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

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9 views

Circumference of hyperbolic circle is $2\pi \sinh r$

I'm looking for a proof that in the Poincare disk model the circumference of a circle of radius $r$ is $2\pi \sinh r$. I have seen this result in many places but I haven't been able to find a proof. ...
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1answer
16 views

Distance in the $y$-axis of the hyperbolic plane

I'm reading Stillwell's Geometry of Surfaces but I'm having a little bit of trouble because my background in calculus isn't great, I'm struggling with these problems: In the upper half-plane model, ...
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31 views

Solving $\frac{\partial}{\partial r} \int_r^x g(y,r) dy = -\frac{\cosh(a\sqrt{x^2-r^2})}{\sqrt{x^2-r^2}}$

I am stuck on this problem. I need to find the correct function, $g(y,r)$, such that $$ \frac{\partial}{\partial r} \int_r^x g(y,r) dy = -\frac{\cosh(a\sqrt{x^2-r^2})}{\sqrt{x^2-r^2}} $$ So far I am ...
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1answer
60 views

Evaluating $ \int_0^\theta \cosh(a\sin x) dx$

The integral below seems quite simple, but I couldn't find anywhere the result. $$ I = \int_0^\theta \cosh(a\sin x) dx$$ I tried to expand it into Taylor expansion series and successfully evaluate the ...
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1answer
21 views

Transformation of parameter between the hyperbolas $xy=1$ and $x^2-y^2=2$ during rotation?

It is fairly straightforward to see that the hyperbola $xy=1$ is simply the hyperbola $x^2-y^2=2$ rotated by $\pi/4$. All we do is apply the corresponding rotation matrix to the vector ...
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1answer
71 views

Integral of $\int_{-\infty}^\infty\frac{e^{-y^2/2} \sinh(cy)^2} {\cosh(Mcy) }dy$

Is it difficult to compute or find a good computable lower bound on the integral \begin{align*} \int_{-\infty}^\infty\frac{e^{-y^2/2} \sinh(cy)^2} {\cosh(Mcy) }dy \end{align*} where $c$ and $M$ are ...
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1answer
295 views

Integral of the ratio of two exponential sums

I am trying to find a lower bound on the following integral \begin{align*} \int_{y=-\infty}^{y=\infty} \frac{ (\sum_{n=[-N..N]/\{0\}}n e^{-\frac{(y-cn)^2}{2}})^2} {\sum_{n=[-N..N]/\{0\}} ...
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1answer
34 views

a hyperbolic summation

Find the value of $$\lim_{n\to\infty} \sum_{k=1}^{n}\frac{1}{\sinh 2^k}$$ Numerical approximations gives me a value of $\frac{2}{e^2-1}$. I tried to write the sum as ...
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3answers
86 views

How to simplify if $a > 0$ and $\cos(a) < 0$ [closed]

$$\sqrt{\cos (a)} \sinh \left(\ln (2) a^{\frac{1}{2} \left(e^{-ia }+e^{ia}\right)}\right)+\sqrt{\cos (a)} \cosh \left(\ln (2) a^{\frac{1}{2} \left(e^{-ia }+e^{ia }\right)}\right)=$$
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1answer
29 views

hyperbolic isometry

I have a project I have to do. In order to do it I need to investigate this book. In page 94 they defined hyperbolic isometry on a metric s.t it possesses no fixed point in the tree. After that they ...
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1answer
27 views

Prove this equality about hyperbolic right triangles

If K is the area of a hyperbolic right triangle ABC in which the right angle is at C, prove that $$ \sin K=\frac{ \sinh a \sinh b}{1+\cosh a\cosh b}$$ My attempt at the solution: I basically need ...
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2answers
99 views

Find: $\int_0^{\infty}\frac{\sinh x}{1+\cosh^2x}dx$

$$\int_0^{\infty}\frac{\sinh x}{1+\cosh^2x}dx$$ Here's what I've attempted: Using the identity $1+\cosh^2x=\sinh^2x$ I got: $$\int_0^{\infty}\frac{\sinh x}{\sinh^2x}dx=\int_0^{\infty}\frac1{\sinh ...
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1answer
24 views

