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Questions tagged [hyperbolic-functions]

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

8
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0answers
75 views

Showing $\int_0^{\int_0^u{\rm sech}vdv}\sec vdv\equiv u$ and $\int_0^{\int_0^u\sec vdv}{\rm sech} vdv\equiv u$

The two following very weird-looking theorems $$\int_0^{\int_0^u\operatorname{sech}\upsilon d\upsilon}\sec\upsilon d\upsilon \equiv u$$ $$\int_0^{\int_0^u\sec\upsilon d\upsilon}\operatorname{sech}\...
3
votes
2answers
79 views

Taylor series of functions

Consider the Taylor series of the function $$\frac{2e^x}{e^{2x}+1} = \sum_{n=0}^{\infty} \frac{E_n}{n!} x^n$$ Prove that $E_0 = 1, E_{2n-1} = 0$ and, for $n \ge 1$, $$E_{2n} = - \sum_{l=0}^{n-1} C_{...
1
vote
1answer
21 views

Discrepancy in solutions of differential equation?

The differential equation at hand is this : $$ \frac{\text{d}\psi}{\text{d}x}+2\tanh(x)\,\psi\left(x\right)=0\ $$ And what I have tried is this : $$ \int_{}^{} \frac{\text{d}\psi}{\psi}=-2\int_{}^{}...
0
votes
0answers
19 views

Simplify trigonometric expression of hyperbolic functions

I have $\cos^2x\cosh^2y - \sin^2x\sinh^2y$. I saw it written simplified as $\cosh^2 y - \sin^2 x$. But I don't get how to get it. My attempts were to write $\cosh^2y -1$ instead of $\sinh^2y$ but ...
1
vote
1answer
21 views

How to evaluate hyperbolic functions, involving inverses, by hand?

How does one evaluate $$\sinh(2{\sinh^{-1}{(2)}})$$ by hand?
0
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1answer
30 views

using sinh(x) to find series representation of arcsinh(x)

From "Complex Variables Demystified", 2008, page 102: Given: $$sinh(z)=\frac{e^z-e^{-z}}{2}$$ find the series representation for arcsinh(x). Solution: (1) The Maclaurin theorem can be used to ...
0
votes
2answers
50 views

Manual approximation of $\operatorname{sech}(0.7)$

In the archive of a midterm exam collection there are some question like the one above. How can we approximate expressions like $$\operatorname{sech}(0.7)$$ without a calculator? Thanks in ...
3
votes
1answer
100 views

Basis formed by hyperbolic functions

I am currently working with separation of variables for different kinds of PDEs and one often uses here the fact that one has the sine base, i.e., $$ \left( \sin(k\pi y) \right)_{k=1}^{\infty} $$ ...
0
votes
1answer
20 views

Why is the domain of the hyperbolic function $\sinh x$ is symmetric about the origin?

My book asked me to prove that the function $\sinh x$ is odd, but in order to be odd I must be sure that the domain of it symmetric about the origin, how can I be sure from this?
1
vote
1answer
29 views

Is the similarity between tanh and normal distribution just coincidence?

So, explaining to someone why tanh is used in machine learning (i.e. it squashes an open range to -1..+1, and changes most rapidly around 0), I brought up $\frac d{dx}$ $tanh(x)$, and it looks just ...
0
votes
1answer
30 views

Solve second order differential equation with cosh using frobenios method

i need to show that the differential equation $y^{''}+(\cosh(2x)-4)y = 0$ has the solution: $ y(x) = x+\frac{1}{2}x^3-\frac{1}{40}x^5 -... $ using Frobenius method. I started by writing cosh(2x) ...
-2
votes
1answer
43 views

Find min distance from y = cosh x to y = x

The full question is this: Point P on curve y = cosh x is such that its perpendicular distance from the line y = x is a minimum. Show P's coordinates are (ln(1 + root 2), root 2). I am completely at ...
1
vote
4answers
345 views

Proof for hyperbolic trigonometric identities [closed]

I've been studying hyperbolic functions and was wondering where the following two identities were derived from: $$\sinh(x) = \frac{e^{x}-e^{-x}}{2}$$ $$\cosh(x) = \frac{e^{x}+e^{-x}}{2}$$ I ...
1
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1answer
23 views

Getting function from four points

I'm facing this problem I can't solve myself. I've got four points on a cartesian place, and I would like to find the function that equates them. Coords are: ...
0
votes
1answer
64 views

Inverse Function Theorem: Proving Global Invertibility.

