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

learn more… | top users | synonyms

1
vote
1answer
42 views

Is that hyperbolic identity correct?

Given the expression: \begin{equation} |x|\cosh(kx)+x\sinh(kx), \;\; k>0 \end{equation} By taking cases for $x$, we have: \begin{equation} \bullet \quad x>0: x(\cosh(kx)+\sinh(kx))=x\left( ...
1
vote
1answer
30 views

Prove this equality about hyperbolic right triangles

If K is the area of a hyperbolic right triangle ABC in which the right angle is at C, prove that $$ \sin K=\frac{ \sinh a \sinh b}{1+\cosh a\cosh b}$$ My attempt at the solution: I basically need ...
1
vote
1answer
61 views

Hard integral of root function and hyperbolic function

I need to calculate this integral: $$\int^A_B\frac{\sqrt{x-B}}{\cosh^2x}dx$$ Is there any way to do this?
1
vote
1answer
35 views

Evaluating $\lim_{x\to\infty}\left(\frac{\sinh x}{x}\right)^{{1}/{x^2}}$

$$\lim_{x\to\infty}\left(\frac{\sinh x}{x}\right)^{\dfrac{1}{x^2}}$$
1
vote
1answer
138 views

Solving the Laplace equation in terms of exponential of hyperbolic trigonometric functions

I'm solving the Laplace equation $U_{xx}+U_{yy}=0$ subject to BC's: \begin{align} U(0,y) &= 0 \\ U(a,y) &= 0 \\ U(x,0) &= 0 \\ U(x,b) & = \left\{x \text{ for } x \in ...
0
votes
1answer
20 views

Showing that hyperbolic trigonometric functions parameterize the unit hyperbola

I know that the same way circular trigonometry is defined over the circle $ x^2 + y^2 = 1 $, hyperbolic trigonometry is defined over the hyperbola $ x^2 - y^2 = 1 $. What I don't know is how deduced ...
0
votes
1answer
39 views

Hyperbolic Trig Functions - Identities

I don't understand how the 3rd step (the 4 divisions) happens? Can someone explain how they arrived at that.
0
votes
1answer
84 views

Simplify a parametric equation with hyperbolic trigonometric functions

I've the following parametric equations for a curve: $$\begin{cases}x(t)=a\cdot \operatorname{sech} (t) \\ y(t)=a\cdot(t-\tanh(t))\end{cases}$$ Now let $\theta(t)=-\arctan(\sinh(t))$ how does the ...
0
votes
1answer
35 views

showing $\sinh(x/2) = \epsilon \sqrt{\frac{1}{2}(\cosh(x) -1)}$

showing $$\sinh(x/2) = \epsilon \sqrt{\frac{1}{2}(\cosh(x) -1)}$$ and I was told to determine the value of $\epsilon$. From identities I reached $ \sinh^2(x) = \dfrac{1}{2}(\cosh(x) -1)$ however ...
0
votes
1answer
49 views

hyperbolic inequality

Calculating some contour integral, I have to prove that $\int^{R+i}_{R}\frac{cosh(az)}{cosh(\pi z)}dz$ goes to zero if R goes to infinity. And we know that $\left|a\right|<\pi$. I want to use the ...
6
votes
0answers
79 views

Can these two indefinite integrals be evaluated in closed form?

I'm wondering whether any of these two indefinite integrals $$\int \frac{1}{\sqrt{1+\alpha \sinh(x)^{-4/3}}}dx$$ $$\int \frac{\sinh(x)^{-4/3}}{\sqrt{1+\alpha\sinh(x)^{-4/3}}}dx$$ can be evaluated in ...
5
votes
0answers
85 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 ...
3
votes
0answers
38 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
54 views

Looking for a bound on a function involving $\sinh$

Fix $T > 0$ and let $t \in (0,T)$ let $c > 1$ be a constant (which may be bigger than $T$). Consider the function $$f(c,t,T) = \frac{\sinh ((T-t)c)}{\sinh (Tc)}.$$ I am looking for a bound of ...
3
votes
0answers
351 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
0answers
45 views