Finding equation of hyperbola with only foci and asymptote

This is a concept we learned in class today, which I still can't seem to grasp. I have no specific question that necessarily has to be done, so I will use one of the examples my book gives me: Given ...
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2answers
30 views

Differentiating inverse hyperbolic function

I am trying to differentiate $\tanh^{−1}\left(x/(1 + x^2)\right)$, but am finding it difficult understanding what to do. I think you have to place the differential of the angle of the hyperbolic ...
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1answer
225 views

Proof of Hyperbolic Functions

Find the proof:  (a) Use the definitions cosh(x)= 1/2(ex +e^−x) , sinh(x)= 1/2(e^x − e^−x) to express sinh(x + y) and cosh(x + y) in terms of cosh(x), sinh(x), cosh(y) and sinh(y). (b) Using the ...
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2answers
187 views

Why is $\sinh$ often pronounced “shine”?

I talked to some guys from the UK and they told me that they would pronounce $\sinh$ as "shine". I am not a native english speaker so I don't know, but in my country we call this function "sintsh" ...
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0answers
22 views

Four simultaneous equations

General form of the function: $$y=d\sinh^{-1}\left(\frac{ax+b}2\right)+c$$ I want the function to pass three points, $(0,0)$, $\left(\frac{t}2,\frac{g}2\right)$ and $(t,g)$, and I want the function to ...
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1answer
493 views

Partials sums of cosh(x) and sinh(x)

Ok, i asked this question yesterday but then hit a snag again. Using these identities $\sinh(x+1) - \sinh (x) = (-1+\cosh(1))\sinh(x) + \sinh(1)\cosh(x)$ $\cosh (x+1) - \cosh (x) = ...
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1answer
204 views

Sum of hyperbolic functions, having problems expressing $\sinh(1)$

Using these identities $\sinh(x+1) - \sinh (x) = (-1+\cosh(1))\sinh(x) + \sinh(1)\cosh(x)$ $\cosh (x+1) - \cosh (x) = (-1+\cosh(1))\cosh(x) + \sinh(1)\sinh(x)$ Express the series $C = \cosh 0 + ...
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0answers
14 views

Simplifying equation using hyperbolic transcendental functions… Ahhh!!!!! [duplicate]

I am having real difficulty in seeing how the two equations seen below relate to one another. Can anybody help?? $$\Omega(\theta)=-b.\coth(\operatorname{arsinh}(\exp a\theta . \sinh(c_0)))$$ ...
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35 views

Prove that $2\cosh \left(\frac{1}{2} a\right) \sin \left(\frac {1}{2} \alpha\right) = 1.$

Suppose a hyperbolic equilateral triangle has side $a$ and angle $\alpha$.Prove that $$2\cosh \left(\frac{1}{2} a\right) \sin \left(\frac {1}{2} \alpha\right) = 1.$$
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8 views

Linearization of hyperbolic function with unknown exponent

I have a graph that is clearly some inverse function of the form $y=Ax^n$ where n is a negative. I want to linearize this graph to give me the values of A and n without merely approximating the ...
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2answers
33 views

$Im(\cosh z)=\sinh x\sinh y$ and $|\sinh z|^2=\sinh^2(z)+\cosh^2(z)$

I have problems in two issues of complex variables ... 1) Prove that $Im(\cosh z)=\sinh x\cosh y$, if $z=x+iy$. I tried to expand $\cosh z= ...
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0answers
44 views

Prove that $\coth ^2(x)-1\equiv \mathrm{cosech}^2 (x)$

Prove that $\coth ^2(x)-1\equiv \mathrm{cosech}^2 (x)$ My attempts, $\coth^2 (x)-1\equiv(\frac{e^x+e^{-x}}{e^x-e^{-x}})^2-1$ $\equiv \frac{e^{2x}+e^{-2x}+2}{e^{2x}+e^{-2x}-2}-1$ $\equiv ...
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0answers
51 views

Derivatives of hyperbolic functions and Osborne's rule.

