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

learn more… | top users | synonyms

47
votes
2answers
1k views

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

Consider the following integral: $$\mathcal{I}=\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\,\pi}{\operatorname{arcoth}x\,-\,\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}dx,$$ where ...
33
votes
2answers
423 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.
14
votes
2answers
267 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, ...
13
votes
3answers
270 views

Closed form for $\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx$

Let $$f(a)=\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx,$$ where $\operatorname{sech}(z)=\frac2{e^z+e^{-z}}$ is the hyperbolic secant. Here are values of $f(a)$ at some ...
11
votes
1answer
547 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)}\\ &=& ...
9
votes
2answers
307 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 ...
8
votes
5answers
488 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
286 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 ...
7
votes
1answer
1k 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?
6
votes
3answers
396 views

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

Show that : $$\sum_{n=1}^{\infty}\frac{\cosh(2nx)}{\cosh(4nx)-\cosh(2x)}=\frac1{4\sinh^2(x)}$$ $$\sum_{n=1}^{\infty}\frac{\sinh\pi}{\cosh(2n\pi)-\cosh\pi}=\frac1{\text{e}^{\pi}-1}$$
5
votes
5answers
164 views

Evaluating the following integral: $ \int \frac{x^2}{\sqrt{x^2 - 1}} \text{ d}x$

For this indefinite integral, I decided to use the substitution $x = \cosh u$ and I've ended up with a $| \sinh u |$ term in the denominator which I'm unsure about dealing with: $$\int ...
5
votes
1answer
175 views

A conjecture $\int_{-\infty}^\infty\frac{\arctan e^x}{\cosh x}\cdot\frac{\tanh\frac{x}2}{x}dx\stackrel?=\frac\pi2\ln2$

I need to find a closed form for this integral: $$\mathcal{I}=\int_{-\infty}^\infty\frac{\arctan e^x}{\cosh x}\cdot\frac{\tanh\frac{x}2}{x}dx.$$ A numerical integration results in an approximation ...
5
votes
1answer
99 views

Weierstrass $ \tanh \frac{\theta}{2} $ substitution confusion.

I'm already familiar with the trigonometric version of this substitution $ t = \tan \frac{\theta}{2} $ and it's geometrical derivation involving the unit circle found here. However, I'm not sure how ...
5
votes
1answer
145 views

Integral with hyperbolic cosine squared

Does anyone can give me a hint how to integrate the following: $$\int_0^\infty{\frac{x^2 {\rm d}x}{\mathrm{cosh}^2(x)}}.$$ The answer is $\frac{\pi^2}{12}$ (taken from the book). I've started with ...
4
votes
4answers
161 views

Evaluating $\int_0^1 x \sinh (x) \ \mathrm{dx}$

I am looking to evaluate $$\int_0^1 x \sinh (x) \ \mathrm{dx}$$
4
votes
4answers
207 views

What is the use of hyperbolic trigonometric functions if they are easily expressible algebraically?

I get that there are uses for $\sin(x)$ and $\cos(x)$ because they are defined with imaginary exponents which aren't as easily worked with but the hyperbolic functions are simply $\frac12(e^x\pm ...
4
votes
1answer
483 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 ...
4
votes
1answer
54 views

solving $2\cosh2x = 13\cosh x - 12$

I've been asked to solve: $2\cosh2x = 13\cosh x - 12$ I showed earlier in the question that $\cosh2x = 2\cosh^2x -1$ So I can say that: $2(2\cosh^2x -1) = 13\cosh x - 12$ $\therefore 4\cosh^2x ...
4
votes
1answer
287 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 ...
4
votes
0answers
38 views

Inverse Fourier transform using Residues for a ratio of hyperbolic functions.

I'm new and glad to be here. I have a problem relating to an inverse Fourier transform. I have $$g(w)= \frac{\sinh{w(a-b)}}{w \cosh{wa}}$$ and want to find $$G(t)$$. I cannot find this in tables so I ...
4
votes
0answers
194 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
4answers
350 views

If $f$ is holomorphic and $\,f'' = f$, then $f(z) = A \cosh z + B \sinh z$

Suppose $f$ is holomorphic in a disk centered at the origin and $f$ satisfies the differential equation $$f'' = f.$$ Show that $f$ is of the form $$f(z)=A \sinh z + B \cosh z,$$ for suitable constants ...
3
votes
3answers
161 views

Find the limit of $S_n=\sum_{i=1}^n \big\{ \cosh\big(\!\!\frac{1}{\sqrt{n+i}}\!\big) -n\big\}$, as $n\to\infty$?

