Questions tagged [hyperbolic-functions]

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

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Books for learning Hyperbolic Dynamical Systems and differentiable manifolds

I am looking for some books/lectures that cover Hyperbolic Dynamical systems and supplemental materials that cover the very basics of differentiable manifolds, enough to understand everything relevant ...
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Is there a hyperbolic variant of the sine-cosine fourier series?

The sine-cosine form of the fourier series is given by: $$ s_{\scriptscriptstyle N}(x) = \frac{a_0}{2} + \sum_{n=1}^N \left(a_n \cos\left(\tfrac{2\pi}{P} nx \right) + b_n \sin\left(\tfrac{2\pi}{P} nx \...
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Derivation of $\operatorname{tgh}^{-1}(x)=\tanh^{-1}(x)=\operatorname{arctanh}(x)={1\over 2}\left(\ln(1+x)-\ln(1-x)\right)$

I think I posted dupe post. $$\begin{align} \color{red}{\tanh^{-1}(x)={1\over 2}\ln\left({1+x\over 1-x}\right)={1\over 2}\left(\ln(1+x)-\ln(1-x)\right)}~~\text{with}~~\left(\left|x\right|<1\right) \...
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2 votes
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Approximate value of hyperpolic tangent in certain case

Reading this interesting Book ( thé nature of magnetism) , I came across a particular approximation of hyperpolic tangent, while in first case T bigger than Tc , it is just Taylor series, in case T ...
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7 votes
2 answers
125 views

Question regarding Solutions to the equation $y'' + y = y^3$

A while ago I had a question given to me by a tutoring student that read as follows: A solution to the differential equation $y'' + y = y^3$ is: A) $y = \tanh(x/\sqrt{2})$ B) $y = \tanh(x\sqrt{2})$ C)...
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2 answers
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Why doesn't this approach to deriving the hyperbolic functions work?

This question may be the closest to what I'm looking for, but the link in the answer uses a CAS to complete the proof. I was thinking I could derive the formulas for $\sinh{x}$ and $\cosh{x}$ using a ...
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Complex fourier expansion of $\cosh(t-1)$ in a given period,

So I attempted this question 'An even function 𝑓(𝑡) is periodic with period $𝑇 = 2$ and $𝑓(𝑡) = \cosh(𝑡 − 1)$ for $0 ≤ 𝑡 ≤ 1$. Find a complex Fourier series representation for 𝑓(𝑡).' my ...
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1 vote
1 answer
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Decompose transmission line matrix as hyperbolic rotation and other simple transformations?

Transmission lines are often modelled using a two-port relationship $$ \begin{bmatrix} V_2 \\ I_2 \end{bmatrix} = \begin{bmatrix} \cosh(x) & Z \sinh(x) \\ \frac{1}{Z} \sinh(x) & \cosh(x) \end{...
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Is it the only solution $\sinh(u-\xi+\eta)=\sinh(\eta)\rightarrow \eta=\pm\frac{i\pi}{3} \hspace{0.3cm} \& \hspace{0.3cm} u-\xi = \pm\frac{i\pi}{3}$

I am working on a problem that maps the 6 vertex model in statistical physics to alternating sign matrices. The main idea is that there is a one to one correspondence between alternating sign matrices ...
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1 vote
1 answer
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Solving $\frac{4}{\pi}\arctan{(1 + x)}\cosh{x} = 1$

This equation was part of a bigger calc question for a weekly assignment. $$\frac{4}{\pi}\arctan{(1 + x)}\cosh{x} = 1$$ I found the solution to be $x = 0$ by inspection, but was wondering if this is ...
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Find parameter to catenary interpolate a specific point

I'm working with the catenary equation and this equation is given by $$ f(x) = a \cdot \cosh\left(\dfrac{x}{a}\right) $$ I know this function pass at the point $(x_0, \ y_0)$ and therefore I want to ...
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Expansion of $\sinh^{-1}$ at $-\infty$

Excuse me if I'm being dense, but how do you derive $$\lim_{x \to -\infty} \sinh^{-1} (x) = -\infty$$ I have $$ \sinh^{-1}(x) = \log \left(x + \sqrt{x ^ 2 + 1}\right) = \log \left(x \left(1 + \...
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1 answer
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Is $ \frac {1}{\sqrt{a^2-b^2}} $ = $\cosh^{-1}(\frac{a}{b})$? [closed]

