Questions tagged [hyperbolic-functions]
For questions related to hyperbolic functions: $\sinh$, $\cosh$, $\tanh$, and so on.
0
votes
0answers
11 views
Undefined arctanh evaluating when implementing hyperbolic CORDIC
I am trying to implement an exponential function using the CORDIC method. I am able to get the 'basic' version to work fine, but that only works for a very limited input range (i.e., inputs smaller ...
1
vote
2answers
40 views
Asymptotic behaviour of Convolution power
We have a function $f(t): [0,\infty) \to \mathbb{R}$. The convolution of $f(t)$ with itself is:
\begin{equation}
(f*f)(t) = \int\limits_0^t \! \mathrm{d}\tau \; f(\tau) f(t-\tau)
\end{equation}
We ...
4
votes
3answers
77 views
Is it possible express $\sinh(nx)$ in terms of $\sinh^k(x)$?
I wonder if it’s possible express $\sinh(nx)$ in terms of $\sinh^k(x)$, that is
$$\sinh(nx)=\sum_{k=0}^{A(n)} A_k\sinh^k(x)$$
Thanks in advance!
3
votes
0answers
37 views
Infinite complex series of hyperbolic sine.
Can it be said that the following equivalence is true?
$$ 0 = \sum_j^\infty (e^{\zeta_j} - e^{-\zeta_j}) \Leftrightarrow \sum_j^\infty e^{\zeta_j} = \sum_j^\infty e^{-\zeta_j} \quad , \quad \...
1
vote
1answer
41 views
How should I solve this limits of a series which goes to infinity
How can I determinate
$$\lim_{n\to \infty}{\frac{\sum_{i=1}^n\sinh\big(\frac{i}{n}\big)}{\sum_{i=1}^n\cosh\big(\frac{i}{n}\big)}}$$
I can't solve how should I solve this?
Thank you.
0
votes
3answers
24 views
asymptotic expansion of $\ln(\cosh(x))$
I am currently looking into some C++ code which approximates the function in the title as follows
$$\ln\cosh(x))\approx x- \ln(2); \qquad x \geq 12$$
The approximation is plausible to me, since ...
2
votes
1answer
76 views
Inequality of $\vert \text{tanh}(x)-x+\frac{x^3}{3}\vert\leq\frac{1}{8}$ for $x\in[-1/4,1/4]$
I have an assignment, that I cannot seem to find a starting point for. I am not looking for the specific answer, but only where to begin. Any pointers are appreciated.
I have to prove the inequality
$...
0
votes
2answers
18 views
Find the general solution of the linear ordinary differential equation
$y'= 6(y - 2.5) \tanh 1.5x$
The general solution of the differential equation is to be found using integrating factor. It is a linear ODE of first order. How do I do it?
1
vote
3answers
49 views
Show that $\frac{1}{\cosh(x)} + \log\left(\frac{\cosh(x)}{1+\cosh(x)}\right) \ge 0$
I want to show that the following is true:
$$\frac{1}{\cosh(x)} + \log(\frac{\cosh(x)}{1+\cosh(x)}) \ge 0$$
Below is how I approached it, but I did not end up with an acceptable answer.
\begin{...
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3answers
54 views
$|\cosh z|^2 = \cosh^2x + \sin^2y$
I'm trying to prove :
$|\cosh z|^2 = \cosh^2x + \sin^2y$
I know that:
$|\cos z|^2 = \cos^2x + \sinh^2y$
My procedure is:
$|\cosh z|^2 = |\cos iz|^2 = \cos^2y + \sinh^2x$
And since:
$\cos^2x + \sin^2x =...
2
votes
1answer
41 views
Proving the Partial Fraction Decomposition of the Hyperbolic Cotangent Function by using Poisson Summation
While skimming through the wonderful post What are some examples of colorful language in serious mathematics papers? on MathOverflow an example given by Ben Green aroused my curiosity. He referred to ...
2
votes
1answer
90 views
Evaluating $\int_{-\infty}^{\infty}\frac{1-a\cosh(\alpha x)}{(\cosh(\alpha x)-a)^2}\cos(\beta x)\,dx$
I would like to solve the following improper integral:
$$\int_{-\infty}^{\infty}\frac{1-a\cosh(\alpha x)}{(\cosh(\alpha x)-a)^2}\cos(\beta x)\,dx$$
where $a$, $\alpha$ and $\beta$ are real constants....
2
votes
3answers
68 views
Primitive of $\frac{1-a\cosh(x)}{(\cosh(x)-a)^2}$
I would like to solve the following primitive:
$$\int\frac{1-a\cosh(x)}{(\cosh(x)-a)^2}\,dx$$
where $a$ is a constant, $0\leq a\leq1$.
I really don't know how to start. I can't relate the numerator ...
