Questions tagged [hyperbolic-functions]

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

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0answers
26 views

Evaluate error of $\tanh{x}$ with inequality [duplicate]

I have to prove the inequality $\vert \tanh(x)-(x-\frac{x^3}{3})\vert\leq\frac{1}{8}$ for $x\in[-1/4,1/4]$ where $x-\frac{x^3}{3}$ is the 3rd order Taylor expansion at $0$. I know I have to somehow ...
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28 views

integral of Bessel function of first kind with Hyperbolic function

I'd like to solve the following equations: $$\int_{0}^{\infty} \frac{J_{0}(z)+J_{2}(z)}{z+a_{0}+a_{1}\left(\operatorname{coth}\left(a_{2}z\right)-1\right)} dz$$ where $J_{0}$ and $J_{2}$ are Bessel ...
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1answer
42 views

$\frac{1}{\cosh{x}}+\log\left ( \frac{\cosh{x}}{1+\cosh{x}} \right )$ for $x \rightarrow \pm \infty$ has a limit

Show from the definition of a limit that $$\frac{1}{\cosh{x}}+\log\left ( \frac{\cosh{x}}{1+\cosh{x}} \right )$$ for $x \rightarrow \pm \infty$ has a limit. My attempt This one is really tough for me. ...
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4answers
81 views

Express $\operatorname{sech}^{-1}(x)$ in terms of logarithms

I'm trying to express the following $\operatorname{sech}^{-1}(x)$ in terms of logarithms, and would warmly appreciate feedback towards my approach. The solution should be : $$\ln\left(\dfrac{1+\sqrt{(...
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2answers
71 views

Parameterizing both branches of a hyperbola

Recently I have been studying parametric equations of surfaces and curves, specifically hyperbolic functions. Given by the equations $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1, \quad\frac{(x-\alpha)^2}{a^2}-\...
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1answer
68 views

Show $\cosh x$ and $\sinh x$ are continious using $\varepsilon - \delta$ proof

I have to prove that $\sinh x$ and $\cosh x$ are continuous functions. I have to use the hyperbolic addition formula, and the inequalities: $|\sinh x| \leq 3|x|, \, |x|<\frac{1}{2}$ $|\cosh x -1| ...
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1answer
22 views

What parameters can be used with the hyperbolic tangent function to enable optimised curve fitting?

If I want to fit a sigmoid curve to data using the logistic function I use something like $$y = \frac{L}{1 + \exp[-k(x-x_{0})]} + b$$ where $L$, $k$, $x_{0}$ and $b$ are functional parameters that can ...
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1answer
98 views

Why is $\sin(\tanh x) + \sinh(\tanh x)$ almost exactly $2\tanh x$?

I was trying to come up with some approximations for the solution to the differential equation $y'' + \operatorname{sgn}(y') + y = 0$ and noticed while I was messing around that $\sin(\tanh x) + \sinh(...
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2answers
59 views

Expansion of $(\sinh(x))^{\frac15}$ around 0 for x > 0

I'm aware of the series expansion of the hyperbolic functions, but how does one expand a fractional power of sinus hyperbolicus, i.e. e.g. $(\sinh(x))^{\frac15}$?
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1answer
109 views

$\epsilon,\delta$-proof for the limit of $\log \sinh{(x^2)}-x^2$ for $x \rightarrow \infty$

I want to use a $\epsilon,\delta$-proof for the existence and value for the limit of $$\log \sinh{(x^2)}-x^2$$ for $x \rightarrow \infty$. Now, I know the definition for such proof to be $\forall \...
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3answers
75 views

Show $\left | \cosh{x}-1 \right |\leq 3 \left | x \right |$ for $\left | x \right |<1/2$

I have to show that $\left | \cosh{x}-1 \right |\leq 3 \left | x \right |$ for $\left | x \right |<1/2$. I can not use derivatives, series of the trigonometric functions etc. I have to use generel ...
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0answers
62 views

Hyperbolic sine strictly increasing [duplicate]

Proof without using derivatives (and that $e^x$ is striclty increasing) that $\sinh{x}$ is strictly increasing on $\mathbb{R}$. I am having troubles with a few things. I know strictly increasing would ...
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1answer
61 views

Proving $\sinh{x}$ is strictly increasing over all the reals and $\cosh{x}$ is strictly decreasing on $(- \infty , 0]$

