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

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21
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0answers
356 views

Curious about an empirically found continued fraction for tanh

First of all, and since this is my first question in this forum, I would like to specify that I am not a professional mathematician (but a philosophy teacher); I apologize by advance if something is ...
6
votes
0answers
90 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
140 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
46 views

Evaluate $\int\frac{1}{x}\coth(ax)\sin^2(\frac{xt}{2})dx$

I am trying to integrate this function: $\int_0^\infty\frac{1}{x}\coth(\frac{\hbar x}{2kT})\sin^2(\frac{xt}{2})dx$ which Wolframalpha (for me) returns nothing, just a blank screen. I thought that it ...
3
votes
0answers
51 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
62 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
409 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
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0answers
105 views

Analytic forms of $\frac{\sinh(2\pi/7)}{\sinh^{2}(3\pi/7)} - \frac{\sinh(\pi/7)}{\sinh^{2}(2\pi/7)} + \frac{\sinh(3\pi/7)}{\sinh^{2}(\pi/7)}$

On page 183 of Berndt's Ramanujan's Notebooks Vol. 4, eq. 32.34 reads: $$ \frac{\sin(2\pi/7)} {\sin^{2}(3\pi/7)} - \frac{\sin(\pi/7)}{\sin^{2}(2\pi/7)} + \frac{\sin(3\pi/7)} {\sin^{2}(\pi/7)} = ...
2
votes
0answers
37 views

Find the positive root of the equation $\cosh x+\cos x-3=0$, other than numerically

I know you are able to find the root of the equation by using Newton-Raphson method. But is there any other way? $$\cosh x+\cos x-3=0$$ I thought maybe you could say that $-1\leq \cos x \leq 1$. So ...
2
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0answers
32 views

Evaluating Hyperbolic Cotangent (coth) Integral

I am working on some simulation, and the paper that I am basing some of the work off of involves several complex integrals. In particular, the one I am trying to solve is $\int_0^\infty ...
2
votes
0answers
123 views

Intriguing Poisson sum with hyperbolic function

I've been playing with lots of Poisson sums lately, and I thought this one to be interesting: $$\sum_{k\in\mathbb{Z}}\left(\frac{1}{(k+x)\sinh{(k+x)\pi q}}-\frac{1}{\pi q (k+x)^2}\right)$$I want to ...
2
votes
0answers
55 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
382 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
85 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 Jürgen Jost. Right now I'm looking at an exercise (12.5) under the ...
1
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0answers
34 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 ...
1
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0answers
61 views

Inverse function of sum of coth and tanh terms

In a publication I found an equation of the form $c_p = B + C \left( \frac{D/T}{\sinh(D/T)} \right)^2 + E \left( \frac{F/T}{\cosh(F/T)} \right)^2$ $c_p$ is the heat capacity, $T$ is the ...
1
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0answers
33 views

Functional equation for $\sum_{n=1}^{\infty} \sinh(cn)^{-s}$?

Does anyone know of any kind of functional equation (or closed form) for $\sum\limits_{n=1}^{\infty}\sinh(cn)^{-s}$, where $c$ is an arbitrary constant? I've been messing around with it off and on for ...
1
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0answers
53 views

Law of Cosines with imaginary arguments?

Does the law of cosines: $c^2 = a^2 + b^2 - 2 a b \cos \theta$ work with imaginary angles? to get something like: $c^2 = a^2 + b^2 - 2 a b \cosh \theta$ Alternatively, is there a hyperbolic ...
1
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0answers
66 views

Examples of integrals solved using hyperbolic functions.

I've read in some questions here that various types of integrals usually solved by involving $\tan$ and $\sec$ into the mix can sometimes be solved in an easier manner using hyperbolic functions, as ...
1
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0answers
49 views

Elliptic formulation of sum with cosine and hyperbolic cosine

Can anyone check my reformulation below?$$\sum_{k\in \mathbb{Z}}\frac{1}{\cosh{(k+z)\pi}+\cos{(k+z)\pi}}=\frac 2\pi \csc{\pi ...
1
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0answers
38 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
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0answers
40 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
158 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
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0answers
34 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
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0answers
29 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
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20 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
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0answers
145 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
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0answers
68 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
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0answers
102 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
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0answers
62 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
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0answers
90 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
179 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
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0answers
53 views

Hyperboloid equation related question?

How to draw this graph please? $$4y^2 -x^2+4z^2-1 \geq 0$$
0
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0answers
38 views

Sum Calculation: $\sum_{n=1}^\infty \left(1- \frac{\cosh^{-1} n}{\log 2x}\right)$

I was investigating the asymptotic properties of the $\cosh$ functions and how they all strongly relate to $e^x$ In my studying, I found out that $\cosh x\sim \frac{e^x}{2}$ By that definition, that ...
0
votes
0answers
25 views

What the inverse function of $_2F_1(a,b;c;z)$

What the inverse function of the function $f(z)$ given by $$ f(z) = \, _2F_1(a,b;c;z), \quad \mid z \mid <1, $$ where is the Gauss hypergeometric function given by $$ ...
0
votes
0answers
20 views

$\frac{d^n}{dx^n}$ in term of $\frac{d}{d t}$ for $x= \frac{1}{\cosh^{2} (t)}$

Let: $$x= \frac{1}{\cosh^{2} (t)},$$ I want to express $\frac{d^n}{dx^n}$ in term of $\frac{d}{d t}$. we have $x = \cosh^{-2} (t)$, so \begin{align*} \frac{d}{dt} &= \frac{d}{dx} \frac{d x ...
0
votes
0answers
15 views

implicit derivation of hyperbolic functions

derive $$\sinh(x+y)=\tanh^{-1}(\frac{x}{y})$$ $$\cosh(x+y)\cdot(1+f')=\frac{\frac{y-xf'}{y^2}}{1-(\frac{x}{y})^2}$$ $$\cosh(x+y)\cdot(1+f')=\frac{\frac{y-xf'}{y^2}}{\frac{y^2-x^2}{y^2}}$$ ...
0
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0answers
21 views

How to prove that $(\frac{1+\tanh x}{1-\tanh x})^3=\cosh 6x+\sinh 6x$

How to prove that $$(\frac{1+\tanh x}{1-\tanh x})^3=\cosh 6x+\sinh 6x$$ I have tried using the Dmoivres theorme
0
votes
0answers
8 views

Is the follow equation representative of a Hyperbolic One dimensional conservation law?

For $y : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ and let $j : \mathbb{R} \to \mathbb{R}$ $$\frac{\partial y}{\partial x_2} + \frac{\partial (j \circ y)}{\partial x_1} = 0$$
0
votes
0answers
17 views

Help me find the values of A,B,C if $Ax^2+By^2+Cz^2=1$ is the equation of a hyperboloid of one sheet that goes through the point $(-1,-4,-3)$?

What are the values of A,B,C if $Ax^2+By^2+Cz^2=1$ is the equation of a hyperboloid of one sheet that goes through the point $(-1,-4,-3)$ Here is what I did: $$Ax^2+By^2+Cz^2=1$$ ...
0
votes
0answers
20 views

How can I get a and b out of these equations involving $tanh$?

For every value $x$, we know: $$\dfrac{tanh(xa+b)}{tanh(a+b)}$$ Is it now possible to know $\dfrac{a}{b}$? The only thing I think I was able to show, was that: ...
0
votes
0answers
63 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
94 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
130 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
41 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
122 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
55 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
123 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 ...