0 answers
32 views

On the distribution of odd/even length symmetric continued fractions.

3 votes
3 answers
146 views

How to evaluate $\,\lim\limits_{x \to 1}\left(x^n-1\right)\ln^m(1-x)\,$ without L'Hopital? [duplicate]

3 votes
1 answer
58 views

How to show that if $P_n^{3-n}< x<(P_{n}+1)^{3-n}$ then $P_n <x^{3^n} < P_{n}+1$?

2 votes
1 answer
144 views

Limit of a strictly increasing sequence bounded by strictly decreasing sequence is sandwiched between the sequences.

2 votes
1 answer
55 views

On the "efficiency" of digit-sum divisibility tricks in other bases (or about the growth rate of the number of divisors function).

2 votes
1 answer
127 views

Can I find an inverse Laplace operator?