I am slightly confused when it comes to Osborne's rule when you take derivatives of hyperbolic functions. For example. The derivative of cotx is -cosec^2x, so there is a product of sines. So should ...
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25 views

Difference between hyperbolic sector, hyperbolic angle and hyperbolic argument

I've been working with hyperbolic functions and am completely confused by the Wikipedia definitions of hyperbolic sectors and angles. Are they the same thing? Based on my trial calculations, they seem ...
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21 views

Hyperbolic functions calculator values differing from the graph

I was exploring hyperbolic functions and noticed something weird while comparing the analytical definition (e^x+e^-x)/2 with the geometrical definition using the hyperbola x^2 - y^2 = 1. For the angle ...
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0answers
38 views

Geometric interpretation of hyperbolic functions and the hyperbolic angle/argument

I've been reading up on hyperbolic functions and was wondering if there was a geometric definition for the hyperbolic angle and hyperbolic function. In particular I was reading this: Definition of ...
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3answers
68 views

Integrating $\frac{1}{(x^2+b)^{3/2}}$?

How to integrate $$\int\frac{1}{(x^2+b)^{3/2}}dx$$ using the hyperbolic sine substitution ?
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3answers
40 views

How to integrate $I = \int_{-a/2}^{a/2}\frac{1}{\sqrt{x^2 + b}}dx$

$$I = \int_{-a/2}^{a/2}\frac{1}{\sqrt{x^2 + b}}dx$$ I tried to integrate by parts but failed. I think I'm supposed to change variables using a hyperbolic sine, but I don't know this method.
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0answers
14 views

Exponential curve with hyperbolic sine behavior on the tails

I have a dataset that I've fitted an exponential curve to that looks like a great fit at midrange values of the domain but is not such a good fit at low and high end domain values. Instead, at these ...
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2answers
68 views

Evaluating: $ \int\sqrt{\tanh(\ln(\sqrt{x}))} dx$ ; $ \int \ln\left(\sqrt{\tanh(\ln(\sqrt{x}))}\right) dx$

I don't have much experience with hyperbolic trig funnctions... So I don't know how to start solving this. How do I evaluate the following integrals? Any advice, hint or well-thought solution will be ...
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2answers
153 views

Closed form for ${\large\int}_0^\infty\frac{x\,\sqrt{e^x-1}}{1-2\cosh x}\,dx$

I was able to calculate $$\int_0^\infty\frac{\sqrt{e^x-1}}{1-2\cosh x}\,dx=-\frac\pi{\sqrt3}.$$ It turns out the integrand even has an elementary antiderivative (see here). Now I'm interested in a ...
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0answers
21 views

Rearrangingg equations using hyperbolic transcendental functions [duplicate]

I have tried and tried but cannot for the life of me see how one equation follows onto the other... can anybody help?? $$\Omega(\theta)=-b.\coth(\operatorname{arsinh}(\exp a\theta . \sinh(c_0)))$$ ...
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31 views

Osborne's rule for hyperbolic functions?

I am confused as to why you only change the sign for powers of sine that are 4n+2. As I understand, $sin(i\theta)=isinh(\theta)$ $sin^2(i\theta)=-sinh^2(\theta)$ $sin^3(i\theta)=-isinh^3(\theta)$ ...
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25 views

Rearranging equations using hyperbolic transcendental functions [duplicate]

I have tried and tried but cannot for the life of me see how one equation follows onto the other... can anybody help?? $$\Omega(\theta)=-b.\coth(\operatorname{arsinh}(\exp a\theta . \sinh(c_0)))$$ ...
3
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2answers
50 views

Computing Residue for a General, Multiple-Poled function?

I'm trying to compute the residue of the following function at $a$. I'm having a little trouble seeing which poles are relevant: Compute $\,Res_f(a)$ for the following function: $$f(z) = ...
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1answer
24 views

Differentiating hyperbolic functions.