0
votes
1answer
22 views
Explain why a hyperbola with center $(m,n)$ has the parametrized curve $r(t) = (m + a cosh, n + b sinh)$ “laying” on it
Struggling with how to approach a task given at university.
The first part of the task asked to show that the equation $$4x^2-32x-9y^2+36y=8$$ resulted in a hyperbola, and to find the center, ...
3
votes
0answers
54 views
Show $ \coth(sx)(\coth(sx)-\frac{1}{s}\coth(x))$ is increasing
Does anyone know how to show that for $s\geq2$ fixed the function
$$ \coth(sx)(\coth(sx)-\frac{1}{s}\coth(x))$$
is increasing for $x>0$ ?
Thanks :)
1
vote
2answers
75 views
Show the inequalities $|\sinh x| \leq 3|x|$ and $| \cosh x - 1 | \leq 3|x|$ for $|x| \lt \frac{1}{2}$
I am working through a real analysis book and one of the exercises want me to show that these inequalities exists.
$|\sinh x| \leq 3|x|$ and $| \cosh x - 1 | \leq 3|x|$ for $|x| \lt \frac{1}{2}$
I ...
1
vote
1answer
36 views
Integral of $\frac{1}{\sqrt{(z-z')^2 + s^2}}$
I have a question about the signs of the antiderivative when one integrate $\frac{1}{\sqrt{(z-z')^2 + s^2}}$.
According to Wolfram Alpha here and here:
If one evaluates $\int \frac{1}{\sqrt{(z-z')^2 ...
0
votes
1answer
26 views
Simplification using hyperbolic trig identities
I'm trying to follow the derivation of equipotential lines surrounding two parallel, oppositely charged wires, found here. But I feel like the source is skipping a few critical steps in the derivation ...
0
votes
1answer
48 views
Is $\tanh^{-1}\left(\sin 2 \left(x +\dfrac {\pi}{4}\right) \right) = \dfrac {1}{\pi} \ln \left( \cot ^2 x \right) $?
https://www.desmos.com/calculator/av124c6vix
As you see the above graph that $artanh\left(\sin 2 \left(x +\frac {\pi}{4}\right) \right)$ is similar to $ \frac {1}{\pi} \ln \left( \cot ^2 x \right) $.
...
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votes
3answers
36 views
What is the value of $\cosh(\sqrt{i})$?
I am puzzling about the value of
$$\cosh(\sqrt{i})$$
I know that
$$\sqrt{i} = \sqrt{\frac{1}{2}}+i\sqrt{\frac{1}{2}}$$
But how to go on with that? Are there also multiple values?
Thank you ...
0
votes
1answer
123 views
Integral $\int_{-\infty}^{\infty} \frac{\sinh(x)}{x [a+\cosh(x)]^2}dx$
I have difficulties with calculating the following integral:
$$I(a)=\int_{-\infty}^{\infty} \frac{\sinh(x)}{x [a+\cosh(x)]^2} \mathrm dx~~~~~~~,\text{where } a>1$$
For the case with $a=1$ the ...
1
vote
3answers
61 views
Evaluate $\int \cosh^3 (x) \sinh^2 (x )dx $
Evaluate $$\int \cosh^3 (x) \sinh^2 (x )dx $$
So my original thought was to apply the identity that $\sinh^2(x)=\cosh^2(x)-1$. This means that my integral becomes
$$\int \cosh^5(x)-\cosh^3(x) dx$$
...
2
votes
0answers
44 views
Some infinite series involving hyperbolic functions
I have been working on a problem in Quantum Field Theory that involves some sums that I haven't been able to get a closed form. They all involve hyperbolic functions:
\begin{gather}
\sum_{m=1}^{\...
5
votes
0answers
50 views
Why is sinh called “sinus hyperbolicus” despite being just a regular e function?
What does the sinus hyperbolicus have to do with the sinus? Their graphs do not look alike at all.
The only similarity I can find is that their exponential representation looks similar.
$sin(x) = \...
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votes
1answer
47 views
Scalar multiplication and vector addition with hyperbolic vectors
I'm representing hyperbolic vectors using the Minowski hyperboloid: $x_0^2 - x_1^2 - x_2^2 = 1$.
I understand that the distance between two hyperbolic vectors of this form is $acosh(B(u, v))$ where $...
0
votes
0answers
17 views
A series identity for $\tanh(z)/z$ [duplicate]
How would one go about proving this identity (for real $z$)?
$$
\sum_{j=1}^\infty \frac{8}{(1 - 2j)^2 \pi^2 + 4z^2} = \frac{\tanh z}{z}
$$
Mathematica assures me that the above is true, but I have no ...
1
vote
1answer
60 views
Definite integral with hyperbolic cosine and circumference segment
I've been struggling with this definite integral:
$$I=\int_{0}^{a}\frac{\sqrt{1-\frac{x^2}{a^2}}}{b+\cosh x} dx,$$
with
$$ a>0,\\ b>1.$$
Mathematica won't solve it
Any idea on how to solve ...
