User HGF

1 vote

0 answers

38 views

Genus of binary quadratic forms

asked 2 days ago

1 vote

1 answer

45 views

Proving $\left[a_1,\frac{b_1+\sqrt\Delta}2\right]\cdot\left[a_2,\frac{b_2+\sqrt\Delta}2\right]=\left[a_1a_2,\frac{b+\sqrt\Delta}2\right]$

asked Aug 31 at 4:57

3 votes

1 answer

145 views

$1^\alpha+2^\alpha+3^\alpha+\cdots+n^\alpha$

asked May 26 at 2:21

0 votes

1 answer

62 views

It a rotation matrix in 3D space a product of three basic rotations?

asked Mar 5 at 14:51

4 votes

2 answers

131 views

Let $f(x)\in C^2[0,2]$, prove that there exists a $\xi\in(0,2)$ such that $\int_0^2f(x)dx=2f(1)+\frac{f''(\xi)}{3}$.

asked Dec 14, 2023 at 0:59

0 votes

1 answer

52 views

About Bessel's Book 1830: Tabulae Regiomontanae reductionum observationum astronomicarum ab anno 1750 usque ad annum 1850 computatae [closed]

asked Sep 3, 2023 at 12:57

0 votes

2 answers

89 views

What is the determinant of Bernoulli matrix?

asked Jun 8, 2023 at 22:36

9 votes

1 answer

204 views

$\sum_{n=1}^\infty\ln(1+u_n)$ converges but $\sum_{n=1}^\infty u_n$ diverges

asked May 12, 2023 at 7:53

0 votes

2 answers

128 views

If the second homotopy group is trivial, surface integral is independent with surface

asked Apr 18, 2023 at 0:43

2 votes

1 answer

89 views

Continuous function of multivariable which has directional derivative nowhere

asked Mar 10, 2023 at 15:38

1 vote

1 answer

294 views

On the conditions of the Implicit Function Theorem

asked Feb 23, 2023 at 1:11

1 vote

0 answers

97 views

$\sum\limits_{n=1}^\infty\frac{a_n}{n}$ is convergent implies $\sum\limits_{n=1}^\infty\frac{a_n}{n^{1+it}}$ is convergent

asked Sep 25, 2022 at 6:07

1 vote

1 answer

301 views

The necessary and sufficient conditions for a complex function to have primitive function

asked Apr 15, 2022 at 2:48

1 vote

1 answer

120 views

Uniform convergence about Dirichlet integral $f(s):=\int_1^\infty\frac{a(x)}{x^s}\,dx =\lim\limits_{T\to\infty}\int_1^T\frac{a(x)}{x^s}\,dx$

asked Mar 24, 2022 at 7:25

20 votes

5 answers

784 views

Find the limit $\lim\limits_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}$

asked Mar 19, 2022 at 4:10

9 votes

1 answer

414 views

$x_n=f\left(\frac{1}{n^2}\right)+f\left(\frac{2}{n^2}\right)+\cdots+f\left(\frac{n}{n^2}\right)$, $\lim_{n\to\infty}x_n=\dfrac{f'(0)}{2}$

asked Nov 16, 2021 at 0:16

2 votes

0 answers

57 views

The function space $C^k[a,b]$ and $C^{k+1}[a,b]$

asked Sep 14, 2021 at 1:54

3 votes

2 answers

118 views

Is the function $f(x)=\sqrt[x+1]{\Gamma(x+2)}-\sqrt[x]{\Gamma(x+1)}$ decreasing?