Is $ \frac {1}{\sqrt{a^2-b^2}} $ = $\cosh^{-1}(\frac{a}{b})$ is this true or false?
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2 votes
2 answers
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How to solve a hyperbolic (cosh) equality when the argument of the cosh function is different

if $ \alpha+\lambda = c \cosh( \frac{a+d}{c})$ , and $ \alpha + \lambda = c \cosh( \frac{-a+d}{c})$ how does one get that $α + λ = c \cosh(\frac{a}{c})$ and $d = 0$ is there a rule that im missing? ...
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Real analysis - Showing that Sinh(x) is strictly increasing with the use of logarithmic hyperbolic formulas

I need to show that $\sinh{x}$ is strictly increasing on all of $\mathbb{R}$ that $\cosh(x)$ is stricly increasing on $[0,\infty)$ and strictly decreasing on $(\infty,0]$ I need to do it through the ...
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How to compute tanh(X) where X is a matrix?

According to the definition of $\tanh(x)$ on a scalar, we have $\tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} = \frac{2}{1 + e^{-2x}} - 1$. Now if X is a matrix instead of a scalar, then is it true ...
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3 votes
2 answers
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Can we be certain that the only nonzero differentiable function satisfying $g(x+y)=\frac{g(x)+g(y)}{1+g(x)g(y)}$ and $|g|\lt 1$ is $\tanh(kx)$?

The following question is a question aimed for the Further Maths UK syllabus (it is a Step $2$ question)- i.e. not very formal, so I am certain my solution to the question is correct as the question ...
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How to prove that the graph of function $y=\frac{x}{\sqrt{3}}+\frac{1}{x}$ is a hyperbola? [closed]

How to prove that the graph of function $y=\frac{x}{\sqrt{3}}+\frac{1}{x}$ is a hyperbola? Actually,I want to know this question can be proved in polar coordinates with rotation? Thanks a lot
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3 votes
1 answer
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Approximate solution of $\sinh ^{2 r}(z)+\cosh ^{2 r}(z)=a$

Making this question more general : find the zero of function $$\color{blue}{f(x) = 1 + x^r - a(x-1)^r }\qquad \text{with}\quad a >2\quad \text{and}\quad r >2 $$ Its maximum value is attained at ...
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1 answer
61 views

Bounds or value of expectation of $\mathrm{sech}(a X)$ where $X$ is Gaussian?

I would like to compute the following integral $$ f(a) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty \mathrm{sech}(ax)~e^{-x^2/2} \, dx, \quad a > 0. $$ It is the expectation of $\mathrm{sech}(aX)$...
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How to rewrite $f\left(x\right)=\frac{A^{2}e^{\frac{1}{2}x}+4e^{-\frac{1}{2}x}}{2A}$ in terms of hyperbolic functions

I was solving for the function whose surface of revolution equals the volume of the solid of revolution. I ended up with two functions $f(x)=0$ and $$f\left(x\right)=\frac{A^{2}e^{\frac{1}{2}x}+4e^{-\...
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1 vote
0 answers
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Limits of integral over a region after substitution

I have the integral of a function $f(x,y,z)$ defined over the region $a<\sqrt{x^2-y^2-z^2}<b$; that is, $$I\equiv \int_{a<\sqrt{x^2-y^2-z^2}<b}dxdydzf(x,y,z).$$ I realized that defining ...
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Prove $\sum_{n=0}^{\infty} (-1)^n \frac{\text{sech}\left((2n+1)\frac{\pi}{2}\right)}{2n+1} = \frac{\pi}{8}$ [duplicate]

Let $$f(x) = \sum_{n=0}^{\infty} (-1)^n \frac{\text{sech}\left((2n+1)x\right)}{2n+1} $$ which converges for all $x\in \mathbb R$. How can it be shown that $f\left(\frac{\pi}{2} \right) = \frac{\pi}{8}$...
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Finding $ \int_0^x \frac{\sinh^{-1}\left(\alpha\tanh(kx)\right)}{\sinh^{-1}⁡(k)} dx $

I'm trying to solve this equation but not quite sure which calculation method to use. $$ \int_0^x \frac{\sinh^{-1}\left(\alpha\tanh(kx)\right)}{\sinh^{-1}⁡(k)} dx $$ where $\alpha$ and $k$ are just ...
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1 vote
4 answers
102 views

Proving that $\sqrt{x^2+1}+x>0$ for all $x$

Recently while dealing with inverse hyperbolic functions, I came across the expression $$\sinh^{-1}x=\ln(x+\sqrt{x^2+1})$$ We know that $f(x)=\sinh^{-1}x$ is defined for all real values of $x$ since ...
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Is there a graph that from any given interval a to b, the area under the graph is equal to the arc length (of the graph in the same interval) squared?

