Questions tagged [hyperbolic-functions]
For questions related to hyperbolic functions: $\sinh$, $\cosh$, $\tanh$, and so on.
1
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0answers
22 views
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 ...
-2
votes
0answers
23 views
Hyperbolic functions given value [on hold]
Given $\sinh x = 5/12$ find the remaining five hyperbolic functions ($\cosh x$, $\tanh x$, etc.).
2
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0answers
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}^{\...
5
votes
0answers
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) = \...
0
votes
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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0answers
17 views
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 ...
1
vote
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 ...
2
votes
0answers
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 ...
3
votes
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) $4\sinh(3) \sinh(1)$
...
5
votes
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 ...
1
vote
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 ...
0
votes
0answers
10 views
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 ...
2
votes
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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0answers
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 $\exp$?
2
votes
0answers
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^...
-1
votes
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 & ...
-1
votes
1answer
19 views
Find the orthogonal trajectory to the family of curves $x^2+y^2=4c^2$ where $c$ is a constant.
Differentiate w.r.t $x$:
$$x+y\frac{dx}{dy}=4c^2$$
8
votes
4answers
172 views
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:$$\int_0^\infty \ln(\tanh(x))\,\,\mathrm{d}x$...
0
votes
1answer
20 views
Geometric interpretation of tanh
Ok so in today's lecture on hyperbolic functions, the lecturer drew the well-known graph of the equilateral hyperbola, which shows sinh(a), cosh(a) and the area which is equal to a/2.
However, when I ...
2
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2answers
31 views
Proving an inequality including tanh functions
For $k_2 \geq k_1 > 0$ and $d\geq 1$, I need to show that
$$ 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(...
0
votes
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 ...
0
votes
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) ...
0
votes
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 $r^4 > p^4-q^4?$
12
votes
2answers
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
$$\int_0^{\int_0^u\operatorname{sech}\upsilon d\upsilon}\sec\upsilon d\upsilon \equiv u$$
$$\int_0^{\int_0^u\sec\upsilon d\upsilon}\operatorname{sech}\...
3
votes
2answers
85 views
Taylor series of functions
Consider the Taylor series of the function
$$\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_{...
1
vote
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 :
$$
\int_{}^{} \frac{\text{d}\psi}{\psi}=-2\int_{}^{}...
0
votes
0answers
22 views
Simplify trigonometric expression of hyperbolic functions
I have $\cos^2x\cosh^2y - \sin^2x\sinh^2y$. I saw it written simplified as $\cosh^2 y - \sin^2 x$. But I don't get how to get it.
My attempts were to write $\cosh^2y -1$ instead of $\sinh^2y$ but ...
1
vote
1answer
24 views
How to evaluate hyperbolic functions, involving inverses, by hand?
How does one evaluate $$\sinh(2{\sinh^{-1}{(2)}})$$
by hand?
0
votes
1answer
49 views
using sinh(x) to find series representation of arcsinh(x)
From "Complex Variables Demystified", 2008, page 102:
Given:
$$sinh(z)=\frac{e^z-e^{-z}}{2}$$
find the series representation for arcsinh(x).
Solution:
(1) The Maclaurin theorem can be used to ...
1
vote
1answer
105 views
Problem wih hyperbolic function
Can we use hyperbolic function to solve the following problems ?
If $(\sqrt {{y^2-x^3}} - x)(\sqrt {{x^2} + y^3} - y) =y^3$ , prove that $x+ y = 0$
If $(\sqrt {{x^2+y^4}} - x)(\sqrt {{y^2} + x^...
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votes
2answers
53 views
Manual approximation of $\operatorname{sech}(0.7)$
In the archive of a midterm exam collection there are some question like the one above.
How can we approximate expressions like
$$\operatorname{sech}(0.7)$$
without a calculator?
Thanks in ...
3
votes
1answer
119 views
Basis formed by hyperbolic functions
I am currently working with separation of variables for different kinds of PDEs and one often uses here the fact that one has the sine base, i.e.,
$$ \left( \sin(k\pi y) \right)_{k=1}^{\infty} $$
...
0
votes
1answer
23 views
Why is the domain of the hyperbolic function $\sinh x$ is symmetric about the origin?
