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

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11
votes
1answer
6k 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?
7
votes
1answer
2k 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
385 views

Alternative definition of hyperbolic cosine without relying on exponential function

Ordinary trigonometric functions are defined independently of exponential function, and then shown to be related to it by Euler's formula. Can one define hyperbolic cosine so that the formula ...
15
votes
2answers
345 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, ...
2
votes
3answers
621 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 $$ \int_C \frac{\exp(az)}{\cosh(z)}\,\mathrm dz, $$ ...
11
votes
1answer
929 views

Can the real and imaginary parts of $\dfrac{\sin z}z$ be simplified?

I have calculated the real and imaginary parts of $\dfrac{\sin z}z.$ I've obtained $$\begin{eqnarray} \frac{\sin z}z&=&\frac{\sin(x+iy)}{(x+iy)}\\ &=& ...
7
votes
3answers
773 views

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

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

What are all functions of the form $\frac{\cosh(\alpha x)}{\cosh x+c}$ self-reciprocal under Fourier transform?

There are some functions that are self reciprocal under cosine Fourier transform: \begin{equation} \frac{1}{\cosh x}, \frac{\cosh x}{\cosh 2x},\frac{1}{1+2\cosh x} \end{equation} It seems that they ...
5
votes
1answer
770 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 ...
0
votes
1answer
144 views

Integral involving the hyperbolic tangent

Find the integral of $ \int \limits _{-\infty} ^{\infty} e^{\Bbb i x \frac h k} \tanh x \ \Bbb dx$. I tried to expand this but that didn't help.
3
votes
5answers
43 views

Limit of function of hyperbolic

How can I - without using derivatives - find the limit of the function $f(x)=\frac{1}{\cosh(x)}+\log \left(\frac{\cosh(x)}{1+\cosh(x)} \right)$ as $x \to \infty$ and as $x \to -\infty$? We know ...
0
votes
1answer
64 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 $$ \cos^2(x)\cosh^2(y) + \sin^2(x)\sinh^2(y) = -1 + \sin^2(x) - \sinh^2(y)? $$
61
votes
2answers
2k 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 ...
35
votes
2answers
545 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.
26
votes
4answers
638 views

Integral ${\large\int}_0^\infty\frac{dx}{\sqrt[4]{7+\cosh x}}$

How to prove the following conjectured identity? $$\int_0^\infty\frac{dx}{\sqrt[4]{7+\cosh x}}\stackrel{\color{#a0a0a0}?}=\frac{\sqrt[4]6}{3\sqrt\pi}\Gamma^2\big(\tfrac14\big)\tag1$$ It holds ...
4
votes
2answers
190 views

Evaluating $\sum_{n=0}^{\infty } 2^{-n} \tanh (2^{-n})$

Reading in some tables pages I found $$\sum _{n=0}^{\infty } 2^{-n} \tanh \left(2^{-n}\right)=\tanh (1) \left(1+\coth ^2(1)-\coth (1)\right)$$ I try to split in two sum using the roots of the ...
9
votes
2answers
152 views

Calculate the sum: $\sum_{x=2}^\infty (x^2 \operatorname{arcoth}(x) \operatorname{arccot} (x) -1)$

$${\color\green{\sum_{x=2}^\infty (x^2 \operatorname{arcoth} (x) \operatorname{arccot} (x) -1)}}$$ This is an impressive sum that has bothered me for a while. Here are the major points behind the ...
7
votes
2answers
335 views

An alternating series identity with a hidden hyperbolic tangent

How does one use the inverse Mellin transform to prove that the following identity holds? $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n(e^{n\pi} + 1)} = \frac{1}{8}(\pi - 5\log(2))$$ The identity follows ...
11
votes
2answers
489 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 ...
10
votes
2answers
296 views

Closed-form of $\int_0^\infty \frac{1}{\left(a+\cosh x\right)^{1/n}} \, dx$ for $a=0,1$

While I was working on this question by @Vladimir Reshetnikov, I've conjectured the following closed-forms. $$ I_0(n)=\int_0^\infty \frac{1}{\left(\cosh x\right)^{1/n}} \, dx \stackrel{?}{=} ...
6
votes
2answers
1k views

Geometric interpretation of hyperbolic functions

When proving identities like $$\cosh(2x)=\cosh^2(x)+\sinh^2(x)$$ $$\cosh^2(x)=\sinh^2(x)+1$$ algebraically, I am beset by the feeling that there should be a geometrical interpretation that makes them ...
3
votes
0answers
91 views

Two complementary continued fractions that are algebraic numbers

Define the two similar continued fractions, $$x=\cfrac{1}{km\color{blue}+\cfrac{(m-1)(m+1)} ...
0
votes
1answer
122 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
1k 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 ?
6
votes
1answer
374 views