After seeing how cosh(x) has the property of the area under cosh(x) from an interval a to b equals the arc length above that area, I wondered if there was a graph with similar properties. Arc length, ...
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2 answers
70 views

The integral of $\frac{x}{16-x^4}$ [closed]

$$\int\frac{x}{16-x^4}\;dx$$ $\because$ $\frac{d}{dx}(\coth^{-1}\frac{x}{a}$)$=\frac{1}{a^2-x^2}$ ,And also $\frac{d}{dx}(\tanh^{-1}\frac{x}{a}$)$=\frac{1}{a^2-x^2}$ $\therefore$ If we say that $u=x^2$...
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0 votes
1 answer
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Parametrising the intersection of a double cone and a plane?

As the title states, I am struggling to parametrise the hyperbola resulting from intersection of a double cone and a plane. The equation of the cone is given as $z^2=x^2+y^2$ and the equation of the ...
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2 answers
45 views

To simplify the logarithmic expression and possibly find its inverse.

While solving an equation, I came across an unexpectedly symmetric solution. $$\frac{\log \left(\frac{a r+h}{\sqrt{(a r+h)^2-1}}+1\right)}{2 a}-\frac{\log \left(1-\frac{a r+h}{\sqrt{(a r+h)^2-1}}\...
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4 votes
1 answer
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How to prove $\frac{d}{dx}\sinh x=\cosh x$ when $\sinh$ and $\cosh$ are defined by an integral?

Define $\sinh$ and $\cosh$ by $$x=\int_0^{\sinh x}\frac{dt}{\sqrt{t^2+1}},\, x\in\mathbb{R}$$ $$x=\int_1^{\cosh x}\frac{dt}{\sqrt{t^2-1}},\, x\ge 0$$ and define $\cosh (-x)=\cosh x$ for $x\lt 0$. By ...
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2 votes
2 answers
112 views

Using addition formulas brings weird results

So, I have this set of equation that I want to solve for x,y $$ A = \tanh(x+y) \\ B = \tanh(x-y) $$ and of course it can be solved by inverting and then summing/differencing, so I get $$ x = \dfrac{1}{...
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2 votes
1 answer
134 views

The value of the sum $\sum_{n=1}^\infty \frac{1}{n\sinh(\pi n /4)}$

Can we compute the exact value of the sum $$S = \sum_{n=1}^\infty \dfrac{1}{n\sinh(\pi n /4)}.$$ WolframAlpha spits out $S = 1.4667$. But I have no clue how it obtains this and I suspect this may be ...
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How to show that $\frac{\sinh(b)}{\sinh(a)} \leq \frac{b}{a}$ for $a\geq b>0$.

I want to prove that $\frac{\sinh(b)}{\sinh(a)} \leq \frac{b}{a}$ where $a\geq b>0$ and $a,b \in \mathbb{R}$. I have tried to set $a = bx$ for $x\geq 1$ but did not manage to show that the ...
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1 vote
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Relation simplification between hyperbolic and ordinary trigonometric functions

Classically, we have the relation between cartesian and cylindrical coordinates given by: \begin{align} x &= r\cos(\theta),\\ y &= r\sin(\theta), \end{align} such that we have: \begin{equation}...
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1 vote
1 answer
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Identity for $1/z$, $\cot(z)$, and $\coth(z)$.

I'm trying to verify a seemingly simple identity that I encountered in a paper. After discarding some irrelevant scale factors it boils down to the following. It comes in three variants, $$ \alpha(z) =...
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1 vote
1 answer
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Arranging $\int_{0}^{\infty}\frac{\sin(x)}{\sinh(x)}dx, \int_{0}^{\infty}\frac{\sin(x)}{\cosh(x)}dx, \int_{0}^{\infty}\frac{\cos(x)}{\cosh(x)}dx$

Let $$I_1=\int_{0}^{\infty}\frac{\sin(x)}{\sinh(x)}dx,$$ $$I_2=\int_{0}^{\infty}\frac{\sin(x)}{\cosh(x)}dx,$$ $$I_3=\int_{0}^{\infty}\frac{\cos(x)}{\cosh(x)}dx.$$ Which of the following is true? $\...
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6 votes
2 answers
120 views

How do I solve for y in $y=\text{tanh}(\frac{x}{y})$?

