Questions tagged [hyperbolic-functions]
For questions related to hyperbolic functions: $\sinh$, $\cosh$, $\tanh$, and so on.
0
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0answers
20 views
can't find solution for hyperbolic equation
I need to convert the hyperbolic equation to canonical form. And I can't find solution for it. My equation:
$$ \partial^2_xu + 10\partial_x\partial_yu + 16\partial^2_yu + 5\partial_xu + 2\partial_yu =...
0
votes
2answers
32 views
Finding the root of a transcendental equation involving $\cosh$
How would I find the value of $a$ from the below equation:
$$a\cosh(\frac a{50})-a=20$$
I have tried doing it by turning the $\cosh$ into its respective exponential form but I ended up in a dead end ...
0
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0answers
23 views
Taylor expansion of fraction involving hyperbolic functions
How can I calculate the Taylor series of this function about $x=0$, $y=0$?
$$f(x,y)= \frac{1}{\coth x + \coth y}$$
I can't seem to work out the limits of the derivatives about the origin?
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votes
2answers
49 views
How do I show this hyperbolic integral: $ \int_0^1 \sinh(2x)dx= \frac14 (e-\frac1e)^2$ [closed]
How do I show that: $$ \int_0^1 \sinh(2x)dx= \frac14 \left(e-\frac1e\right)^2$$
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2answers
27 views
Get arsinh from sinh
I need to establish the inverse function of the hyperbolic sine:
I am trying to do this by setting $y = \sinh(x)$ and solving for $x$, however I got stuck at this:
$$
y=\frac{e^x -e^{-x}}{2}
$$
$$
2y=...
0
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0answers
38 views
Exponential of Matrix Rewritten as Hyperbolic Functions
Working through a paper and cannot seem to confirm the following equality:
$\exp(-\beta\hbar\omega_i\hat{S_{iz}}) = \cosh(\frac{\beta\hbar\omega_i}{2})-2\hat{S_{iz}}\sinh(\frac{\beta\hbar\omega_i}{2})...
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1answer
28 views
Getting the general solution to $G''(x) - c G(x) = 0$, for $(c\neq 0)$, in the form of hyperbolic/trigonometric functions
I have this second order PDE: $$G''(x) - c G(x) = 0$$ where c is constant.
To find a general solution of this we have to consider three cases:
$1)$ $c=0$, then we have $G''(x) = 0$, then $G(...
3
votes
3answers
37 views
Showing $\lvert\cos z\rvert^2=a\cos2x+b\cosh2y$, where $z=x+iy$, for numbers $a$ and $b$ to be determined
Show that, for $z=x+iy$,
$$\lvert\cos z\rvert^2=a \cos2x+b\cosh2y$$
where $a$ and $b$ are numbers to be determined.
My incomplete solution:
$$\begin{align}
\lvert z\rvert &= \sqrt{z^*z} \\
\...
1
vote
2answers
48 views
Completing a proof of a formula for the area of a hyperbolic right triangle
Note. The answer that inspired this question had a sign error. That error has been corrected; I'm making the corresponding correction to the formula shown here. (Existing answers acknowledge the error....
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3answers
51 views
Prove that $\sum_{n=0}^\infty {\frac{x^{2n+1}}{(2n+1)!}} = \frac{e^x - e^{-x}}{2}$
I have been trying to show:
$\sum_{n=0}^\infty {\frac{x^{2n+1}}{(2n+1)!}} = \left(\frac{e^x - e^{-x}}{2} \right)$
I have come so far as to show:
$\begin{aligned}
\sum_{n=0}^{\infty} {\frac{x^{2n+1}}...
1
vote
0answers
50 views
Evaluate the following integral using the substitution $x=9\sinh^2\theta$ $\int_0^1\sqrt{\frac{x+9}{x}}dx$
Evaluate the following integral using the substitution $x=9\sinh^2\theta$
$$\int_0^1\sqrt{\frac{x+9}{x}}dx$$
So I have attempted this integral but, I've made a mistake somewhere and I cannot see ...