My question states: Prove that the following coordinate transformation is invertible everywhere, at all values of $(x, y)$ . $$u = \arctan(x - y)$$ $$v = \sinh(3x) + 2\sinh(y)$$ That is x and y ...
1
vote
2answers
103 views

Fourier transform of $x / \tanh(x)$

I have problems to calculate analytically the (inverse) Fourier transform of $x / \tanh(x)$: $$\frac{1}{\sqrt{2\pi}}\int_\mathbb{R} \frac{x}{\tanh(x)} \mathrm{e}^{- \mathrm{i} x k} \mathrm{d} x$$ ...
0
votes
0answers
18 views

Problems to find the correct limit of a logarithm of an hyperbolic function

I want to find the next limits $$ f(x) = \frac{1}{x}\coth\left(\frac{1}{x}\right) - \ln\left(2 \sinh\left(\frac{1}{x}\right)\right)\\ \lim_{x\to 0} f(x)\\ \lim_{x\to\infty} f(x)$$ when I evaluate ...
2
votes
3answers
85 views

Prove formula $\operatorname{arctanh} x = \frac12\,\ln \left(\frac{1+x}{1-x}\right)$

Problem Prove formula $\operatorname{arctanh} x = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right)$ Attempt to solve To start off with definition of functions $\sinh(x)$ and $\cosh(x)$ $$ \cosh(x)=\...
1
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2answers
45 views

Principal part of Laurent Series for $\frac{1}{(1-\cosh(z))^2}$

in this exercise I am asked to provide the principal part of the Laurent series of $$\frac{1}{(1-\cosh(z))^2}$$ And i am kinda struggling with fonding a solution or even a pattern towards one Thanks ...
4
votes
4answers
814 views

Range of real values of sin(z) [closed]

Given $f(z) = \sin(z)$ such that $z$ is an element of the complex numbers is the range of the real part of $f(z)$ all the reals? Is the range of the real part of $f(z)$ all reals given that the ...
0
votes
2answers
47 views

Proving $\frac{\sinh\tau+\sinh i\sigma}{\cosh\tau+\cosh i\sigma }=-\coth\left(i\frac{\sigma+i\tau }{2}\right)$ for bipolar coordinates $(\sigma,\tau)$

I am having trouble proving the following identity: $$\frac{\sinh \tau +\sinh i\sigma }{\cosh \tau +\cosh i\sigma }=-\coth \left(i \frac{\sigma +i\tau }{2}\right)$$ I have tried using identities ...
-1
votes
2answers
188 views

Integral of $\arccos(x + 1)$

I'm trying to work out how to find the indefinite integral of $\operatorname{arccosh}(x + 1)$ I have been using integration by parts to get it down to $$x\operatorname{arccosh}(x + 1) - \int \frac{...
0
votes
1answer
23 views

What is the relation to $\sinh{x},\cosh{x}$ and $\sin{x},\cos{x}$ [duplicate]

I've learned what $\sinh{x},\cosh{x}$ (the hyperbolic trig functions) are defined as formula, but how is it related to $\sin{x},\cos{x}?$ The only thing I've noticed is that $\cosh^2(x)-\sinh^2(x)=1.$
0
votes
1answer
56 views

Relationship between hyperbolic functions and complex analysis

As you know, hyperbolic functions are defined in terms of $e$. For example, the hyperbolic cosine: $$\cosh x = \frac{e^x + e^{-x}}{2}$$ There is a result in complex analysis that looks uncannily ...
0
votes
1answer
28 views

Segmentations with equal length on a hyperbola [closed]

How can I divide a hyperbola or one of its branches into many segmentations with equal length = l along the curve. hyperbola can be express as: $x^2/a^2 - y^2/b^2 = 1$ How to compute the x, y for the ...
0
votes
1answer
54 views

Find the Taylor series of argtanh(x) using sinh(x) and cosh(x)

I just finished my exam a few hours a go, and there was 1 question I couldn't answer. I was asked to derive the Taylor series of $\arg\tanh(x)$ using the fact that $$\tanh(x)=\frac{\sinh(x)}{\cosh(x)},...
0
votes
0answers
31 views

Inverse Laplace, residue, simple or essential pole from bivaluated hyperbolic trigonometric function?