$\DeclareMathOperator{\sech}{sech}$Can anyhow me how to differentiate the following? I already tried using the product rule, but I can't quiet seem to succeed. $\sech^{2} x$. $2\bigl(\cosh(2x) - ...
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1answer
34 views

Hyperbolic Trig Functions - Identities

I don't understand how the 3rd step (the 4 divisions) happens? Can someone explain how they arrived at that.
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3answers
51 views

Solving an equation with hyperbolic functions

I'm trying to prove that for a given $s,t\in\mathbb{R}$ there exists $w\in\mathbb{R}$ such that $\cosh(t)e^{i(s+w)}+\sinh(t)e^{i(s-w)}\in\mathbb{R}$. How to solve this?
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2answers
228 views

Integral $ \int_{0}^{\pi/2} \frac{\pi^{(x^{e})}\sin(x)\tan^{-1}(x)}{\sinh^{-1}\left({1+\cos(x)}\right)} dx$

I need help in evaluating the following integral :- $$ \int_{0}^{\pi/2} \frac{\pi^{\displaystyle (x^{e})}\sin(x)\tan^{-1}(x)}{\sinh^{-1}\left({1+\cos(x)}\right)} dx$$ A brief solution would be very ...
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2answers
44 views

Proving $\ln \cosh x\leq \frac{x^2}{2}$ for $x\in\mathbb{R}$

It seems to me that $\ln \cosh x\leq \frac{x^2}{2}$ for $x\in\mathbb{R}$, as suggested by graphing the difference between both functions as well as the fact that the Taylor series expansion of ...
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1answer
36 views

What is the $n$th derivative of $\coth(x)$?

I would like to know the $n$th derivative of the Hyperbolic Cotangent, i. e., $\frac{\partial^n}{\partial x^n} \coth( x )$. So far, I have only found an expression for the $n$th derivative of the ...
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3answers
97 views

A level Integration: $\int\frac{x^3}{\sqrt{x^2-1}}dx$

Using the substitution $x=\cosh (t)$ or otherwise, find $$\int\frac{x^3}{\sqrt{x^2-1}}dx$$ The correct answer is apparently $$\frac{1}{3}\sqrt{x^2-1}(x^2+2)$$ I seem to have gone very wrong ...
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1answer
28 views

Solve for a hyperbolic Laplace Transform by expressing as exponents and shiftig on s-axis (5.3-21)

I cannot get past a certain point on this problem as shall be shown. I need guidance in order to complete the problem. The exercise as stated in the text: Represent the hyperbolic function in terms ...
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24 views

Volumes of revolutions question

The point $P(a,b)$ lies on the curve $y=arsinh x$. $R$ is the region bounded by the curve, the $x$- and $y$-axes and the line $x=a$. When $R$ is rotated 2$π$ radians about the $x$-axes the solid ...
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3answers
39 views

How can I calculte the probability of $X$ with a Generlized Hyperbolic Distribution?

I would like to know how to calculate the probability of $X$ when I have fitted a Generalized Hyperbolic Distribution to my data set. The depth of my knowledge is basic t-tests and z-tests. I am ...
2
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1answer
30 views

Stuck on an integration question…

$$\int x^{-\frac{1}{2}}\cosh^{-1}(\frac{x}{2}+1)dx$$ The answer I should get is $$2x^{\frac{1}{2}}\cosh^{-1}(\frac{x}{2}+1)-4(x+4)^{\frac{1}{2}}$$ but I keep going wrong. Can someone show me how to ...
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0answers
30 views

Complex Mapping of $\mathrm{cosh}(w)=z$

Mapping in complex analysis has not been very easy for me unfortunately. I am having difficult trying to find the mapping between the z and w plane. I attempted to simply write that ...
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1answer
19 views

Generalizing 1D function to higher dimensions

I have a function in 1D given by $f(x) = \tanh(x-x_1) + \tanh(x-x_2)$. I want to generalize this to two dimensions, such that it describes a circle. The function $f(x,y)$ has to have a form such that ...