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Questions tagged [hyperbolic-functions]

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

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Burger's Equation

I have solved Burger's equation using Total Variation Diminishing for initial conditions $\sin(2\pi x)$, and the results are as shown here: Do the results look okay? How would I find the exact ...
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Hyperbolic functions given value [on hold]

Given $\sinh x = 5/12$ find the remaining five hyperbolic functions ($\cosh x$, $\tanh x$, etc.).
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35 views

Some infinite series involving hyperbolic functions

I have been working on a problem in Quantum Field Theory that involves some sums that I haven't been able to get a closed form. They all involve hyperbolic functions: \begin{gather} \sum_{m=1}^{\...
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48 views

Why is sinh called “sinus hyperbolicus” despite being just a regular e function?

What does the sinus hyperbolicus have to do with the sinus? Their graphs do not look alike at all. The only similarity I can find is that their exponential representation looks similar. $sin(x) = \...
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1answer
38 views

Scalar multiplication and vector addition with hyperbolic vectors

I'm representing hyperbolic vectors using the Minowski hyperboloid: $x_0^2 - x_1^2 - x_2^2 = 1$. I understand that the distance between two hyperbolic vectors of this form is $acosh(B(u, v))$ where $...
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A series identity for $\tanh(z)/z$ [duplicate]

How would one go about proving this identity (for real $z$)? $$ \sum_{j=1}^\infty \frac{8}{(1 - 2j)^2 \pi^2 + 4z^2} = \frac{\tanh z}{z} $$ Mathematica assures me that the above is true, but I have no ...
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1answer
55 views

Definite integral with hyperbolic cosine and circumference segment

I've been struggling with this definite integral: $$I=\int_{0}^{a}\frac{\sqrt{1-\frac{x^2}{a^2}}}{b+\cosh x} dx,$$ with $$ a>0,\\ b>1.$$ Mathematica won't solve it Any idea on how to solve ...
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50 views

Prove the size of a hyperbolic angle is twice the area of its hyperbolic sector.

I'm trying to figure out how the hyperbolic functions are derived using a unit hyperbola. According to this walkthrough, argument u in (cosh(u), sinh(u)) should be equal to 2A, where A is the area ...
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1answer
77 views

Is $ \sinh(x) \sinh(y)=\sinh(y) \sinh(x)$?

Is $ \sinh(x) \sinh(y)=\sinh(y) \sinh(x)$? While evaluation a question on multiple integral I have got answer $4\sinh(3) \sinh(1)$. It was a multiple choice questions with a) $4\sinh(3) \sinh(1)$ ...
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4answers
201 views

Evaluate $\int_0^1 \frac{\operatorname{arctanh}^3(x)}{x}dx$

I'm trying to evaluate the following integral: $$\int_0^1 \frac{\operatorname{arctanh}^3(x)}{x}dx$$ I was playing around trying to numerically approximate the answer with known constants and found ...
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1answer
32 views

Equivalence of two antiderivatives involving trigonometric/hyperbolic functions

I am struggling to see how two antiderivatives of the same function—obtained in two different ways—are equivalent (what I mean by equivalent is that they differ from just a constant), if they even are ...
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Solution of Transcendental equations, trigonometric ones

I am studying vibration of beams, with continuous properties; and I arrived to some kind of trascendental equations. The book I am using, (Chopra, 2014), says that one of the solution is the numerical ...
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1answer
40 views

How to solve this functional equation involving hyperbolic functions?

I'm reading this (physics) book. They have the recurrence relation (book eq. 14.2.14) $$f(K_1,0)=-\frac{1}{2}\ln\{2\sqrt{\cosh(2K_1)}\}+\frac{1}{2}f(\ln\sqrt{\cosh(2K_1)},0).\qquad(1)$$ They give ...
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27 views

Is there an analogue of the Fourier transform based on hyperbolic trig functions?