$S_n=\sum_{i=1}^n\big\{ \cosh\big(\frac{1}{\sqrt{n+i}}\!\big) -n\big\}$ as $n\to\infty$ I stumbled on this question as an reading about Riemannian sums as in $$ \int_a^b f(x)\,dx =\lim_{x\to ...
3
votes
2answers
60 views

Why are hyperbolic functions significant?

I'm currently covering Stewart's Early Transcendentals, and there is a whole section dedicated to defining and differentiating hyperbolic functions. The same amount of space is used to cover other ...
3
votes
4answers
90 views

Hyperbolic cosine

I have an A level exam question I'm not too sure how to approach: a) Show $1+\frac{1}{2}x^2>x, \forall x \in \mathbb{R}$ b) Deduce $ \cosh x > x$ c) Find the point P such that it lies on ...
3
votes
1answer
68 views

given $\cosh u = x$ find $\sinh u$

I'm asked to show that:$\newcommand{\arcosh}{\operatorname{arcosh}}$ $\int{x \arcosh x}dx = \frac{1}{4}(2x^2 -1)\arcosh x - \frac{1}{4}x\sqrt{x^2 -1} + C$ If I integrate by parts: let $u = \arcosh ...
3
votes
1answer
123 views

Why do we get two solutions when inverting $y = \sinh x$?

Using the definition $\sinh x = \dfrac{e^x-e^{-x}}{2},\;$ let's say we want to solve $\;y = \sinh x \;$ for $x$. It's not hard to show that $\;\sinh x \;$ is bijective, so this should have exactly ...
3
votes
1answer
161 views

Evaluation of integral involving $ \tanh(ax) $

Is it possible to evaluate in closed form the integral $$ \int_{-\sqrt{x}}^{\sqrt{x}}\frac{r\tanh(ar)}{\sqrt{x-r^{2}}}dr=2\int_{0}^{\sqrt{x}}\frac{r\tanh(ar)}{\sqrt{x-r^{2}}}dr$$ here $a$ is a ...
3
votes
0answers
75 views

Closed form for $\int^1_0 \mathrm{arccsch}\left( \frac{x^2-x-1}{x^2+x+1}\right)\;\mathrm dx$

Find a closed form for $$\int^1_0 \mathrm{arccsch}\left( \frac{x^2-x-1}{x^2+x+1}\right)\;\mathrm dx$$ What I have tried Expanding the $\mathrm{arccsch}$ into its logarithmic form, however I ...
3
votes
0answers
228 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 ...
2
votes
4answers
98 views

Inverse of $(e^x - e^{-x})/2$

What is the inverse of the function $f(x)=\frac{e^x - e^{-x}}2$? I tried replacing $e^x$ by a variable but I still can't get it.
2
votes
1answer
148 views

Simplifying $\cosh \mathrm{arcsinh} \ x$

How can I simplify the following: $$\cosh \mathrm{arcsinh} \ x$$ I know that an expression of the form $f(g^{-1}(x))$ where $f$ and $g$ are trigonometric functions can be simplified by constructing a ...
2
votes
1answer
299 views

A hard integral with hyperbolic function

I was self studying integral. I meet a difficult problem here: $$\int_{0}^{\infty }{{{x}^{n}}\frac{\sinh ax}{\cosh bx}}\text{d}x=\frac{\pi }{2b}\cdot \frac{{{\text{d}}^{n}}}{\text{d}{{a}^{n}}}\tan ...
2
votes
1answer
80 views

definite Integration

A integration is given, $$M = \int_{- \infty}^{\infty} \left[\frac{1}{2} \left(\frac{d\phi}{dx}\right)^2 + \frac{\lambda}{4}(\phi^2-v^2)^2\right] dx,$$ where $$m=v\sqrt\lambda$$ and $$ \phi(x)= ...
2
votes
1answer
42 views

Inverse trig and trigh in integration?

