Question

Is the reciprocal of the inverse tangent function, $\frac1{\arctan x}$, a (logarithmically) completely monotonic function on $(0,\infty)$?

score 474 · Answer 1 · 2026-04-12 11:56:39

474

Views

Is the reciprocal of the inverse tangent function, $\frac1{\arctan x}$, a (logarithmically) completely monotonic function on $(0,\infty)$?

Published on 12 Apr 2026 - 11:56

score 332 · Answer 2 · 2026-04-16 09:11:35

332

Views

Prove the roots of these exponential functions are integers?

Published on 16 Apr 2026 - 9:11

score 50 · Answer 3 · 2026-04-13 00:20:18

50

Views

Ask for a proof of an inequality bounding the ratio of two consecutive geometric means of the first finitely many natural numbers

Published on 13 Apr 2026 - 0:20

score 154 · Answer 4 · 2026-04-16 20:43:47

154

Views

Partial Derivative of Incomplete Gamma Function

Published on 16 Apr 2026 - 20:43

score 105 · Answer 5 · 2026-04-13 10:47:51

105

Views

Range reduction for the exponential integral?

Published on 13 Apr 2026 - 10:47

score 333 · Answer 6 · 2026-04-13 07:20:14

333

Views

Is this up bound of Bessel function right?

Published on 13 Apr 2026 - 7:20

score 55 · Answer 7 · 2026-04-17 06:34:09

55

Views

Generalizing infinite q - Pochammer symbols to more q symbols

Published on 17 Apr 2026 - 6:34

score 324 · Answer 8 · 2026-04-12 00:03:13

324

Views

Does $\mathrm{\sum\limits_{-\infty}^\infty Ai(x)=1}$? Also on $\sum\limits_{-\infty}^0 \mathrm{Bi(x)}$.

Published on 12 Apr 2026 - 0:03

score 465 · Answer 9 · 2026-04-15 08:07:08

465

Views

Evaluation of ${\sum\limits_{\Bbb N} (\text {ker}(x)+i\text{kei(x)})=\sum\limits_1^\infty \text K_0\left(\sqrt ix\right)= 0.133691… - 0.7256312… i}$?

Published on 15 Apr 2026 - 8:07

score 515 · Answer 10 · 2026-04-14 08:32:33

515

Views

How to evaluate $\sum\limits_{x=0}^\infty \text{erfc}(x)= 1.1619990479471263635323…?$

Published on 14 Apr 2026 - 8:32

score 120 · Answer 11 · 2026-03-25 20:36:05

120

Views

Ask for a proof of logarithmically complete monotonicity of a power-exponential function involving the difference of the psi and logarithmic functions

Published on 25 Mar 2026 - 20:36

score 120 · Answer 12 · 2026-04-17 13:29:57

120

Views

A lower bound for the Gamma function : $\Gamma(x)\geq (f(1-x))^x$ on $1\leq x \leq 2$

Published on 17 Apr 2026 - 13:29

score 59 · Answer 13 · 2026-04-15 20:30:49

59

Views

Closed-form formula for $u(t):= \mathbb E[h(X_1)h(tX_1+(1-t^2)^{1/2} X_2)]$, where $(X_1,X_2,\ldots,X_d)$ is uniform on sphere and $h(x):=(x+c)^k$

Published on 15 Apr 2026 - 20:30

score 123 · Answer 14 · 2026-04-15 20:22:14

123

Views

Integral incomplete gamma:$\int_0^\infty x^{k-1} \frac{\gamma(x,t)}{\Gamma(x)} dx$, $k\in \{1,2,3...\}$ and $t>0$.

Published on 15 Apr 2026 - 20:22

score 145 · Answer 15 · 2026-04-11 16:10:10

145

Views

Are there extensions of Euler's infinite product for sine function?

Published on 11 Apr 2026 - 16:10

List Question

Is the reciprocal of the inverse tangent function, $\frac1{\arctan x}$, a (logarithmically) completely monotonic function on $(0,\infty)$?

Prove the roots of these exponential functions are integers?

Ask for a proof of an inequality bounding the ratio of two consecutive geometric means of the first finitely many natural numbers

Partial Derivative of Incomplete Gamma Function

Range reduction for the exponential integral?

Is this up bound of Bessel function right?

Generalizing infinite q - Pochammer symbols to more q symbols

Does $\mathrm{\sum\limits_{-\infty}^\infty Ai(x)=1}$? Also on $\sum\limits_{-\infty}^0 \mathrm{Bi(x)}$.

Evaluation of ${\sum\limits_{\Bbb N} (\text {ker}(x)+i\text{kei(x)})=\sum\limits_1^\infty \text K_0\left(\sqrt ix\right)= 0.133691… - 0.7256312… i}$?

How to evaluate $\sum\limits_{x=0}^\infty \text{erfc}(x)= 1.1619990479471263635323…?$

Ask for a proof of logarithmically complete monotonicity of a power-exponential function involving the difference of the psi and logarithmic functions

A lower bound for the Gamma function : $\Gamma(x)\geq (f(1-x))^x$ on $1\leq x \leq 2$

Closed-form formula for $u(t):= \mathbb E[h(X_1)h(tX_1+(1-t^2)^{1/2} X_2)]$, where $(X_1,X_2,\ldots,X_d)$ is uniform on sphere and $h(x):=(x+c)^k$

Integral incomplete gamma:$\int_0^\infty x^{k-1} \frac{\gamma(x,t)}{\Gamma(x)} dx$, $k\in \{1,2,3...\}$ and $t>0$.

Are there extensions of Euler's infinite product for sine function?

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