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

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7
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0answers
101 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
148 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
85 views

How to prove sum related to hyperbolic tangents $\sum_{k=0}^{n-1}\frac{\tanh(…)}{1+\frac{\tanh^2x}{\tan^2(…)}}=\tanh(2nx)$

I have no Idea how to start I think to switch it to definite integral, use complex analysis, or some real analysis tricks and at the end I failed to make any progress. $$ \displaystyle \sum_{k=0}^{...
3
votes
0answers
93 views

Two complementary continued fractions that are algebraic numbers

Define the two similar continued fractions, $$x=\cfrac{1}{km\color{blue}+\cfrac{(m-1)(m+1)} {3km\color{blue}+\cfrac{(2m-1)(2m+1)}{5km\color{blue}+\cfrac{(3m-1)(3m+1)}{7km\color{blue}+\ddots}}}}\tag1$$...
3
votes
0answers
52 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
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) & \cos(\theta)\...
3
votes
0answers
70 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
447 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
0answers
109 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\...
2
votes
0answers
30 views

Distance between points lying on a hyperbola?

The question is rather simple but I can't find the answer I'm looking for anywhere. On an ordinary 1-dimensional hyperbola, given two points on the hyperbola, what is the length of the path between ...
2
votes
0answers
43 views

Summation of series involving $\sinh$ of a square root

In a physics problem that I assigned in one of my classes recently, the following series arises: $$ S = \sum_{\text{odd } n} \sum_{\text{odd } m} \frac{(-1)^{(n+m)/2}}{nm} \frac{\sinh( \pi \sqrt{n^2 + ...
2
votes
0answers
36 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 \frac{1}{\...
2
votes
0answers
134 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 \frac{e^{...
2
votes
0answers
425 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
60 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
87 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
29 views

Conformal Map from Domain $D$(where $D$ is bounded by $\{z=x+iy,y=1/x,x>0\}$, $2+2i\in D$) to upper Half Plane

I am trying to find the Conformal Map from Domain $D$(where $D$ is bounded by $\{z=x+iy,y=1/x,x>0\}$, $2+2i\in D$) to upper Half Plane $\mathbb{H}:=\{w:Im(w)>0\}$, I am using $z+1/z$ map but ...
1
vote
0answers
34 views

Integral (Tanh and Normal)

I am trying to evaluate the following: The expectation of the hyperbolic tangent of an arbitrary normal random variable. $\mathbb{E}[\mathrm{tanh}(\phi)]; \phi \sim N(\mu, \sigma^2)$ Equivalently: $...
1
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0answers
21 views

Reduction of expression algebraically

I have asked this question before and it helped me get a little further, but not at a solution. I have to algebraically reduce the expression: $\sinh(2 \cdot \sinh^{-1}(y))$ Now i had the idea of ...
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 $a=\frac{g^2+1}{...
1
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0answers
69 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 temperature,...
1
vote
0answers
36 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
vote
0answers
57 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
vote
0answers
96 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 I'...
1
vote
0answers
55 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 z}D_z\left[\tan^{-1}\left(\frac{\theta(z/2,2)}{\vartheta(z/2,2)}\right)\...
1
vote
0answers
99 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 ...
1
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0answers
40 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
50 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
262 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
37 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 $w=\cosh^{-1}{z}$...
1
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0answers
21 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}& \frac{k}{\...
1
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0answers
175 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
71 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? $$\frac\pi2+8\sum_{n=1}^\infty\...
1
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0answers
114 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
64 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
94 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
189 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
53 views

Hyperboloid equation related question?

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

Complex conjugate of the logarithm of the hyperbolic tangent

Given the Schwarz reflection principle, I would aytomaticaly write down that the complex conjugate of the following function: $$ ln[tanh(z)] $$, where z is a complex number, is: $$ln[tanh(\bar{z})] $$....
0
votes
0answers
39 views

Simplifying $\frac{2\sinh^2 x}{(\cosh x+1)^3}-\frac{\cosh x}{(\cosh x+1)^2}$

Could you explain me how we simplified this trigonometric expression? $$\frac{2\sinh^2 x}{(\cosh x+1)^3}-\frac{\cosh x}{(\cosh x+1)^2}\qquad\to\qquad\frac{\cosh x -2}{(\cosh x +1)^2}$$ Thanks.
0
votes
0answers
29 views

Infinite sum of a product of hyperbolic functions, help!!

Let $g_{a,b}=\mathrm{csch}(n(a-b))$ when $a$ is different from $b$ and $0$ if $a=b$. $n$ is a positive real. I am trying to compute the following sum \begin{equation} \sum_{k=0}^{\infty}(2k-a)g_{0,k}...
0
votes
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
29 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 $$ _2F_1(a,b;c;z)=\sum_{n=0}^{\...
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 }{dt}...
0
votes
0answers
15 views

implicit derivation of hyperbolic functions

derive $$\sinh(x+y)=\tanh^{-1}(\frac{x}{y})$$ $$\cosh(x+y)\cdot(1+f')=\frac{\frac{y-xf'}{y^2}}{1-(\frac{x}{y})^2}$$ $$\cosh(x+y)\cdot(1+f')=\frac{\frac{y-xf'}{y^2}}{\frac{y^2-x^2}{y^2}}$$ $$\cosh(...
0
votes
0answers
28 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 $j : \mathbb{R} \to \mathbb{R}$ $$\frac{\partial y}{\partial x_2} + \frac{\partial (j \circ y)}{\partial x_1} = 0$$