Question

Closed form of $\frac1s\int_0^\infty e^{-s(\tanh(t)+at)}dt=\frac1s\int_0^1e^{-st}(1-t)^{\frac{as}2-1}(1+t)^{-\frac{as}2-1}dt$ to invert $\tanh(x)+ax$?

score 232 · Answer 1 · 2023-07-09 02:11:02

232

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Closed form of $\frac1s\int_0^\infty e^{-s(\tanh(t)+at)}dt=\frac1s\int_0^1e^{-st}(1-t)^{\frac{as}2-1}(1+t)^{-\frac{as}2-1}dt$ to invert $\tanh(x)+ax$?

Published on 09 Jul 2023 - 2:11

score 58 · Answer 2 · 2023-07-10 15:53:32

58

Views

Square root function hyperbolic estimator

Published on 10 Jul 2023 - 15:53

score 45 · Answer 3 · 2023-07-11 10:36:31

45

Views

Expression for real part of ArcTanh(x) for x > 1

Published on 11 Jul 2023 - 10:36

score 24 · Answer 4 · 2023-07-11 13:48:02

24

Views

Value of $t$ for a point on the hyperbola with given parametric equation $P(t)$

Published on 11 Jul 2023 - 13:48

score 116 · Answer 5 · 2023-07-15 11:44:01

116

Views

Intergrate $\text{arccosh}(\sqrt{x^2 + 1})$ via hyperbolic substituation.

Published on 15 Jul 2023 - 11:44

score 66 · Answer 6 · 2023-07-15 15:20:00

66

Views

A particular case of the Jacobi's triple product with an inequality and hyperbolic/trigonometric function

Published on 15 Jul 2023 - 15:20

score 24 · Answer 7 · 2023-07-17 19:31:58

24

Views

Showing that $\sqrt{\frac{a}{k}-1}\cosh(\frac12(z-ax))\tan(\frac12z\sqrt{\frac{a}{k}-1})$, with conditions, is increasing in $z$

Published on 17 Jul 2023 - 19:31

score 36 · Answer 8 · 2023-07-18 15:37:15

36

Views

How do I rotate a 2D catenary in 3D space?

Published on 18 Jul 2023 - 15:37

score 81 · Answer 9 · 2023-07-21 01:40:29

81

Views

Why do these answer differ when integrating the square root of $\tan x$ using complex numbers?

Published on 21 Jul 2023 - 1:40

score 68 · Answer 10 · 2026-03-27 05:38:14

68

Views

Demonstration of hyperbola in polar coordinates

Published on 27 Mar 2026 - 5:38

score 375 · Answer 11 · 2023-08-05 05:43:44

375

Views

Hyper-closed-forms of series involving hyperbolic functions:$\sum_{n=1}^{\infty}\frac1{n^2\cosh(\pi n)^2},\frac{\tanh(\pi n)}{n^3}$ and the like.

Published on 05 Aug 2023 - 5:43

score 113 · Answer 12 · 2026-02-23 06:23:15

113

Views

Alternate expression for $\operatorname{atanh}(x)$ for $\lvert x \rvert \to 1$

Published on 23 Feb 2026 - 6:23

score 93 · Answer 13 · 2023-08-19 10:47:34

How taut must a stretchable, horizontally-oriented string be in order for a straight line to approximate the string to within a given margin of error?

score 129 · Answer 14 · 2023-08-25 08:25:04

129

Views

Compact form of solution of $\displaystyle\int_0^{\frac{\pi}{2}}\ln\left(1+\alpha^n\sin(x)^{2n}\right)\mathrm{d}x$

Published on 25 Aug 2023 - 8:25

score 101 · Answer 15 · 2023-08-27 21:47:12

101

Views

If $\frac1{\cosh x}+\frac1{\cosh y}+\frac1{\cosh z}=1$, prove that $\sinh x \cdot \sinh y \cdot \sinh z \geq 16 \sqrt{2}$.

Published on 27 Aug 2023 - 21:47

List Question

Closed form of $\frac1s\int_0^\infty e^{-s(\tanh(t)+at)}dt=\frac1s\int_0^1e^{-st}(1-t)^{\frac{as}2-1}(1+t)^{-\frac{as}2-1}dt$ to invert $\tanh(x)+ax$?

Square root function hyperbolic estimator

Expression for real part of ArcTanh(x) for x > 1

Value of $t$ for a point on the hyperbola with given parametric equation $P(t)$

Intergrate $\text{arccosh}(\sqrt{x^2 + 1})$ via hyperbolic substituation.

A particular case of the Jacobi's triple product with an inequality and hyperbolic/trigonometric function

Showing that $\sqrt{\frac{a}{k}-1}\cosh(\frac12(z-ax))\tan(\frac12z\sqrt{\frac{a}{k}-1})$, with conditions, is increasing in $z$

How do I rotate a 2D catenary in 3D space?

Why do these answer differ when integrating the square root of $\tan x$ using complex numbers?

Demonstration of hyperbola in polar coordinates

Hyper-closed-forms of series involving hyperbolic functions:$\sum_{n=1}^{\infty}\frac1{n^2\cosh(\pi n)^2},\frac{\tanh(\pi n)}{n^3}$ and the like.

Alternate expression for $\operatorname{atanh}(x)$ for $\lvert x \rvert \to 1$

How taut must a stretchable, horizontally-oriented string be in order for a straight line to approximate the string to within a given margin of error?

Compact form of solution of $\displaystyle\int_0^{\frac{\pi}{2}}\ln\left(1+\alpha^n\sin(x)^{2n}\right)\mathrm{d}x$

If $\frac1{\cosh x}+\frac1{\cosh y}+\frac1{\cosh z}=1$, prove that $\sinh x \cdot \sinh y \cdot \sinh z \geq 16 \sqrt{2}$.

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