1
vote
0answers
17 views
Time difference of arrival - obtaining a hyperbola from equations, I have the starting values but I'm super confused with the equations.
I need help figuring out some calculations relating to Time Difference Of Arrival, I have two reference points (mics in the real wold), mic 1 and mic 2 are 1.01706 (feet) away from each other, this ...
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3answers
106 views
Integration of $1/(1-x^2)$ to prove $\operatorname{arctanh}(x)$
Just for some background knowledge, I am doing this because I am trying to show that the derivative of arctanh(x) = $\frac{1}{1-x^2}$
How do I prove that the integral below is equal to arctanh(x)?
\...
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1answer
32 views
Know any alternative function names?
I know $e^x$ can be written as $exp(x)$ and $ln(x)$ can be written as $log_e(x)$. I wanted to know whether there are any alternative names/syntax for $sinh(x)$, $cosh(x)$, $|x|$, $|Re(z) + iIm(z)|$, ...
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2answers
37 views
Solving $4\sinh(2x)=\cosh(2x)$
Solve $$4\sinh(2x)=\cosh(2x)$$
So my method that I have used brings me to the answer of $x=0$ but this ins't correct and I cannot see what I've done wrong. My method is:
$$4\sinh(2x)-\cosh(2x)=0$$
$$\...
1
vote
1answer
51 views
How can i prove the following Equality? involving these infinite products
$$\prod_{n=2}^{\infty} \left(1-\frac{1}{n^3}\right)= \frac{\cosh(\frac{\pi}{2}\sqrt3)}{3\pi} $$
This can be found here (http://mathworld.wolfram.com/InfiniteProduct.html)
Line 22
It is known that $$...
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1answer
51 views
Exercise XIX number 15 - Calculus Made Easy
$$
\text{Use substitution}\quad\frac{1}{x}=\frac{b}{a}cosh\;u\quad\text{to show that}\quad\\
\int\;\frac{dx}{x\sqrt{a^2-b^2x^2}}=\frac{1}{a}\ln\frac{a-\sqrt{a^2-b^2x^2}}{x}\;+\;C.\\
\text{My way:}\\
\...
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votes
0answers
80 views
Solve $\sinh z = z$ [duplicate]
I have an equation,
$$\sinh z =z, \ z\in\mathbb{C}$$
How to solve it?
My question
It's trivial to see that the $z=0$ is a solution of the equation. But I want to know if there exists a non-zero ...
0
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1answer
26 views
Calculate $\oint_{\gamma} \tanh(z) dz$ on the curve $|z - \frac {\pi}{4}i|=\frac 12$
Calculate $\oint_{\gamma} \tanh(z) dz$ on the curve $|z - \frac {\pi}{4}i|=\frac 12$. I am not sure if I computed this correctly:
I tried to rewrite $\tanh(z)= \frac {\frac {e^z-e^{-z}}{2i}}{\frac {e^...
0
votes
1answer
25 views
Definite integrals: calculating exact value of $\operatorname{arcsinh}$
They didn't teach us much about hyperbolic functions. Pretty much we've been only told that they exist.
Integral's result of such form:
$$\int \frac{A}{\sqrt{x^2 + b}}dx$$
can be expressed either ...
1
vote
2answers
33 views
show $\int \frac{1}{\cosh(x)}dx = \arctan(\sinh(x))$ using the substitution $u=\sinh(x)$
I am trying to show that $$\int \frac{1}{\cosh(x)}dx = \arctan(\sinh(x))$$ Using the substitution $u=\sinh(x)$
So if $u=\sinh(x)$, then $$\frac{du}{dx}=\cosh(x)$$ thus
$$\int \frac{1}{\cosh(x)} \...