Integral of the ratio of two exponential sums

I am trying to find a lower bound on the following integral \begin{align*} \int_{y=-\infty}^{y=\infty} \frac{ (\sum_{n=[-N..N]/\{0\}}n e^{-\frac{(y-cn)^2}{2}})^2} {\sum_{n=[-N..N]/\{0\}} ...
3
votes
1answer
59 views

Reduction of $\tanh(a \tanh^{-1}(x))$

Given $x\in \Re$, $a \in \Re$ where $-1 \le x \le 1$ and $0 \le a \le 4$, is it possible to reduce the following expression: $\tanh(a \tanh^{-1}(x))$ E.g. to some kind of polynomium? I know that if ...
0
votes
3answers
93 views

How to simplify if $a > 0$ and $\cos(a) < 0$ [closed]

$$\sqrt{\cos (a)} \sinh \left(\ln (2) a^{\frac{1}{2} \left(e^{-ia }+e^{ia}\right)}\right)+\sqrt{\cos (a)} \cosh \left(\ln (2) a^{\frac{1}{2} \left(e^{-ia }+e^{ia }\right)}\right)=$$
0
votes
2answers
257 views

Integral $ \int_{0}^{\pi/2} \frac{\pi^{(x^{e})}\sin(x)\tan^{-1}(x)}{\sinh^{-1}\left({1+\cos(x)}\right)} dx$

I need help in evaluating the following integral :- $$ \int_{0}^{\pi/2} \frac{\pi^{\displaystyle (x^{e})}\sin(x)\tan^{-1}(x)}{\sinh^{-1}\left({1+\cos(x)}\right)} dx$$ A brief solution would be very ...
8
votes
1answer
669 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 ...
2
votes
1answer
43 views

Simplifying hyperbolic compositions like $\sinh (N \operatorname{acosh} a)$

In many occasions, we may meet hyperbolic functions, as well as their combined ones. I want to simplify expressions like $$ \tanh\left( N\left(\textrm{acosh}~ a\right)\right) $$ and $$ \sinh\left( ...
1
vote
0answers
37 views

Prove the identity $\tanh(N\textrm{acosh}\;a) = \vert \frac{g^{2N}-1}{g^{2N}+1}\vert$

During my recent study, I found an Identity which is of the form $$ \tanh(N\textrm{acosh}\;a) = \left\vert \frac{g^{2N}-1}{g^{2N}+1}\right\vert $$ where $a\geq1$ and $g>0$ satisfy ...
0
votes
1answer
57 views

Identity between $x=y+z$ and $\tan\left(\frac{\theta}{2}\right)=\tan\left(\frac{\nu}{2}\right)\tan\left(\frac{\pi/2-\epsilon}{2}\right) $

I would like to prove that (1) $$\begin{equation} \tan\left(\frac{\theta}{2}\right)=\tan\left(\frac{\nu}{2}\right)\tan\left(\frac{\pi/2-\epsilon}{2}\right) \end{equation}$$ can transformed to (2) ...
0
votes
4answers
59 views

How to prove $\tanh ^{-1} (\sin \theta)=\cosh^{-1} (\sec \theta)$

As the question says How to prove $$\tanh ^{-1} (\sin \theta)=\cosh^{-1} (\sec \theta)$$ I have tried to solve it The end result that got for RHS $$=\log \frac{1+\tan\frac{\theta}{2}}{1-\tan ...
3
votes
0answers
52 views

Representation of hyperbolas.

I am well aware of the matrix representation for rotation of points on a circle with reals $$M_{R(\theta)} = \left(\begin{array}{rr} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & ...
3
votes
0answers
445 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 ...
3
votes
1answer
2k views

Integration Using Hyperbolic Substitution

I'm supposed to prove the following by using the hyperbolic sine double angle identity: $\sinh(2x)=2\sinh x\cosh x$ and archsinhx formula: $\ln(x+\sqrt{x^2+1})$ but can't seem to figure out the steps. ...
1
vote
1answer
61 views

Distance in the $y$-axis of the hyperbolic plane

I'm reading Stillwell's Geometry of Surfaces but I'm having a little bit of trouble because my background in calculus isn't great, I'm struggling with these problems: In the upper half-plane model, ...
0
votes
1answer
98 views

Newton-Raphson For Integer Factorization

Per my earlier question on Naive Grouping for factorization here, below is the modified Newton-Raphson method (integers only) for the polynomial $N -x^2 - yx - x = 0$. ...
0
votes
0answers
66 views

Transform complex trigonometric expression with $\arccos$

In the proof that the poles of a Chebyshev filter lie on an ellipse, there is the following transformation, for the $s$ values correspondant to the poles. From (1) $$s_{pm} = j \cos \left[ ...
0
votes
0answers
97 views

Naive Grouping for Factorization

I have a naive grouping method for factorization. I am curious as to its novelty and aspects of the code below that will increase its efficiency. The method is best described with an example: For n ...