How would I be proving $\sinh{x}$ is strictly increasing and $\cosh{x}$ is strictly decreasing on $(- \infty , 0]$ I succeded in showing $\cosh{x}$ is strictly increasing on the interval $[0, \infty)$ ...
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1answer
20 views

Jacobi $sd(u,m)$ hyperbolic approximation for $m \to 1$

I am following a paper (Fink 1976); the details are a mathematical problem. We are given a solution (paraphrased from the paper for clarity) $$y^2 = \alpha m_{1n} sd^2(u,\sqrt{m_n}),$$ where $sd$ is a ...
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2answers
48 views

Proving $\cosh(\sqrt{1+x^2})$ is not uniformly continuous in $\mathbb{R}$

Given $f(x) = \cosh(\sqrt{1+x^2})$ I am trying to show that $f(x)$ is not uniformly continuous. Specifically: $\exists\varepsilon\ \forall\delta\ \exists x,y \in \mathbb{R}: \ |x-y| < \delta \wedge ...
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0answers
76 views

This (rather long) implicit equation has a short explicit solution, but how can it be found?

I am curious if a method exists for solving for $k$ or $h$ in this implicit equation: $$\frac{k^2}{h} \mathrm{sech}^2(k) \sqrt{1 + \left(\frac{k}{h} \tanh(k)\right)^2} = \ln\left( \frac{k}{h} \tanh(k) ...
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0answers
44 views

integration of the product of shifted functions

There is an integral having a following form : $\int f(x) f(ax-b) dx$ . Is there any general way/substitution that makes this kind of integrals easy to solve? Particularly, my functions are shifted ...
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0answers
69 views

Vector Addition/Translation in Hyperboloid model

I have problems understanding vector addition in Hyperbolic space. In the Poincaré ball model, vector addition/translation is the Möbis addition and defined as: $$ x \oplus_c y = \frac{(1+2c\langle x,...
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0answers
41 views

Cosine Transform of Hyperbolic Functions

Does anyone know a closed form of the integral: $\int^{\infty}_{0}\text{d}x \cos(k x) \frac{\sinh(\frac{3\gamma-\pi}{4} x)}{\sinh(\frac{\gamma}{4} x)\cosh(\frac{2\gamma-\pi}{4} x)},$ where $\frac{\pi}{...
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2answers
97 views

How do you prove Osborn's rule?

Given a trigonometric identity written in terms of sine and cosine, it is possible to write down the corresponding hyperbolic identity using Osborn's rule: Replace $\cos$ with $\cosh$ Replace $\sin$ ...
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2answers
33 views

Evaluate the flux of $F=\langle\sin(xyz), x^2y, z^2e^{x/5}\rangle$ through surface $S$ … $4y^2+z^2=4, \space x\in [-2,2]$

I am trying to find the flux of $\vec F=\langle\sin(xyz), x^2y, z^2e^{x/5}\rangle$ through the surface $S$ where $S$ consists of the elliptical cylinder defined by $S$ ... $4y^2+z^2=4, \space x\in [-...
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3answers
47 views

Showing $\lim_{\nu\to\infty}\ln(\coth(\frac\nu2))\to2e^{-\nu}$ and $ \lim_{\nu\to0}\ln(\coth(\frac\nu2))\to-\ln(\frac\nu2)$

I am struggling to derive a couple limits I have come across in a paper I am reading. Both involve the natural log of the hyperbolic cotangent. The paper seems to be saying the two terms trend the ...
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1answer
38 views

What is wrong with my derivation of the exponential form of $\sinh(x)$

I know the exponential form of $\sin(x)=\dfrac{e^{ix}-e^{-ix}}{2i}$. I wish to derive the exponential form of $\sinh(x)=\dfrac{e^x-e^{-x}}{2}$, and I am stuck. I try to replace $\sin(x)$ with $\sin(i^...
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6answers
893 views

Unifying the connections between the trigonometric and hyperbolic functions

There are many, many connections between the trigonometric and hyperbolic functions, some of which are listed here. It is probably too optimistic to expect that a single insight could explain all of ...
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2answers
81 views

What is a slower decay rate than hyperbolic?