19
votes
4answers
438 views
A tricky integral involving hyperbolic functions
Can anyone suggest a method for solving the integral below? I've tried numerous things but have had no luck yet. To be honest I'm not sure an analytical solution actually exists.
$$I=\int\cosh(2x)\...
4
votes
2answers
95 views
Is this inequality true? $\coth x\leq x^{-1}+x$
By looking at the graph of RHS-LHS, I believe the following inequality holds:
$$\coth x\;\leq\; x^{-1}+x \quad\text{for } x>0$$
I can't think of a way to prove it right now, and I would love a ...
2
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1answer
35 views
Why $ \Im \frac{1}{1+e^{-s + i a }}=\frac{\sin(a)}{\cos(a)+\cosh(s)} $?
Why this $$ \Im \frac{-2}{1+e^{-s + i a }} $$
equals to this expression:
$$\\\ \frac{\sin(a)}{\cos(a)+\cosh(s)}
$$
I was trying to evaluate the Fourier transform of a hyperbolic function and my ...
0
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0answers
13 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
68 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
88 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!
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0answers
42 views
Infinite complex series of hyperbolic sine.
Can it be said that the following equivalence is true?
$$ 0 = \sum_j^\infty (e^{\zeta_j} - e^{-\zeta_j}) \Leftrightarrow \sum_j^\infty e^{\zeta_j} = \sum_j^\infty e^{-\zeta_j} \quad , \quad \...
1
vote
1answer
43 views
How should I solve this limits of a series which goes to infinity
How can I determinate
$$\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)}}$$
I can't solve how should I solve this?
Thank you.
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3answers
55 views
asymptotic expansion of $\ln(\cosh(x))$
I am currently looking into some C++ code which approximates the function in the title as follows
$$\ln\cosh(x))\approx x- \ln(2); \qquad x \geq 12$$
The approximation is plausible to me, since ...
2
votes
1answer
85 views
Inequality of $\vert \text{tanh}(x)-x+\frac{x^3}{3}\vert\leq\frac{1}{8}$ for $x\in[-1/4,1/4]$
I have an assignment, that I cannot seem to find a starting point for. I am not looking for the specific answer, but only where to begin. Any pointers are appreciated.
I have to prove the inequality
$...
0
votes
2answers
26 views
Find the general solution of the linear ordinary differential equation
$y'= 6(y - 2.5) \tanh 1.5x$
The general solution of the differential equation is to be found using integrating factor. It is a linear ODE of first order. How do I do it?
1
vote
3answers
56 views
Show that $\frac{1}{\cosh(x)} + \log\left(\frac{\cosh(x)}{1+\cosh(x)}\right) \ge 0$
I want to show that the following is true:
$$\frac{1}{\cosh(x)} + \log(\frac{\cosh(x)}{1+\cosh(x)}) \ge 0$$
Below is how I approached it, but I did not end up with an acceptable answer.
\begin{...
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votes
3answers
69 views
$|\cosh z|^2 = \cosh^2x + \sin^2y$
I'm trying to prove :
$|\cosh z|^2 = \cosh^2x + \sin^2y$
I know that:
$|\cos z|^2 = \cos^2x + \sinh^2y$
My procedure is:
$|\cosh z|^2 = |\cos iz|^2 = \cos^2y + \sinh^2x$
And since:
$\cos^2x + \sin^2x =...
2
votes
1answer
67 views
Proving the Partial Fraction Decomposition of the Hyperbolic Cotangent Function by using Poisson Summation
While skimming through the wonderful post What are some examples of colorful language in serious mathematics papers? on MathOverflow an example given by Ben Green aroused my curiosity. He referred to ...
2
votes
1answer
97 views
Evaluating $\int_{-\infty}^{\infty}\frac{1-a\cosh(\alpha x)}{(\cosh(\alpha x)-a)^2}\cos(\beta x)\,dx$
I would like to solve the following improper integral:
$$\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$$ resulted in a hyperbola, and to find the center, ...
