For questions related to hyperbolic functions: $\sinh$, $\cosh$, $\tanh$, and so on.

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14
votes
2answers
301 views

$\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx$

Is there any closed-form representation for the following integral? $$\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx,$$ where $\mathrm{sech}\,x$ is the hyperbolic secant, ...
7
votes
1answer
2k views

How were Hyperbolic functions derived/discovered?

Trig functions are simple ratios, but what does Cosh, Sinh and Tanh compute? How are they related to euler's number anyway?
5
votes
1answer
549 views

Geometric construction of hyperbolic trigonometric functions

If we have a circle we can geometrically construct the trigonometric functions as shown. The functions all derive from sin and cos. If we say that the circle is a conic section and imagine it on the ...
2
votes
3answers
316 views

Use residues to evaluate $\int_0^\infty \frac{\cosh(ax)}{\cosh(x)}\,\mathrm{d}x$, where $|a|<1$

Use residues to evaluate $$ \int_0^\infty \frac{\cosh(ax)}{\cosh(x)}\,\mathrm{d}x $$ where $|a|<1$. Try considering the integral of the form $$ \int_C \frac{\exp(az)}{\cosh(z)}\,\mathrm dz, $$ ...
0
votes
1answer
37 views

Proof that $\cos^2(x)\cosh^2(y) + \sin^2(x)\sinh^2(y) = -1 + \sin^2(x) - \sinh^2(y)$

Could anyone offer a proof that $$ \cos^2(x)\cosh^2(y) + \sin^2(x)\sinh^2(y) = -1 + \sin^2(x) - \sinh^2(y)? $$
50
votes
2answers
1k views

Integral $\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}\mathrm dx$

Consider the following integral: $$\mathcal{I}=\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\,\pi}{\operatorname{arcoth}x\,-\,\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}\mathrm dx,$$ where ...
33
votes
2answers
440 views

Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$

Please help me to find a closed form for the infinite product $$\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$$ where $\tanh(z)=\frac{e^z-e^{-z}}{e^z+e^{-z}}$ is the hyperbolic tangent.
9
votes
2answers
315 views

Series $\sum\limits_{n=1}^\infty \frac{1}{\cosh(\pi n)}= \frac{1}{2} \left(\frac{\sqrt{\pi}}{\Gamma^2 \left( \frac{3}{4}\right)}-1\right)$

I was playing around with Mathematica and found that $$\sum_{n=1}^\infty\frac1{\cosh(\pi n)} = \frac12\left(\frac{\sqrt{\pi}}{\Gamma \left(\tfrac34\right)^2}-1\right)$$ Does anybody know how to ...
6
votes
3answers
404 views

Show $\sum_{n=1}^{\infty}\frac{\sinh\pi}{\cosh(2n\pi)-\cosh\pi}=\frac1{\text{e}^{\pi}-1}$ and another

Show that : $$\sum_{n=1}^{\infty}\frac{\cosh(2nx)}{\cosh(4nx)-\cosh(2x)}=\frac1{4\sinh^2(x)}$$ $$\sum_{n=1}^{\infty}\frac{\sinh\pi}{\cosh(2n\pi)-\cosh\pi}=\frac1{\text{e}^{\pi}-1}$$
0
votes
1answer
97 views

Questions about the hyperbolic system of equations

$$u_t+A(x,t,u)u_x=b(x,t,u) \tag 1$$ $$u=(u_1, \dots, u_n), b=(b_1, \dots, b_n)$$ $$A=[a_{ij}], i,j = 1, \dots, n$$ $$$$ We set the question if there are characteristic directions at the path of which ...
8
votes
5answers
556 views

Geometric meanings of hyperbolic cosinus and sinus

In euclidean geometry, $\cos$ and $\sin$ are used for angles in trigonometry. Is there an equivalent for $\cosh$ and $\sinh$ the hyperbolic cosine and sine, and not cosinus and sinus ?
8
votes
1answer
309 views

Geometric definitions of hyperbolic functions

I've learned in school that all the trigonometric functions can be constructed geometrically in terms of a unit circle: Can the hyperbolic functions be constructed geometrically as well? I know ...
4
votes
1answer
320 views

fourier expansion of $\coth$ and justifying an identity

The problem: Justify the following equalities: $$\cot x = i\coth (ix) = i \sum^\infty_{n=-\infty} \frac{ix}{(ix)^2+(n\pi)^2}=\sum^\infty_{n=-\infty}\frac{x}{x^2+(n\pi)^2}$$ I am trying to figure ...
3
votes
0answers
239 views

Integrating a fractional power of a rational function

I am currently working on a project where I stumbled upon the integral $$ \int \frac{\sinh \left(\frac{R}{2}\right)}{(\coth R - 6R \coth\left(\frac{R}{2}\right) + 9)^{1/4}} \,dR $$ where $R$ is a ...