I would like to compute the following function: $$f(t)=\mathcal{L}^{-1}\Big[\frac{1}{s(e^{a+\text{arcosh}(s+\cosh a)}-1)}\Big](t)$$ However, it seems that there is no other pole than the pole of ...
-2
votes
1answer
73 views

Prove $\forall x \in \mathbb{R}$: $[\sinh(x)+\cosh(x)]^n = \cosh(nx)+\sinh(nx)$ ; $ n\in \mathbb{Q}$ [closed]

How to prove? Prove $\forall x \in \mathbb{R}$: $[\sinh(x)+\cosh(x)]^n = \cosh(nx)+\sinh(nx)$ ; $ n\in \mathbb{Q}$
2
votes
2answers
100 views

Series expansion of $\tan^2$ and $\tanh^2$

Are there known closed formula expressions for their power series expansion at the origin? I couldn't find anything online. Edit: (to clarify) Of course we could simply take the series expansion of ...
0
votes
1answer
31 views

Two equations of hyperbolic partial differential equation in wiki

The link is here A common hyperbolic partial differential equation is the one-way wave equation So I understand But in the wiki, Why is it a hyperbolic partial differential equation? Is the ...
7
votes
3answers
85 views

Analogies between $(\tan, \sec)$ and $(\sinh, \cosh)$

The pair of functions $(\tan, \sec)$ shares some interesting properties with the pair $(\sinh, \cosh)$. First of all, they satisfy the same quadratic equation, namely $$\sec^2 x - \tan^2 x = 1 \qquad ...
1
vote
1answer
84 views

In trig substitutions, why favor $\sin$, $\tan$, $\sec$, $\sinh$, $\cosh$, $\tanh$ over $\cos$, $\cot$, $\csc$, etc?

Exemplified by this question and its comment, there seems to be a near-universal preference for the substitutions $\sin(\theta)$, $\tan(\theta)$, $\sec(\theta)$, $\sinh(\theta)$, $\cosh(\theta)$, and $...
1
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3answers
123 views

The interconnection between Hyperbolic functions and Euler's Formula

From Euler's identity one may obtain that, $$\sin x=\dfrac{e^{ix}-e^{-ix}}{2i}$$ $$\cos x=\dfrac{e^{ix}+e^{-ix}}{2}$$ However, it looks quite same to the hyperbolic functions such as $$\sinh x=\dfrac{...
0
votes
1answer
57 views

Help with hyperbolic integral $\int_0^\infty x\frac{\cosh(bx)}{\sinh(x)}dx.$

The Integral Calculator couldn't help me with the following integral: $$\int_0^\infty x\frac{\cosh(bx)}{\sinh(x)}dx.$$ From some mathematical physics considerations, I get that the answer should ...
0
votes
0answers
20 views

Bounded similarity measure for points in hyperbolic space

given two points $x, y$ on an n-sphere (embedded in $\mathbb{R}^{n+1})$, their inner product $\langle x, y \rangle = x_1y_1 + \dots + x_{n+1}y_{n+1}$ will be a scalar in $[-1, 1]$ which can be ...
1
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1answer
51 views

Finding the asymptotic behavior of a function series

I solved the shape of an elastic sheet annulus clamped on the inner and outer circle with a point load (the figure below shows the cross section of an example): Solve Green function of an annulus to ...
0
votes
1answer
131 views

