# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 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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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? $\... 6 votes 2 answers 120 views ### How do I solve for y in$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 ... 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, ... 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 ... 2 votes 1 answer 36 views ### 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 ... 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$\text{arctanh}(2^{-k})$: ... 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 $$(1-x)\left(... 0 votes 1 answer 32 views ### Solve function involving cosh for x I need help solving the following function for x$$g(x) = x - x \cdot \cosh\left(\frac{1}{2x}\right)$$As I have never used hyperbolic functions, all my attempts at solving this have failed ... 0 votes 1 answer 43 views ### Integration using x = 2\cosh u I'm working on the problems in this booklet: https://mcs-notes2.open.ac.uk/files/MScDiagnosticquiz.pdf In question 1.2.1(f) the integration is:$$\int_{2}^{3}\frac{x+1}{\sqrt{x^2-4}} \,dx$$Later on ... 1 vote 2 answers 83 views ### Area Under Unit Hyperbola? Going through Strang's Calculus right now and don't understand a seemingly basic homework question. It asks to integrate under the unit hyperbola, from (1,0) to (\cosh t, \sinh t). The answer in ... 0 votes 0 answers 29 views ### Show \coth(m\theta)\left(\coth(m\theta)-\frac{1}{m}\coth(\theta)\right)\geq \frac{1}{3}\frac{m^2-1}{m^2} I would like to show that the function$$ 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_{... 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$Zis standard normal Consider the following function: \begin{align} f(x)= \frac{1}{2} E\left[ \tanh \left( \frac{x+Z}{2} \right) \right]-\tanh(x)+\frac{x}{2}, \end{align} whereZ$is standard normal. Question: How to ... 1 vote 0 answers 77 views ### 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$then you get the solutions that have the golden ratio in them. These are the solutions Wolfram calculation: https://www.wolframalpha.com/input/?i=xy%3Di%2C+x%... 0 votes 1 answer 63 views ### Why are the domain and image of$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{...
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 \$\...