3
votes
0answers
60 views
Show $ \coth(sx)(\coth(sx)-\frac{1}{s}\coth(x))$ is increasing
Does anyone know how to show that for $s\geq2$ fixed the function
$$ \coth(sx)(\coth(sx)-\frac{1}{s}\coth(x))$$
is increasing for $x>0$ ?
Thanks :)
1
vote
2answers
101 views
Show the inequalities $|\sinh x| \leq 3|x|$ and $| \cosh x - 1 | \leq 3|x|$ for $|x| \lt \frac{1}{2}$
I am working through a real analysis book and one of the exercises want me to show that these inequalities exists.
$|\sinh x| \leq 3|x|$ and $| \cosh x - 1 | \leq 3|x|$ for $|x| \lt \frac{1}{2}$
I ...
1
vote
1answer
37 views
Integral of $\frac{1}{\sqrt{(z-z')^2 + s^2}}$
I have a question about the signs of the antiderivative when one integrate $\frac{1}{\sqrt{(z-z')^2 + s^2}}$.
According to Wolfram Alpha here and here:
If one evaluates $\int \frac{1}{\sqrt{(z-z')^2 ...
0
votes
1answer
32 views
Simplification using hyperbolic trig identities
I'm trying to follow the derivation of equipotential lines surrounding two parallel, oppositely charged wires, found here. But I feel like the source is skipping a few critical steps in the derivation ...
0
votes
1answer
52 views
Is $\tanh^{-1}\left(\sin 2 \left(x +\dfrac {\pi}{4}\right) \right) = \dfrac {1}{\pi} \ln \left( \cot ^2 x \right) $?
https://www.desmos.com/calculator/av124c6vix
As you see the above graph that $artanh\left(\sin 2 \left(x +\frac {\pi}{4}\right) \right)$ is similar to $ \frac {1}{\pi} \ln \left( \cot ^2 x \right) $.
...
0
votes
3answers
44 views
What is the value of $\cosh(\sqrt{i})$?
I am puzzling about the value of
$$\cosh(\sqrt{i})$$
I know that
$$\sqrt{i} = \sqrt{\frac{1}{2}}+i\sqrt{\frac{1}{2}}$$
But how to go on with that? Are there also multiple values?
Thank you ...
1
vote
1answer
133 views
Integral $\int_{-\infty}^{\infty} \frac{\sinh(x)}{x [a+\cosh(x)]^2}dx$
I have difficulties with calculating the following integral:
$$I(a)=\int_{-\infty}^{\infty} \frac{\sinh(x)}{x [a+\cosh(x)]^2} \mathrm dx~~~~~~~,\text{where } a>1$$
For the case with $a=1$ the ...
1
vote
3answers
85 views
Evaluate $\int \cosh^3 (x) \sinh^2 (x )dx $
Evaluate $$\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$$
...
2
votes
0answers
50 views
Some infinite series involving hyperbolic functions
I have been working on a problem in Quantum Field Theory that involves some sums that I haven't been able to get a closed form. They all involve hyperbolic functions:
\begin{gather}
\sum_{m=1}^{\...
5
votes
0answers
53 views
Why is sinh called “sinus hyperbolicus” despite being just a regular e function?
What does the sinus hyperbolicus have to do with the sinus? Their graphs do not look alike at all.
The only similarity I can find is that their exponential representation looks similar.
$sin(x) = \...
0
votes
1answer
62 views
Scalar multiplication and vector addition with hyperbolic vectors
I'm representing hyperbolic vectors using the Minowski hyperboloid: $x_0^2 - x_1^2 - x_2^2 = 1$.
I understand that the distance between two hyperbolic vectors of this form is $acosh(B(u, v))$ where $...
0
votes
0answers
17 views
A series identity for $\tanh(z)/z$ [duplicate]
How would one go about proving this identity (for real $z$)?
$$
\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 ...