I have just done part (iii) of this question and can get the right answer but am a bit confused why do we take arcosh i.e. just the principle value of cosh and not the other value. I presume this is ...
2
votes
1answer
65 views

definite integral involing hyperbolic and trigonometric functions

Trying to prove the following: $$ \int_0^\infty xe^{-c x^2}\sinh(a x)\cos(bx)\,dx = ...
2
votes
1answer
186 views

How to convert between the hyperboloid model and the Poincare patch for $\mathbb{H}_n$?

This question is motivated by the results in this paper, http://calvino.polito.it/~camporesi/JMP94.pdf In this paper some of its most important results about the asymptotics of symmetric traceless ...
2
votes
2answers
100 views

proving that the differences of squares of hyperbolic sin/cos is an integer.

The hyperbolic sine and cosine are defined as following: $\sinh(x)=\dfrac{e^x-e^{-x}}{2}$ $\cosh(x)=\dfrac{e^x+e^{-x}}{2}$ How do I show that their differences of squares are always an integer for ...
2
votes
3answers
80 views

What is wrong? Symmetric function

I need some advice here. What is $y(\ln(4))$ if the function $y$ satisfies: $$\frac{dy}{dx} = 1-y^2$$ and is symmetric about the point $(\ln(9),0)$. Solving that equation I end up with: ...
2
votes
1answer
461 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. ...
2
votes
1answer
143 views

Evaluation of an integral involving hyperbolic sine and exponential

I am wondering if the following integral can be reduced to either a closed form involving elementary functions, or well-known special functions (such as $\operatorname{erf}$, Bessel functions, etc.): ...
2
votes
1answer
71 views

Some estimate concerning hyperbolic functions

I want to show that $|\sinh(az)|\leq|\sinh(z)|$ for all $z\in\mathbf{C}$ (or at least for all $z\in\mathbf{H}$, the upper half plane), provided that $0<a<1$ However, I am not even certain ...
2
votes
1answer
172 views

Definite integral involving hyperbolic cosine

I have had no experience so far with hyperbolic functions so any help will be appreciated. This is on the chapter of complex integration but I would especially appreciate it if you could turn this ...
2
votes
3answers
298 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 $$ \int_C \frac{\exp(az)}{\cosh(z)}\,\mathrm dz, $$ ...
2
votes
0answers
30 views

Integral Evaluation: Exponential of and Hyperbolic Function

I'm trying to evaluate $$G^{\pm} = \frac{-i}{8\pi^2 X} \partial_X \int_{-\infty}^\infty d\phi e^{i m \left[X \sinh \phi \pm T \cosh \phi \right]}$$ for $T = \pm X$. Where $T, X, m \in \mathbb{R}$ ...
2
votes
0answers
226 views

Parametric Equation of a Hyperbolic Paraboloid

I need to make two trace plots of the hyperbolic paraboloid $z=x^2-y^2$. In the first plot, we set $z$ equal to a constant $k$, $z=k$. How do I find the parametric equation for this representation of ...
2
votes
0answers
46 views

Integral with $\cosh$ and $\log$ in the integrand

I am trying to find a good way to simplify (or even solve) the following integral: $$ \int_0^{-\log \varepsilon} \cosh^3r \frac{\partial^2}{\partial r^2} \log (\det E) dr $$ where $r > 0$ is a ...
2
votes
0answers
74 views

Solution of nonlinear waves( breathers)

The sine-Gordon equation is known as $$\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} + \sin u = 0,$$ Can you please derive the equation which is known as breather equation ...
2
votes
0answers
41 views

hyperbolic group ; showing the existence of a ration function with a certain condition

I'm currently working out of a book called differentialgeometry and minimal surfaces written by Jost-Hinrich Eschenburg und Jürgen Jost. Right now I'm looking at an exercise (12.5) under the ...
1
vote
2answers
121 views

The derivative of $\tanh x$

I'm trying to calculate the derivative of $\displaystyle\tanh h = \frac{e^h-e^{-h}}{e^h+e^{-h}}$. Could someone verify if I got it right or not, if I forgot something etc. Here goes my try: ...