3
votes
1answer
103 views
Prove the size of a hyperbolic angle is twice the area of its hyperbolic sector.
I'm trying to figure out how the hyperbolic functions are derived using a unit hyperbola.
According to this walkthrough, argument $u$ in $(\cosh(u), \sinh(u))$ should be equal to $2A$, where $A$ is ...
3
votes
1answer
79 views
Is $ \sinh(x) \sinh(y)=\sinh(y) \sinh(x)$?
Is $ \sinh(x) \sinh(y)=\sinh(y) \sinh(x)$? While evaluation a question on multiple integral I have got answer $4\sinh(3) \sinh(1)$.
It was a multiple choice questions with
a) $4\sinh(3) \sinh(1)$
...
5
votes
4answers
226 views
Evaluate $\int_0^1 \frac{\operatorname{arctanh}^3(x)}{x}dx$
I'm trying to evaluate the following integral:
$$\int_0^1 \frac{\operatorname{arctanh}^3(x)}{x}dx$$
I was playing around trying to numerically approximate the answer with known constants and found ...
1
vote
1answer
33 views
Equivalence of two antiderivatives involving trigonometric/hyperbolic functions
I am struggling to see how two antiderivatives of the same function—obtained in two different ways—are equivalent (what I mean by equivalent is that they differ from just a constant), if they even are ...
0
votes
0answers
11 views
Solution of Transcendental equations, trigonometric ones
I am studying vibration of beams, with continuous properties; and I arrived to some kind of trascendental equations.
The book I am using, (Chopra, 2014), says that one of the solution is the numerical ...
2
votes
1answer
43 views
How to solve this functional equation involving hyperbolic functions?
I'm reading this (physics) book. They have the recurrence relation (book eq. 14.2.14)
$$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)$$
They give ...
1
vote
0answers
31 views
Is there an analogue of the Fourier transform based on hyperbolic trig functions?
Is there something analogous to Fourier series or the Fourier transform but which is based on hyperbolic trig functions rather than $\sin, \cos$, and $\exp$?
2
votes
0answers
52 views
Solve the equation $\frac{x^7}{7}=1+\sqrt[7]{10}x\left(x^2-\sqrt[7]{10}\right)^2$
Solve in $\mathbb R$ the following equation.
$$\frac{x^7}{7}=1+\sqrt[7]{10}x\left(x^2-\sqrt[7]{10}\right)^2$$
Solution
Setting $x=2\times 10^{\frac 1{14}}y$, the equation becomes $64y^7-112y^5+56y^...
-1
votes
1answer
59 views
A simple Variation on the Imaginary Unit i
I think a more appropriate tag would have been 'quasicomplex numbers' rather than 'hypercomplex numbers'.
I'm normally perfectly comfortable with the correspondence between hyperbolic functions & ...
-1
votes
1answer
25 views
Find the orthogonal trajectory to the family of curves $x^2+y^2=4c^2$ where $c$ is a constant.
Differentiate w.r.t $x$:
$$x+y\frac{dx}{dy}=4c^2$$
8
votes
4answers
205 views
Integral of $\ln(\tanh(x))$
I'd like a hint toward how I could evaluate this definite integral. I'm aware it's likely to be non elementary and I haven't found a way to evaluate it yet:$$\int_0^\infty \ln(\tanh(x))\,\,\mathrm{d}x$...
0
votes
1answer
26 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
votes
2answers
33 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 $$ f(...
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votes
1answer
40 views
Prove that $\mathrm{arcsinh}(x)$ is an odd function
The inverse hyperbolic sine $\sin^{-1}(x) = \mathrm{arcsinh}(x)$ is an odd function. This can be proved by manipulating the expression $\mathrm{arcsinh}(-x) = y$ as shown here.
But how to prove it ...
0
votes
1answer
27 views
Equivalence of two methods to obtain $\sin \theta \geq 1$ with a complex $\theta$
When evaluating the function $\sin \theta$ with a complex angle $\theta$, a real value $A \geq 1$ can be obtained in two ways.
Considering $\theta = i \log \left[ -i \left( A + \sqrt{A^2 - 1} \right) ...
0
votes
1answer
34 views
Hyperbolic functions problem
If $p^2\sinh x+q^2\cosh x = r^2$ has at least one root, how do I show that $r^4 > p^4-q^4?$
12
votes
2answers
129 views
Showing $\int_0^{\int_0^u{\rm sech}vdv}\sec vdv\equiv u$ and $\int_0^{\int_0^u\sec vdv}{\rm sech} vdv\equiv u$
The two following very weird-looking theorems
$$\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
89 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
25 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
98 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
107 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^...