asked Jul 9, 2021 at 4:08

2 votes

1 answer

258 views

Must a continuous function with finitely many strict extrema on a closed interval have bounded variation? And Pointwise convergence of Fourier series

asked Jun 19, 2021 at 13:59

0 votes

0 answers

42 views

A question about the regular condition of line integral

asked Apr 26, 2021 at 7:48

2 votes

1 answer

583 views

A question about Riemann Integral, the function is not integrable, but the limit of Riemann sum exists

asked Apr 25, 2021 at 6:37

0 votes

2 answers

91 views

Double Integrals in Polar Coordinates and the geometric meaning

asked Apr 12, 2021 at 15:38

2 votes

0 answers

86 views

Great Picard's Theorem and infinite number of poles

asked Feb 8, 2021 at 2:10

1 vote

0 answers

77 views

Can we prove that $\int_0^1\int_0^1 \sin(mx)\sin(ny)|y-x|^{\alpha} \,dx\,dy$ is nonzero?

asked Sep 3, 2020 at 4:00

4 votes

1 answer

214 views

Tauberian Theorem and Prime Number Theorem

asked Jul 20, 2020 at 9:01

2 votes

1 answer

67 views

Let $U$ be a subset of $\mathbb{R}^2\setminus\{0\}$, $i_\ast(\pi_1(U))=0$ or $\mathbb{Z}$

asked Mar 24, 2020 at 16:50

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26 Questions

Genus of binary quadratic forms

Proving $\left[a_1,\frac{b_1+\sqrt\Delta}2\right]\cdot\left[a_2,\frac{b_2+\sqrt\Delta}2\right]=\left[a_1a_2,\frac{b+\sqrt\Delta}2\right]$

$1^\alpha+2^\alpha+3^\alpha+\cdots+n^\alpha$

It a rotation matrix in 3D space a product of three basic rotations?

Let $f(x)\in C^2[0,2]$, prove that there exists a $\xi\in(0,2)$ such that $\int_0^2f(x)dx=2f(1)+\frac{f''(\xi)}{3}$.

About Bessel's Book 1830: Tabulae Regiomontanae reductionum observationum astronomicarum ab anno 1750 usque ad annum 1850 computatae [closed]

What is the determinant of Bernoulli matrix?

$\sum_{n=1}^\infty\ln(1+u_n)$ converges but $\sum_{n=1}^\infty u_n$ diverges

If the second homotopy group is trivial, surface integral is independent with surface

Continuous function of multivariable which has directional derivative nowhere

On the conditions of the Implicit Function Theorem

$\sum\limits_{n=1}^\infty\frac{a_n}{n}$ is convergent implies $\sum\limits_{n=1}^\infty\frac{a_n}{n^{1+it}}$ is convergent

The necessary and sufficient conditions for a complex function to have primitive function

Uniform convergence about Dirichlet integral $f(s):=\int_1^\infty\frac{a(x)}{x^s}\,dx =\lim\limits_{T\to\infty}\int_1^T\frac{a(x)}{x^s}\,dx$

Find the limit $\lim\limits_{s\to0^+}\sum_{n=1}^\infty\frac{\sin n}{n^s}$

$x_n=f\left(\frac{1}{n^2}\right)+f\left(\frac{2}{n^2}\right)+\cdots+f\left(\frac{n}{n^2}\right)$, $\lim_{n\to\infty}x_n=\dfrac{f'(0)}{2}$

The function space $C^k[a,b]$ and $C^{k+1}[a,b]$

Is the function $f(x)=\sqrt[x+1]{\Gamma(x+2)}-\sqrt[x]{\Gamma(x+1)}$ decreasing?

Must a continuous function with finitely many strict extrema on a closed interval have bounded variation? And Pointwise convergence of Fourier series

A question about the regular condition of line integral

A question about Riemann Integral, the function is not integrable, but the limit of Riemann sum exists

Double Integrals in Polar Coordinates and the geometric meaning

Great Picard's Theorem and infinite number of poles

Can we prove that $\int_0^1\int_0^1 \sin(mx)\sin(ny)|y-x|^{\alpha} \,dx\,dy$ is nonzero?

Tauberian Theorem and Prime Number Theorem

Let $U$ be a subset of $\mathbb{R}^2\setminus\{0\}$, $i_\ast(\pi_1(U))=0$ or $\mathbb{Z}$