Prove that $\coth ^2(x)-1\equiv \mathrm{cosech}^2 (x)$

Prove that $\coth ^2(x)-1\equiv \mathrm{cosech}^2 (x)$ My attempts, $\coth^2 (x)-1\equiv(\frac{e^x+e^{-x}}{e^x-e^{-x}})^2-1$ $\equiv \frac{e^{2x}+e^{-2x}+2}{e^{2x}+e^{-2x}-2}-1$ $\equiv ...
2
votes
0answers
340 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
57 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
79 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
46 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 J├╝rgen Jost. Right now I'm looking at an exercise (12.5) under the ...
1
vote
0answers
30 views

Exponential and Hyperbolic Functions Before Power Series

Is there a textbook that covers the exponential function and the possibly the hyperbolic functions from a precalculus geometrical viewpoint? I'm essentially asking for something like a trigonometry ...
1
vote
0answers
24 views

Four simultaneous equations

General form of the function: $$y=d\sinh^{-1}\left(\frac{ax+b}2\right)+c$$ I want the function to pass three points, $(0,0)$, $\left(\frac{t}2,\frac{g}2\right)$ and $(t,g)$, and I want the function to ...
1
vote
0answers
48 views

Difference between hyperbolic sector, hyperbolic angle and hyperbolic argument

I've been working with hyperbolic functions and am completely confused by the Wikipedia definitions of hyperbolic sectors and angles. Are they the same thing? Based on my trial calculations, they seem ...
1
vote
0answers
30 views

Complex Mapping of $\mathrm{cosh}(w)=z$

Mapping in complex analysis has not been very easy for me unfortunately. I am having difficult trying to find the mapping between the z and w plane. I attempted to simply write that ...
1
vote
0answers
26 views

Hyperbolic trigonometric inequality

Is the following hyperbolic trigonometric inequality correct and if so, is there a simple derivation? $$\tanh A- \tanh B \geq (A-B)(\operatorname{sech}{^2}(A)), \qquad \forall A\ge B\ge 0.$$
1
vote
0answers
17 views

matrix in normal coordinates

Writing the matrix $ \begin{pmatrix} -\frac{k}{\gamma} & \frac{k}{\gamma}&0&0&0&\cdots&0&0&0&0 \\ \frac{k}{\gamma} &-2\frac{k}{\gamma}& ...
1
vote
0answers
105 views

Poles (and order of) of $(\cosh(1/(z-\pi)))^2$ and Residue at $z=\pi$

Hi I need help finding the poles and the order of the poles of the following function: $$\left(\cosh\frac1{z-\pi}\right)^2$$ and the residue at $z=\pi$. I have tried a number of different methods ...
1
vote
0answers
57 views

How to find the value of this sum?

The sum below numerically (to 13th digit at least) is the same as $\ln 2$. So there should be a way to prove it analytically, but I haven't succeeded. Any suggestions? ...
1
vote
0answers
83 views

Why are hyperbolic trigonometric functions avoided in (my) high school and early post-secondary school?

I remember seeing hyperbolic trigonometric functions (sinh, cosh, tanh, etc.) in my precalculus textbook back in high school and see them today in my calculus textbook. However, I have not had a ...
1
vote
0answers
60 views

$\lim\limits_{n\to\infty}\operatorname{arsinh}(1 + \operatorname{arsinh}(2 + \operatorname{arsinh}(3 + \dots\operatorname{arsinh}(n+\dots)\dots)))=?$

Does the limit $$\lim\limits_{n\to\infty}\operatorname{arsinh}(1 + \operatorname{arsinh}(2 + \operatorname{arsinh}(3 + \operatorname{arsinh}(4+\dots\operatorname{arsinh}(n+\dots)\dots))))$$ exist ...
1
vote
0answers
87 views

Solution by of nonlinear equation

$$\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} + \sin u = 0$$ From the sine-Gordon equation we can easily solve, \begin{equation} \phi(x) = \pm 4 \tan^{-1}\left[e^{\frac{x-t ...
1
vote
0answers
167 views

closed-form solution for 1/tanh(x) - 1/x that can be evaluated at/near x=0?

I'm looking to evaluate $\frac{1}{\tanh x}-\frac{1}{x}$ over a range that includes x=0. Is there an alternate form that is both exact, and numerically stable at/near x=0? For now I'm using the Taylor ...
1
vote
0answers
51 views

Hyperboloid equation related question?

