# Tagged Questions

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

104 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 ...
152 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 ...
87 views

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 ...
449 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 ...
110 views

37 views

428 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 ...
61 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 ...
89 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 ...
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 ...
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 ...
38 views

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,...
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}...
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 ...
59 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 ...
98 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'...
56 views

121 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 ...
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 ...
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, \phi(x) = \pm 4 \tan^{-1}\left[e^{\frac{x-t ...
190 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 ...
53 views

### Hyperboloid equation related question?

How to draw this graph please? $$4y^2 -x^2+4z^2-1 \geq 0$$
28 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: $$ln[tanh(\bar{z})]$$....
41 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.
30 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 \sum_{k=0}^{\infty}(2k-a)g_{0,k}...
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 ...
29 views

29 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
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 $j : \mathbb{R} \to \mathbb{R}$ $$\frac{\partial y}{\partial x_2} + \frac{\partial (j \circ y)}{\partial x_1} = 0$$
19 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: $$Ax^2+By^2+Cz^2=1$$ A(-1)^2+B(-4)...