Hyperbolic decay $$f_\alpha(x)=\frac{1}{\alpha x + 1} $$ is slower than exponential decay $$f(x) = e^{-x}$$ where $\alpha > 0$ is a scaling factor. The larger that $\alpha$ is, the steeper the ...
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2answers
126 views

How to simplify $cosh(arccoth(x))$? [closed]

I was given the problem to simplify $\cosh(\text {arccoth} (x))$ for $|x| > 1$, and I was just wondering how I would do that.
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1answer
139 views

Fourier transform of hyperbolic function $\frac{d}{dx}\log({\sinh{x}})$

I am trying to calculate the FT of $\frac{d}{dx}\log({\sinh{x}})$. I did not calculate it directly but I used the result of its derivative. This is a tentative: I found the following result in the ...
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1answer
35 views

What do sinh and cosh have to do with exp? [closed]

My friend told me that $\sinh$ and $\cosh$ result from an exponential function, but I can't figure out why
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1answer
66 views

Show that $ 2 \sinh(z)=\exp(z)-\exp(-z)$ [closed]

$ 2 \sinh(z)=\exp(z)-\exp(-z)$; $ 2 \cosh(z)=\exp(z)+\exp(-z)$ where $z \in \mathbb{C} $ $$\sin(z) := \sum_{k=0}^{\infty}\frac{(-1)^k}{2k+1!} z^{2k+1}$$ $$\cos(z) := \sum_{k=0}^{\infty}\frac{(-1)^k}{...
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1answer
40 views

Indefinite integral of $\int \frac 1 x \operatorname{arsech} \frac x a \, \mathrm d x$

Spiegel's "Mathematical Handbook of Formulas and Tables" (Schaum, 1968), item $14.668$ gives the indefinite integral of the area hyperbolic secant (that is, the "inverse" ...
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2answers
16 views

Range of Real Inverse Hyperbolic Cosine — can it be negative?

There are several results in Spiegel's "Mathematical Handbook of Formulas and Tables" (Schaum, 1968) concerning $\cosh^{-1}$ which are presented in the following format: $$\cosh^{-1} x = \pm ...
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5answers
71 views

Maclaurin expansion for sech$(x)$

I am a bit unsure where I have gone wrong in working this out. Sech$(x)=2/(e^x+e^{-x}).$ Maclaurin expansions: $e^x = 1+ x + x^2/2+ x^3/6 + x^4/24;\; e^{-x} = 1- x + x^2/2 - x^3/6 - x^4/24;$ so sech$...
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1answer
36 views

When proving hyperbolic identities why do we add one or two

Example: $\cosh^2x+\sinh^2x=\cosh2x$ Proof: $$\frac{1}{2}(e^x+e^{-x})^2+\frac{1}{2}(e^x-e^{-x})^2$$ Where does this two's come from? $$\frac{1}{4}(e^{2x}+e^{-2x}+2)+\frac{1}{4}(e^{2x}+e^{-2x}-2)$$
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3answers
42 views

Hyperbolic functions simplifying

How do you simplify $$\cosh(\sinh^{-1}(x))$$ to become $$(1+x^2)^{1/2}$$ I have managed to get $(1+\sinh^2(\sinh^{-1}(x))^{1/2}$ but haven't been able to progress from there.
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0answers
20 views

If $f,g \in PSL(2,\mathbb{C})$ then $\langle f,g \rangle$ is not discrete

Let $H^3 = \mathbb{C} \times \mathbb{R}^+$, the $Isom(H^3) = PSL(2,\mathbb{C})$ Now. I try to show that if $f,h \in PSL(2,\mathbb{C})$ with a exactly one common fixed point then the group $\langle f,h\...
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1answer
32 views

How to deduce that $\arctan\left(\frac{a}{i}\right)=-i\cdot \text{ arctanh}(a)$

I was doing some calculations in wolframalpha and I found the following equality: $$\arctan\left(\frac{a}{i}\right)=-i\cdot \text{ arctanh}(a)$$ This is the first time I've seen this equality. How is ...
2
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1answer
46 views

Verifying the hyperbolic law of cosines and law of sines..