I want to solve the following equation as a function of purely $x$: $$y=\text{tanh}\left(\frac{x}{y}\right)$$ My best guess up to this point has been to rearrange the equation using inverse hyperbolic ...
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0 votes
2 answers
72 views

Inverse of $\tanh^{\prime\prime}(x)$ [closed]

I have problems finding the inverse function of the second derivative of the hyperbolic tangent. I know it is not invertible on the whole of $\mathbb{R}$, but having a closed form for the inverse on, ...
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0 votes
2 answers
52 views

Solve the equation $xy=x+2y+2009$ in integers

I know that the left side is a hyperbola and the right hand side is a line. So they have at most 2 solutions. I set $xy=k$ and solved for $y$, and after that substituted it on the right side. The ...
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2 votes
1 answer
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Hyperbolic functions as polynomials

I have recently found that the change of variable $t\to 2 \arctan (t)$ makes $\cos(2 \arctan (t)) = \dfrac{1-t^2}{1+t^2}$ and $\sin (2\arctan (t) ) = \dfrac{2t}{1+t^2}$ for certain values of $t$. I ...
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5 votes
2 answers
230 views

Why does $\text{arctanh}(2^{-k})$ approach powers of $2$?

This is from a piece of verilog code I generated for $\text{arctanh}(2^{-k})$: ...
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4 votes
3 answers
90 views

Faster way to find the first four non-zero terms of the Maclaurin series for $\frac{1-x}{1+x}\cosh\sqrt{x}$

I want to find the first 4 non-zero terms for : $$\frac{1-x}{1+x}\cosh\sqrt{x}$$ Before expanding, I rewrite this as $$(1-x)\left(\frac{1}{1+x}\right)\cosh\sqrt{x}$$ Then I expand to get $$(1-x)\left(...
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0 votes
1 answer
32 views

Solve function involving cosh for x

I need help solving the following function for $x$ $$g(x) = x - x \cdot \cosh\left(\frac{1}{2x}\right)$$ As I have never used hyperbolic functions, all my attempts at solving this have failed ...
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0 votes
1 answer
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Integration using $x = 2\cosh u$

I'm working on the problems in this booklet: https://mcs-notes2.open.ac.uk/files/MScDiagnosticquiz.pdf In question 1.2.1(f) the integration is: $$\int_{2}^{3}\frac{x+1}{\sqrt{x^2-4}} \,dx$$ Later on ...
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1 vote
2 answers
83 views

Area Under Unit Hyperbola?

Going through Strang's Calculus right now and don't understand a seemingly basic homework question. It asks to integrate under the unit hyperbola, from $(1,0)$ to $(\cosh t, \sinh t)$. The answer in ...
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29 views

Show $\coth(m\theta)\left(\coth(m\theta)-\frac{1}{m}\coth(\theta)\right)\geq \frac{1}{3}\frac{m^2-1}{m^2}$

I would like to show that the function $$ f_m(\theta)=\coth(m\theta)\left(\coth(m\theta)-\frac{1}{m}\coth(\theta)\right), $$ with $m\in\mathbb{N}$, satisfies for $\theta>0$ $$ f_m(\theta)\geq \lim_{...
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  • 101
11 votes
3 answers
431 views

Number of zeros of $f(x)= \frac{1}{2} E\left[ \tanh \left( \frac{x+Z}{2} \right) \right]-\tanh(x)+\frac{x}{2}$ where $Z$ is standard normal

Consider the following function: \begin{align} f(x)= \frac{1}{2} E\left[ \tanh \left( \frac{x+Z}{2} \right) \right]-\tanh(x)+\frac{x}{2}, \end{align} where $Z$ is standard normal. Question: How to ...
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1 vote
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What does the golden ratio have to do with complex hyperbola and real circle

If you have $xy = i $ and $x^2 + y^2 = 1$ then you get the solutions that have the golden ratio in them. These are the solutions Wolfram calculation: https://www.wolframalpha.com/input/?i=xy%3Di%2C+x%...
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0 votes
1 answer
63 views

Why are the domain and image of $F(x) = \sqrt{1-\cosh(x)}$ only $\{0\}$? [closed]

I came across this weird function $$F(x) = \sqrt{1-\cosh(x)}$$ When you study the range of his composite function, you will find that the domain is $\mathrm{D}(F): x = \{0\}$ and the image is $\mathrm{...
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1 vote
0 answers
157 views

Improper Integral Involving Hyperbolic Cotangent

I am trying to evaluate the following integral: $$\int_0^\infty \frac{\frac{1}{x}-\pi\coth(\pi x)}{x^2+4}dx$$ I'm not sure if a closed-form exists, so far I only know the decimal approximation to be $\...
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