How to draw this graph please? $$4y^2 -x^2+4z^2-1 \geq 0$$
0
votes
0answers
40 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
81 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 ...
0
votes
0answers
44 views

Circumference of hyperbolic circle is $2\pi \sinh r$

I'm looking for a proof that in the Poincare disk model the circumference of a circle of radius $r$ is $2\pi \sinh r$. I have seen this result in many places but I haven't been able to find a proof. ...
0
votes
0answers
39 views

Prove that $2\cosh \left(\frac{1}{2} a\right) \sin \left(\frac {1}{2} \alpha\right) = 1.$

Suppose a hyperbolic equilateral triangle has side $a$ and angle $\alpha$.Prove that $$2\cosh \left(\frac{1}{2} a\right) \sin \left(\frac {1}{2} \alpha\right) = 1.$$
0
votes
0answers
12 views

Linearization of hyperbolic function with unknown exponent

I have a graph that is clearly some inverse function of the form $y=Ax^n$ where n is a negative. I want to linearize this graph to give me the values of A and n without merely approximating the ...
0
votes
0answers
85 views

Derivatives of hyperbolic functions and Osborne's rule.

I am slightly confused when it comes to Osborne's rule when you take derivatives of hyperbolic functions. For example. The derivative of cotx is -cosec^2x, so there is a product of sines. So should ...
0
votes
0answers
34 views

Hyperbolic functions calculator values differing from the graph

I was exploring hyperbolic functions and noticed something weird while comparing the analytical definition (e^x+e^-x)/2 with the geometrical definition using the hyperbola x^2 - y^2 = 1. For the angle ...
0
votes
0answers
59 views

Geometric interpretation of hyperbolic functions and the hyperbolic angle/argument

I've been reading up on hyperbolic functions and was wondering if there was a geometric definition for the hyperbolic angle and hyperbolic function. In particular I was reading this: Definition of ...
0
votes
0answers
20 views

Exponential curve with hyperbolic sine behavior on the tails

I have a dataset that I've fitted an exponential curve to that looks like a great fit at midrange values of the domain but is not such a good fit at low and high end domain values. Instead, at these ...
0
votes
0answers
41 views

Osborne's rule for hyperbolic functions?

I am confused as to why you only change the sign for powers of sine that are 4n+2. As I understand, $sin(i\theta)=isinh(\theta)$ $sin^2(i\theta)=-sinh^2(\theta)$ $sin^3(i\theta)=-isinh^3(\theta)$ ...
0
votes
0answers
27 views

inverse hyperbolic function of a complex argument

It is not too difficult to prove that $f(z)=\cosh z$ is a bijection from $$\def\C{{\Bbb C}}D=\{\,z=x+iy\in\C\mid 0<y<\pi\,\}$$ to $$R=\{\,w=u+iv\in\C\mid v\ne0\,\}\cup\{\,w\in\C\mid ...
0
votes
0answers
134 views

Hyperbolic sinc function

Cardinal sine function or sinc function is defined by: \begin{equation} \mathrm{sinc}x=\begin{cases}\frac{\sin x}{x}, & x \neq 0,\\ 1, & x = 0,\end{cases} \end{equation} Is there any ...
0
votes
0answers
17 views

How can I find a hyperbolic function denoting zoom levels?

I'm working between two values. The first ($m$) represents the number of meters wide an estimate of location accuracy is, and the other ($z$) represents a vague level of zoom as described below. This ...
0
votes
0answers
72 views

Mathematica Integrate gives back the integrand

i'm trying to Integrate the following function: (q (1 + q) - E^-q Sinh[q])/(-q + Cosh[q] Sinh[q]) - ( 2 q Tanh[q])/(-q + Cosh[q] Sinh[q]) I already solved ...
0
votes
0answers
67 views

Exact Values of Hyperbolic Trig Functions

There are some well-known exact values for trig functions, such as $$\sin\frac{\pi}{6}=\frac{1}{2},\quad \tan\frac{\pi}{3}=\sqrt 3, \quad\text{etc.}$$ Are there comparable special values for the ...
-1
votes
0answers
17 views

Catenary calculation

I have a catenary wire with the following known factors: P: horizontal force in wire u: weight/mtr s: catenary length v: height difference between y1, y2 From statics follows: y= a cosh (x/a) ...
-1
votes
0answers
24 views

Hyperbolic distance with natural log

We have been learning a lot about hyperbolic lines, midpoints and distances. Would someone be able to assist me how to solve for this problem? The hyperbolic line consists of positive real numbers ...