Is there something analogous to Fourier series or the Fourier transform but which is based on hyperbolic trig functions rather than $\sin, \cos$, and $\exp$?
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46 views

Solve the equation $\frac{x^7}{7}=1+\sqrt[7]{10}x\left(x^2-\sqrt[7]{10}\right)^2$

Solve in $\mathbb R$ the following equation. $$\frac{x^7}{7}=1+\sqrt[7]{10}x\left(x^2-\sqrt[7]{10}\right)^2$$ Solution Setting $x=2\times 10^{\frac 1{14}}y$, the equation becomes $64y^7-112y^5+56y^...
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1answer
53 views

A simple Variation on the Imaginary Unit i

I think a more appropriate tag would have been 'quasicomplex numbers' rather than 'hypercomplex numbers'. I'm normally perfectly comfortable with the correspondence between hyperbolic functions & ...
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1answer
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Integral of $\ln(\tanh(x))$

I'd like a hint toward how I could evaluate this definite integral. I'm aware it's likely to be non elementary and I haven't found a way to evaluate it yet:$$\int_0^\infty \ln(\tanh(x))\,\,\mathrm{d}x$...
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1answer
20 views

Geometric interpretation of tanh

Ok so in today's lecture on hyperbolic functions, the lecturer drew the well-known graph of the equilateral hyperbola, which shows sinh(a), cosh(a) and the area which is equal to a/2. However, when I ...
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2answers
31 views

Proving an inequality including tanh functions

For $k_2 \geq k_1 > 0$ and $d\geq 1$, I need to show that $$ k_1\tanh(k_1d) + k_2\tanh(k_2d) - 2\sqrt{k_1k_2\tanh(k_1d)\tanh(k_2d)} \leq (k_2-k_1)\tanh((k_2-k_1)d). $$ I've started by letting $$ f(...
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1answer
37 views

Prove that $\mathrm{arcsinh}(x)$ is an odd function

The inverse hyperbolic sine $\sin^{-1}(x) = \mathrm{arcsinh}(x)$ is an odd function. This can be proved by manipulating the expression $\mathrm{arcsinh}(-x) = y$ as shown here. But how to prove it ...
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1answer
27 views

Equivalence of two methods to obtain $\sin \theta \geq 1$ with a complex $\theta$

When evaluating the function $\sin \theta$ with a complex angle $\theta$, a real value $A \geq 1$ can be obtained in two ways. Considering $\theta = i \log \left[ -i \left( A + \sqrt{A^2 - 1} \right) ...
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1answer
24 views

Hyperbolic functions problem

If $p^2\sinh x+q^2\cosh x = r^2$ has at least one root, how do I show that $r^4 > p^4-q^4?$
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127 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}\...
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2answers
85 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_{...
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1answer
22 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_{}^{}...
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0answers
22 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 ...
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1answer
24 views

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

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

Problem wih hyperbolic function

Can we use hyperbolic function to solve the following problems ? If $(\sqrt {{y^2-x^3}} - x)(\sqrt {{x^2} + y^3} - y) =y^3$ , prove that $x+ y = 0$ If $(\sqrt {{x^2+y^4}} - x)(\sqrt {{y^2} + x^...
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2answers
53 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 ...
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1answer
119 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., $$ \left( \sin(k\pi y) \right)_{k=1}^{\infty} $$ ...
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1answer
23 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?
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1answer
33 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 ...
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1answer
33 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) ...
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1answer
53 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 ...
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4answers
384 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 ...
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1answer
25 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: ...
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1answer
110 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 ...
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2answers
147 views

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

I have problems to calculate analytically the (inverse) Fourier transform of $x / \tanh(x)$: $$\frac{1}{\sqrt{2\pi}}\int_\mathbb{R} \frac{x}{\tanh(x)} \mathrm{e}^{- \mathrm{i} x k} \mathrm{d} x$$ ...
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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 ...
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3answers
105 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)=\...
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2answers
55 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 ...
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4answers
833 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 ...
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2answers
54 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 ...
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2answers
211 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{...
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1answer
30 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 $\cosh^2(x)-\sinh^2(x)=1.$
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1answer
144 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 ...
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1answer
29 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 ...
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1answer
97 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)},...