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Questions tagged [hyperbolic-functions]

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

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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 =...
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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 ...
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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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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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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=...
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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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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(...
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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} \\ \...
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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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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}}...
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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 ...
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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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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: $$4\sinh(2x)-\cosh(2x)=0$$ $$\...
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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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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 ...
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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^...
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1answer
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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 ...
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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)} \...
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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)\...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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 \...
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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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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 ...
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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 $...
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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?
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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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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 =...
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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 ...
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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....
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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 ...
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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, ...
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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 :)
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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 ...
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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 ...
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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 ...
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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) $. ...
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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 ...
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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 ...
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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 $$\int \cosh^5(x)-\cosh^3(x) dx$$ ...
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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}^{\...
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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) = \...
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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 $...
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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 ...