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

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A Method For Calculating Large Exponents Quickly

I've derived a formula for calculating large exponents quickly: $$a^b = 2 \cosh( - b \log( a ) )$$ My question is: Has anyone seen anything similar? I am curious if either it's novel OR if I have ...
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34 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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28 views

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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6 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 ...
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0answers
23 views

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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1answer
32 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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3answers
66 views

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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2answers
364 views

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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4answers
213 views

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
16 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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59 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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49 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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2answers
81 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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35 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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1answer
38 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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1answer
42 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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180 views

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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2answers
67 views

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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1answer
96 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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1answer
48 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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2answers
37 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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2answers
20 views

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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1answer
32 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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1answer
50 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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38 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 ...
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1answer
27 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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1answer
70 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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1answer
53 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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Integral Evaluation: Exponential of and Hyperbolic Function

I'm trying to evaluate $$G^{\pm} = \frac{-i}{8\pi^2 X} \partial_X \int_{-\infty}^\infty d\phi e^{i m \left[X \sinh \phi \pm T \cosh \phi \right]}$$ for $T = \pm X$. Where $T, X, m \in \mathbb{R}$ ...
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1answer
64 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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1answer
35 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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2answers
37 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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1answer
43 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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246 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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1answer
40 views

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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2answers
38 views

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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1answer
40 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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2answers
46 views

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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1answer
57 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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85 views

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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58 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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1answer
87 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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1answer
101 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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516 views

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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44 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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1answer
139 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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47 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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1answer
71 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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1answer
109 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 ...