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

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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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here is a theorem that the isometries of the Hyperbolic plane are generated by $PSL(2, \mathbb{R})$ and $z \rightarrow - \overline{z}$.

There is a theorem that the isometries of the Hyperbolic plane are generated by $PSL(2, \mathbb{R})$ and $z \rightarrow - \overline{z}$. My question is, isn't $z \rightarrow kz$ an isometry for ...
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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 ...
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Proof that function is real part of $\sec(z)$ [duplicate]

I'm working on the following problem: I've deduced that the key is to show that $u$ is the real part of $\sec(z)$. But, I'm getting stuck in the algebra and am hoping someone can point me in the ...
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43 views

Proof that $\cos^2(x)\cosh^2(y) + \sin^2(x)\sinh^2(y) = -1 + \sin^2(x) - \sinh^2(y)$

Could anyone offer a proof that $$ \cos^2(x)\cosh^2(y) + \sin^2(x)\sinh^2(y) = -1 + \sin^2(x) - \sinh^2(y)? $$
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How does $\frac{1}{2}\cosh(2x) -1 = \sinh^2(x)$?

Using hyperbolic trigonometric function identities is there a way to prove the following equation? $$\frac{1}{2} (\cosh(2x)-1) = \sinh^2(x)$$
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Calculate cosh(x) given sinh(x)

Given the value of sinh(x) for example sinh(x) = 3/2 How can I calculate the value of cosh(x) ?
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What is the use of hyperbolic trigonometric functions if they are easily expressible algebraically?

I get that there are uses for $\sin(x)$ and $\cos(x)$ because they are defined with imaginary exponents which aren't as easily worked with but the hyperbolic functions are simply $\frac12(e^x\pm ...
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1answer
25 views

rotated hyperbolic cylinder parameterization

A hyperbolic cylinder is given by $\frac{x^2}{a^2}-\frac{y^2}{b^2} = -1$, but thats a hyperbolic cylinder that goes along the Z-axis. How do you parametrize a hyperbolic cylinder that goes instead of ...
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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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Prove that $\lim\limits_{x \to 0} \sinh(x)/x =1$.

Prove that $ \lim\limits_{x \to 0} \frac{\sinh x}{x} =1.$ I am having some trouble proving this without derivative. Some help would be much appreciate!
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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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351 views

Hyperbolic function identity proof?

On a question i am working thru it says: Obtain the formula:$$ \sinh 2x - \sinh 2y = 2\cosh(x+y)\sinh(x-y) $$and prove that $$\coshθ + \cosh2θ +...+\cosh nθ ...
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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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1answer
44 views

How to trace the graphic of $\cos(x) + \cosh(y) = k$?

Is there some systematic way to trace the graphic of $\cos(x) + \cosh(y) = k$ given a fixed value for $k$? Suppose $k = 1$: if I choose empirically $y = 1.2$, I know that should be $\cos(x) = - ...
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43 views

Exponential function to prove [closed]

how would you prove that $Ae^x+Be^{-x}=A \sinh x+B\cosh x$ Thank you.
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Evaluating the following integral: $ \int \frac{x^2}{\sqrt{x^2 - 1}} \text{ d}x$

For this indefinite integral, I decided to use the substitution $x = \cosh u$ and I've ended up with a $| \sinh u |$ term in the denominator which I'm unsure about dealing with: $$\int ...
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Why are hyperbolic functions significant?

I'm currently covering Stewart's Early Transcendentals, and there is a whole section dedicated to defining and differentiating hyperbolic functions. The same amount of space is used to cover other ...
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105 views

Questions about the hyperbolic system of equations

$$u_t+A(x,t,u)u_x=b(x,t,u) \tag 1$$ $$u=(u_1, \dots, u_n), b=(b_1, \dots, b_n)$$ $$A=[a_{ij}], i,j = 1, \dots, n$$ $$$$ We set the question if there are characteristic directions at the path of which ...
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56 views

Hyperbolic equations-characteristic system

Let the system: $$\alpha(x,t,u)u_t+\beta(x,t,u)u_x=f(x,t,u)$$ To find the characteristic equations: $$\frac{du}{ds}=\frac{\partial{u}}{\partial{t}} \frac{dt}{ds}+\frac{\partial{u}}{\partial{t}} ...
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44 views

Find $\sinh x$ in terms of $\tanh x$.

Given that $\tanh(x) = u$, find an expression for $\sinh(x)$ in terms of $u$. I don't really know what the question wants from me here. Any help would be great.
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Find the normalisation constant

I am having problems finding the normalisation constant $N$. I have tried this so far use the substitution $x=a tan(u)$ so $dx=a sec^2(u)du$, so $\displaystyle 1=\int_{-\infty}^{\infty}N^2 ...
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66 views

Hyperbolic growth, deriving from hyperbolic functions

When a quantity grows towards infinity in a finite-time, it is said to undergo hyperbolic growth. An example being a quantity that every time it doubles, the growth rate itself also doubles. ...
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60 views

