Question

Prove that $\sum_{n\geq 1} \frac{1}{x+n^2}=\frac{\pi\sqrt{x}\operatorname{coth}(\pi\sqrt{x})-1}{2x}$ for all $x>0$

score 757 · Answer 1 · 2026-04-16 02:45:41

757

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Prove that $\sum_{n\geq 1} \frac{1}{x+n^2}=\frac{\pi\sqrt{x}\operatorname{coth}(\pi\sqrt{x})-1}{2x}$ for all $x>0$

Published on 16 Apr 2026 - 2:45

score 64 · Answer 2 · 2026-04-15 09:19:46

64

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Closed Form of the following infinite series

Published on 15 Apr 2026 - 9:19

score 411 · Answer 3 · 2026-04-17 16:03:40

411

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Evaluate $\int_{0}^{\infty} \frac{x \exp\left\{-\beta^2 x^2\right\}}{\sinh \left(\frac{\pi x}{2}\right)} \mathrm{d} x$

Published on 17 Apr 2026 - 16:03

score 116 · Answer 4 · 2026-05-10 15:52:46

116

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Series Expansion of $\int_{\mathbb{R}} \tanh\left(\beta x\right) x \exp\left(-\frac{x^2}{2}\right) \mathrm{d} x$

Published on 10 May 2026 - 15:52

score 182 · Answer 5 · 2026-04-11 05:36:53

182

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Geometric method to calculate $\tanh(\cosh^{-1}(x))$

Published on 11 Apr 2026 - 5:36

score 1.3k · Answer 6 · 2026-04-11 19:07:31

1.3k

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What are Hyperbolic Trig Functions Functions of?

Published on 11 Apr 2026 - 19:07

score 1.6k · Answer 7 · 2026-04-13 23:26:51

1.6k

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Fitting hyperbolic functions ($\tanh$) on data.

Published on 13 Apr 2026 - 23:26

score 856 · Answer 8 · 2026-04-13 20:49:49

856

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Hyperbolic inverse Function

Published on 13 Apr 2026 - 20:49

score 1.2k · Answer 9 · 2026-04-17 08:14:30

1.2k

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Find $\lim_{x\to 0} \frac{\sinh x -\sin x}{\cosh x - \cos x}$

Published on 17 Apr 2026 - 8:14

score 2.3k · Answer 10 · 2026-04-13 19:50:43

2.3k

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Solving hyperbolic functions

Published on 13 Apr 2026 - 19:50

score 410 · Answer 11 · 2026-04-09 01:42:41

410

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Hyperbolic tangent expansion in temperature

Published on 09 Apr 2026 - 1:42

score 224 · Answer 12 · 2026-05-10 17:14:59

224

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Proving that $\cosh(x)\cosh(y) \geq \sqrt{x^2 + y^2}$

Published on 10 May 2026 - 17:14

score 433 · Answer 13 · 2026-04-16 12:24:49

433

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Proving that $\cosh^{-1} (\cosh(x)\cosh(y) ) \geq \sqrt{x^2 + y^2}$ and the CAT(0) inequality

Published on 16 Apr 2026 - 12:24

score 520 · Answer 14 · 2026-05-10 17:39:39

520

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What is the period of the composition of hyperbolic tangent and hyperbolic arcsine?

Published on 10 May 2026 - 17:39

score 5k · Answer 15 · 2026-05-10 15:16:20

5k

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Infinite Product Expansion of Hyperbolic Functions

Published on 10 May 2026 - 15:16

List Question

Prove that $\sum_{n\geq 1} \frac{1}{x+n^2}=\frac{\pi\sqrt{x}\operatorname{coth}(\pi\sqrt{x})-1}{2x}$ for all $x>0$

Closed Form of the following infinite series

Evaluate $\int_{0}^{\infty} \frac{x \exp\left\{-\beta^2 x^2\right\}}{\sinh \left(\frac{\pi x}{2}\right)} \mathrm{d} x$

Series Expansion of $\int_{\mathbb{R}} \tanh\left(\beta x\right) x \exp\left(-\frac{x^2}{2}\right) \mathrm{d} x$

Geometric method to calculate $\tanh(\cosh^{-1}(x))$

What are Hyperbolic Trig Functions Functions of?

Fitting hyperbolic functions ($\tanh$) on data.

Hyperbolic inverse Function

Find $\lim_{x\to 0} \frac{\sinh x -\sin x}{\cosh x - \cos x}$

Solving hyperbolic functions

Hyperbolic tangent expansion in temperature

Proving that $\cosh(x)\cosh(y) \geq \sqrt{x^2 + y^2}$

Proving that $\cosh^{-1} (\cosh(x)\cosh(y) ) \geq \sqrt{x^2 + y^2}$ and the CAT(0) inequality

What is the period of the composition of hyperbolic tangent and hyperbolic arcsine?

Infinite Product Expansion of Hyperbolic Functions

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