I am trying to understand hyperbolic triangles. These two laws are supposedly universal in the hyperbolic plane. However, whenever I've tried to verify them using applets which let me construct ...
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0answers
25 views

Existence of two constants related to the prime counting function

Well it's the continuation of Conjecture : A lower bound for the prime counting function . . Rearranging some terms and adding a fixed exponent we work with the expression : $$S(n)=\sum_{k=1}^{n}\frac{...
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1answer
73 views

Good approximations for $\int_{-\infty}^0 \bigg(\tanh(ax+b)\tanh(cx+d) - 1\bigg)~\mathrm{d}x$

I have an integral $$I(a,b,c,d) = \int_{-\infty}^0 \bigg(\tanh(ax+b)\tanh(cx+d) - 1\bigg)~\mathrm{d}x$$ which I need to approximate analytically (to use it in further steps within a machine learning ...
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0answers
22 views

Find the Volume V of the solid obtained by rotating the region bounded by the given curves about the y-axis: zy=8, z=0, y=1, y=2

I am aware that I am finding the volume of a hyperbola about the y-axis. I will probably do this using the shell method: $V=2\pi \int_a ^b x(f(x))\,dx$. Any help would be appreciated
4
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1answer
49 views

Conjecture : A lower bound for the prime counting function .

Well it take my a little bit of time to find it but now I think that my conjecture is ready . Conjecture : Let $n\geq 100$ and then define the sum : $$S(n)=\sum_{k=1}^{n}\frac{1}{\operatorname{...
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1answer
56 views

Integrating $\text{sech}(x)$ using a hyperbolic substitution method

I have been tasked to find $\int{\text{sech}(x)dx}$ using both hyperbolic and trig substitutions, for the trig substitution method I did the following. $$I=\int{\frac{2e^x}{e^{2x}+1}dx} $$ $$\text{Let}...
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1answer
52 views

Prove the following and use it to evaluate the integral: [duplicate]

I want to prove that:$$\int_{-\infty}^\infty f(x)dx=\int_{-\infty}^\infty f\left(x-\frac1x\right)dx$$ And use the result of this proof to evaluate:$$\int_{-\infty}^\infty\frac{x^2}{x^4+1}dx$$
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32 views

Show that the hyperbolic sine function is bijective

I am supposed to show that sinh : R -> R is bijective and I'm a bit lost. I suppose the part of showing that it's injective isn't so bad but I'm having some problems with showing that it's ...
2
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1answer
45 views

Find $\int\cosh^\frac{1}{2}u~du$ and/or $\int\frac{1}{\cosh^\frac{1}{2}\theta}~d\theta$

I am trying to find a function of the form $y=f(x)$ such that the volume of the solid generated by the function between any two points around the $x$ axis is numerically equal to its length bewteen ...
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0answers
42 views

Prove $\tanh^{-1}\left(\sqrt{\tanh(x)}\right)-\tan^{-1}\left(\sqrt{\tanh(x)}\right)\geq 0$ without using derivative

Claim : Let $0\leq x$ then prove (without using derivative) that : $$\tanh^{-1}\left(\sqrt{\tanh(x)}\right)-\tan^{-1}\left(\sqrt{\tanh(x)}\right)\geq 0$$ Trick : We have using integral : $$\tanh^{-1}\...
1
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1answer
29 views

Prove $\coth\;2v = \frac{x^2 + y^2 + 1}{2y}$

Q: Given that $x + jy =\tan (u + jv)$, prove that $$\coth\;2v = \frac{x^2 + y^2 + 1}{2y}$$ I would like to ask this question, how can we prove it? I had tried to expand the equation $x + jy =\tan (u + ...
2
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1answer
53 views

Proving that $|\sin(z)|^2 = \sin(x)^2+\sinh(y)^2$

Goal: Prove $|\sin(z)|^2 = \sin(x)^2+\sinh(y)^2$ I have \begin{align*} \sin(x+iy) &=\sin(x)\cos(iy)+\cos(x)\sin(iy)\\ &=\sin(x)\cosh(y)+i\cos(x)\sinh(y)\\ |\sin(z)|^2 &=\sin(x)^2 \cosh(x)^...
0
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0answers
12 views

Symmetry of piecewise defined function

Let $$ f(x) = \begin{cases} 2-\cosh(x), & \text{for } |x| \leq d \\ \alpha e^{-|x|}, & \text{for } |x| > d \end{cases} $$ with a positive constant $d$ and parameter $\alpha$. How can I ...
3
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0answers
124 views

$L\{f(x)\}^{-1}=\int^x_a \frac{dx}{f(x)}$

let $a<b\in \overline {\mathbb R}$ such that $\lim_{x\to a}f(x)=0$, Let $f:(a,b) \to \mathbb R$ be continuous and positive on $(a,b)$ $$L\{f(x)\}^{-1}=\int^x_a \frac{dx}{f(x)}$$ Where $f(x)^{-1}$ ...

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