My book asked me to prove that the function $\sinh x$ is odd, but in order to be odd I must be sure that the domain of it symmetric about the origin, how can I be sure from this?
1
vote
1answer
33 views
Is the similarity between tanh and normal distribution just coincidence?
So, explaining to someone why tanh is used in machine learning (i.e. it squashes an open range to -1..+1, and changes most rapidly around 0), I brought up $\frac d{dx}$ $tanh(x)$, and it looks just ...
0
votes
1answer
33 views
Solve second order differential equation with cosh using frobenios method
i need to show that the differential equation
$y^{''}+(\cosh(2x)-4)y = 0$
has the solution:
$ y(x) = x+\frac{1}{2}x^3-\frac{1}{40}x^5 -... $
using Frobenius method.
I started by writing cosh(2x) ...
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votes
1answer
53 views
Find min distance from y = cosh x to y = x
The full question is this: Point P on curve y = cosh x is such that its perpendicular distance from the line y = x is a minimum. Show P's coordinates are (ln(1 + root 2), root 2).
I am completely at ...
1
vote
4answers
384 views
Proof for hyperbolic trigonometric identities [closed]
I've been studying hyperbolic functions and was wondering where the following two identities were derived from:
$$\sinh(x) = \frac{e^{x}-e^{-x}}{2}$$
$$\cosh(x) = \frac{e^{x}+e^{-x}}{2}$$
I ...
1
vote
1answer
25 views
Getting function from four points
I'm facing this problem I can't solve myself. I've got four points on a cartesian place, and I would like to find the function that equates them.
Coords are:
...
0
votes
1answer
110 views
Inverse Function Theorem: Proving Global Invertibility.
My question states: Prove that the following coordinate transformation is invertible everywhere, at all values of $(x, y)$ .
$$u = \arctan(x - y)$$
$$v = \sinh(3x) + 2\sinh(y)$$
That is x and y ...
1
vote
2answers
147 views
Fourier transform of $x / \tanh(x)$
I have problems to calculate analytically the (inverse) Fourier transform of $x / \tanh(x)$:
$$\frac{1}{\sqrt{2\pi}}\int_\mathbb{R} \frac{x}{\tanh(x)} \mathrm{e}^{- \mathrm{i} x k} \mathrm{d} x$$
...
0
votes
0answers
20 views
Problems to find the correct limit of a logarithm of an hyperbolic function
I want to find the next limits
$$
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)$$
when I evaluate ...
2
votes
3answers
105 views
Prove formula $\operatorname{arctanh} x = \frac12\,\ln \left(\frac{1+x}{1-x}\right)$
Problem
Prove formula $\operatorname{arctanh} x = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right)$
Attempt to solve
To start off with definition of functions $\sinh(x)$ and $\cosh(x)$
$$ \cosh(x)=\...
1
vote
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 ...
4
votes
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 ...
0
votes
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 ...
-1
votes
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 $$x\operatorname{arccosh}(x + 1) -
\int \frac{...
0
votes
1answer
30 views
What is the relation to $\sinh{x},\cosh{x}$ and $\sin{x},\cos{x}$ [duplicate]
I've learned what $\sinh{x},\cosh{x}$ (the hyperbolic trig functions) are defined as formula, but how is it related to $\sin{x},\cos{x}?$
The only thing I've noticed is that $\cosh^2(x)-\sinh^2(x)=1.$
0
votes
1answer
144 views
Relationship between hyperbolic functions and complex analysis
As you know, hyperbolic functions are defined in terms of $e$. For example, the hyperbolic cosine:
$$\cosh x = \frac{e^x + e^{-x}}{2}$$
There is a result in complex analysis that looks uncannily ...
0
votes
1answer
29 views
Segmentations with equal length on a hyperbola [closed]
How can I divide a hyperbola or one of its branches into many segmentations with equal length = l along the curve. hyperbola can be express as:
$x^2/a^2 - y^2/b^2 = 1$
How to compute the x, y for the ...
0
votes
1answer
97 views
Find the Taylor series of argtanh(x) using sinh(x) and cosh(x)
I just finished my exam a few hours a go, and there was 1 question I couldn't answer. I was asked to derive the Taylor series of $\arg\tanh(x)$ using the fact that $$\tanh(x)=\frac{\sinh(x)}{\cosh(x)},...
