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

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5
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346 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 ...
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2answers
50 views

Taking limit with hyperbolic functions

I have a problem with evaluating $$\sinh^{-1}(C \sinh (ax))\bigg|_{-\infty}^{+\infty}$$ where $C$ and $a$ are real positive constants.
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2answers
46 views

Complex Numbers and Hyperbolic Functions

How would you evaluate: $\mathfrak{R}\left[(1+i)\sin\left(\dfrac{(2+i)\pi}{4}\right)\right]$? I know that $\cos x = \dfrac{e^{ix}+e^{-ix}}{2}$ and $\sin x = \dfrac{e^{ix}-e^{-ix}}{2i}$. I have also ...
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2answers
150 views

Evaluating a sum with $\cosh$

In my integration adventures, I ran into this sum: $$\sum_{n=1}^{\infty}\frac{1}{\cosh(\pi an)(4n^{2}-1)}$$ I know that $\sum_{n=1}^\infty \frac{1}{\cosh(\pi n)}$ has a nice closed form, so I was ...
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2answers
26 views

Tangent - point of contact

Question: Tangent to the curve $y = x^2 + 6$ at point P(1, 7) touches the circle $x^2 + y^2 + 16x + 12y + c = 0$ at a point Q. Then the coordinates of Q are: 1) (-6, -11) 2) (-9, -13) 3) (-10, -15) 4) ...
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2answers
94 views

Differentiate $y=\cosh^{3} 4x$.

Differentiate $y=\cosh^{3} 4x$. $$\frac{dy}{dx} = 3 \cosh^{2} (4x) \sinh (4x)\cdot 4$$ These are the parts that I don't quite understand: \begin{align*} \frac {dy}{dx} &=12 \cosh^{2} ...
6
votes
1answer
83 views
+50

Finding taylor expansion for $\tanh(x)$

I am a high school student and am trying to find the taylor expansion of $\tanh(x)$ in terms of a summation form. I have gotten this far, and am aware it might get complicated very quickly. If someone ...
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1answer
83 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 ...
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1answer
43 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
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1answer
28 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
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1answer
44 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 ...
0
votes
1answer
73 views

How to find the angle subtended to the origin by the unit hyperbola through the point (1,0)?

I'm trying to find the angle subtended by the unit hyperbola through the point (1,0). I think that I should be integrating something, but I'm not sure how to set it up. I've been trying to think of ...
0
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1answer
231 views

Find area of a curvilinear triangle that includes hyperbolic functions

We were given this question in class and I tried to compute it and it looks to be pretty crazy. Can anyone take a look and let me know if I did it correctly? I would really appreciate it. ...
6
votes
0answers
90 views

Solving a problem of Ramanujan's interest

I am Brian Diaz, and I am new to the math.stackexchange community. I have been struggling with attempting to find a closed form of the following series: $$ \varphi(\theta) = 1 + 2\sum_{n=1}^{\infty} ...
5
votes
0answers
63 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 ...
4
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0answers
51 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
275 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
263 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
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0answers
50 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
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0answers
77 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
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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 J├╝rgen Jost. Right now I'm looking at an exercise (12.5) under the ...
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0answers
14 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}& ...
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0answers
77 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 ...
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0answers
50 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? ...
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0answers
65 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 ...
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0answers
58 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 ...
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0answers
84 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 ...
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0answers
148 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 ...
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0answers
46 views

Hyperboloid equation related question?

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

limit of a tricky multi-variable function

I am reading a paper on the sine-gordon model. The solution for a two kink solution is given as: ...
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0answers
16 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 ...
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0answers
30 views

$d/dx((\sinh^{-1}(\tan x)))$

I'm just wondering if I did everything correctly for this question, I know the answer is correct but I'm not 100% sure the steps I took to get there are valid: \begin{align} d/dx(\sinh^{-1}(tanx)) ...
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0answers
27 views

How to find z of cosh(z) = -2 & choosing value

$\cosh(z) = -2$ $z = \cosh^{-1}(-2)$ $z = \ln(-2 \pm i\sqrt{4-1})$ $z = \ln(-2 \pm \sqrt{3}) $ -> I wolfram this and it choose only $-2 - \sqrt{3}$ not $-2 + \sqrt{3}$ I would like to know what is ...
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0answers
74 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 ...
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0answers
10 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
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0answers
67 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 ...
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0answers
56 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 ...
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0answers
49 views

Verifying the triangle inequality for a metric for hyperbolic space

I read that the formula $d(x,y)=\mathrm{arccosh}(1+\frac{2||x-y||^{2}}{(1-||x||^{2})(1-||y||^{2})})$, where $x,y$ are in the open unit ball of $\mathbb{R}^{n}$ and $||\cdot||$ denotes Euclidean norm, ...
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0answers
31 views

How are the sine functions along with the hyperbolic functions visualized with imaginary rotations?

Since we know that: cos(t)=cosh(it) and isin(t)=sinh(it) I've been thinking about this, and obviously this is referring to how if you move at a right angle from a circle on a conic section, you end ...
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0answers
583 views

Solving system of equations with hyperbolic functions

Do you maybe know how to solve a system of equations with hyperbolic functions? Imagine the problem of the form: $$ x=\textrm{sech}(x^2+y^2) \\ y=1-\textrm{sech}^2 (x+y) $$ Any ideas how to solve it ...