Closed form of $\int_0^{+\infty}\frac{x\pi}{x\pi+2\sinh(x\pi)} \, dx$

I have numerically computed the integral $\int\limits_0^{+\infty}\frac{x\pi}{x\pi+2\sinh(x\pi)} \, dx$ such that it's value is a rational number and it's equal $0.298549$. An inverse symbolic ...
7
votes
2answers
452 views

Canceling the Integral

Suppose we begin to integrate $e^xcosh(x)\ dx$ by parts. Regardless of our $u$ and $dv$ choices, we arrive at $$\int e^x\cosh x\ dx = e^x\cosh(x) - e^x\sinh(x) + \int e^x\cosh x\ dx\tag{Equation 1}$$ ...
4
votes
1answer
52 views

Prove this integral related to the Ising model

I came across this integral when learning the Ising model. Without external field Onsager's solution of a 2D square lattice with $J_2=0$ should equal the solution of a 1D Ising model, which leads to ...
0
votes
1answer
79 views

Infinite Product Expansion of Hyperbolic Functions

the following equation is from "[1970] Goodson - Distributed system simulation using infinite product expansions": \begin{align*} \cosh z + \left( c z+ \frac{d}{z} \right) \sinh z & = (1 + d) \...
0
votes
1answer
25 views

What is the period of the composition of hyperbolic tangent and hyperbolic arcsine?

Consider the following indefinite integral: $ \int \mathrm d s = \tanh \left ( \mathrm {arcsinh} \frac{\beta}{\alpha} \right ) \, $ where $ 0 < \alpha \in \mathbb R \; $ is constant and $ \beta \in ...
0
votes
1answer
47 views

Proving that $\cosh^{-1} (\cosh(x)\cosh(y) ) \geq \sqrt{x^2 + y^2}$ and the CAT(0) inequality

Not long ago, I asked how to prove that $\cosh(x)\cosh(y) \geq \sqrt{x^2 + y^2}$. People told me to use Taylor expansions and the Arithmetic-Geometric Inequality. Now I'm trying to prove that $\cosh^...
4
votes
3answers
84 views

Proving that $\cosh(x)\cosh(y) \geq \sqrt{x^2 + y^2}$

I'm trying to prove the inequality : $\cosh(x)\cosh(y) \geq \sqrt{x^2 + y^2}$ I played a bit with Geogebra and it looks true. I've tried proving it via convexity but so far I'm unsuccesful. Thank ...
0
votes
2answers
32 views

Hyperbolic tangent expansion in temperature

I'm struggling with the following type of integral $$ \int \limits_0^\infty \mathrm{d} x \, f(x) \tanh \frac{x}{T} $$ I'm desperately trying to somehow "expand" the hyperbolic tangent for low ...
5
votes
2answers
126 views

Solving hyperbolic functions

I have 2 questions with regards to solving of hyperbolic functions. I have presented my current solutions to the best of my ability. Q1: Show that the real solution $x$ of $\tanh(x) = \operatorname{...
-3
votes
2answers
49 views

Find $\lim_{x\to 0} \frac{\sinh x -\sin x}{\cosh x - \cos x}$ [closed]

Find $$\lim_{x\to 0} \frac{\sinh x -\sin x}{\cosh x - \cos x}$$ I used Hopital and couldn't find the answer which is $0$.
-1
votes
1answer
36 views

Hyperbolic inverse Function

how to find the inverse function of the arcsinh x also the domainof arcsinh
0
votes
1answer
196 views

Fitting hyperbolic functions ($\tanh$) on data.

Suppose i have some data dependant on 2 variables n and T. If i plot this data choosing n= 5 and letting T vary from 1 to 100 i get the following curve: Where the blue curve is my real data and the ...
0
votes
0answers
40 views

How to derive the metric formula in the hyperboloid model?

I think I understand where the Minkowski quadratic and bilinear form come from and why it is used in the formula that computes the distance between two points in the hyperboloid model: $d(u,\,v) = \...
4
votes
1answer
106 views

What are Hyperbolic Trig Functions Functions of?

Circular trig functions take in an angle and spit out a ratio. What do hyperbolic functions take in (I know it's a number, but what geometrically does it represent)? I've seen images that suggest ...