Infinite Product Identity for Hyperbolic Sine

Prove $\prod_{n\in\mathbb{N}\backslash\left\{ 0\right\} }\left(1+\left(\frac{\alpha}{\pi n}\right)^{2}\right)=\frac{\sinh\left(\alpha\right)}{\alpha}$. I saw this formula in a book and have no idea ...
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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 ...
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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 ...
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79 views

given $\cosh u = x$ find $\sinh u$

I'm asked to show that:$\newcommand{\arcosh}{\operatorname{arcosh}}$ $\int{x \arcosh x}dx = \frac{1}{4}(2x^2 -1)\arcosh x - \frac{1}{4}x\sqrt{x^2 -1} + C$ If I integrate by parts: let $u = \arcosh ...
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63 views

Limit of a Cosh function

Evaluate $$\lim_{t\to\infty} (\cosh x)^{1/x}.$$ I tried to use L'Hopital's but I think I made a mess of the differentiation, and the differentiation doesn't seem like it'll help much.
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83 views

Laplace's equation-separation of variables

I am looking at the $2$-D Laplace's equation $$\nabla^2u=u_{xx}+u_{yy}=0$$ $$u(x,0)=f(x), x \in (0,a)$$ $$u(x,b)=0, x \in (0, a)$$ $$u(0,y)=u(a,y)=0, y \in (0,b)$$ The solution is in the form ...
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147 views

Definition of hyperbolic cosine and its relation with exponential function

Ordinary trigonometric functions are defined independently of exponential function, and then shown to be related to it by Euler's formula. Can one define hyperbolic cosine so that the formula ...
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40 views

Disappearing negative signs when evaluating a sinh^-1 integral

$$\int_{-2}^{6} \frac{1}{\sqrt{1+(-x)^2}} \, dx$$ When performing this integral on paper, I get $$\sinh^{-1}(6) - \sinh^{-1}(-2) $$ But when I type it on wolframalpha, I get the unintuitive answer ...
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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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48 views

Inverse trig and trigh in integration?

I have just done part (iii) of this question and can get the right answer but am a bit confused why do we take arcosh i.e. just the principle value of cosh and not the other value. I presume this is ...
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665 views

Separate Into Real and Imaginary Parts

Separate the following trigonometric function into Real and Imaginary Parts $$\tan^{-1}e^{i\theta} $$ or $$\tan^{-1}(\cos\theta+i\sin\theta)$$ I Have made till here Assuming $x+iy$ is the final ...
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Find the value of hyperbolic $\tanh x$ function from the equation

If $\sinh x-\cosh x=5$, find $\tanh x$ I have done till the following steps but dont know how to proceed further from solving this equation in Euler's form ...
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Does the definite integral of (1 - tanh t) from 0 to x diverge as x goes to infinity?

Decidedly in the category of things I used to know how to prove but have forgotten: Does $$ \int_0^x (1 - \tanh t) \,dt $$ converge or diverge as $x \to \infty$? (I know that the indefinite ...
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45 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 ...
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Is this a typo, or am I missing something?

I have a handout for my precalc II class. It says $\sinh(-x) = -\sin(x)$ It should be $\sinh(-x) = -\sinh(x)$ right? I don't see how a negative input could make a hyperbolic function circular.
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90 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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Calculate ch(0.2) to the nearest 0.01

Help me calculate ch(0.2) to the nearest 0.01. I tried to rewrite ch as a series but I still don't know how to evaluate it and what to do with factorial Help me please. it's very important
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I cannot find the following integral in an integral table.

In the appendix A of this paper there is an integral that the author says can be solved using any good integral table. However I cannot seem to find it on any integral table (ex: gradshteyn and ...
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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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106 views

How do I solve this integral with hyperbolic functions?

I was studying mechanics when I f ound a problem that lead to an integral that I can't solve. Basically the problem asked to find the period of oscillation function of the energy $E$ of a particle ...
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217 views

Problem with Taylor (asymptotic) expansion of hyperbolic functions at infinity

(Note: I chose a general title, because I believe this discussion will be applicable to all other hyperbolic functions having an asymptote at infinity, but I will specifically be focusing on ...
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Geometric meanings of hyperbolic cosinus and sinus

In euclidean geometry, $\cos$ and $\sin$ are used for angles in trigonometry. Is there an equivalent for $\cosh$ and $\sinh$ the hyperbolic cosine and sine, and not cosinus and sinus ?
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73 views

Write the following as an algebraic expression of x; sinh(lnx)

So I have a questions asking for $sinh(log_ex)$ sinh(x) = $\frac{e^x-e^{-x}}{2}$ So $sinh(log_x)$ = $\frac{e^{lnx}-e^{-lnx}}{2}$ $\frac{e^{lnx}-e^{lnx^{-1}}}{2}$ = $\frac{x - x^{-1}}{2}$ ...
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208 views

Maximum value of tanh Z

$$ f(z) = \tanh (z) = \dfrac{e^z - e^{-z}}{e^z + e^{-z}} $$ Find the point $z$ with $|z| \leq1$ where $|f(z)|$ attain its maximum. I figured out that the maximum is probably at the edge (concluded ...
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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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90 views

definite integral involing hyperbolic and trigonometric functions

Trying to prove the following: $$ \int_0^\infty xe^{-c x^2}\sinh(a x)\cos(bx)\,dx = ...
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133 views

Weierstrass $ \tanh \frac{\theta}{2} $ substitution confusion.

I'm already familiar with the trigonometric version of this substitution $ t = \tan \frac{\theta}{2} $ and it's geometrical derivation involving the unit circle found here. However, I'm not sure how ...