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

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An integral involving hyperbolic functions

$$ \large \displaystyle \int_0^\infty {\dfrac{e^{-2x} \tanh\frac{x}{2}}{x \cosh x}dx} = 2 \ln \frac{\pi}{2\sqrt{2}} $$ How to prove the above integral? What I tried : $\displaystyle I(s) = ...
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3answers
367 views

Integral $\int_0^\infty\frac{\tanh^2(x)}{x^2}dx$

It appears that $$\int_0^\infty\frac{\tanh^2(x)}{x^2}dx\stackrel{\color{gray}?}=\frac{14\,\zeta(3)}{\pi^2}.\tag1$$ (so far I have about $1000$ decimal digits to confirm that). After changing variable ...
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0answers
30 views

Integral (Tanh and Normal)

I am trying to evaluate the following: The expectation of the hyperbolic tangent of an arbitrary normal random variable. $\mathbb{E}[\mathrm{tanh}(\phi)]; \phi \sim N(\mu, \sigma^2)$ Equivalently: ...
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2answers
45 views

How to find $\int \frac {sinh(lnx)} {x}$

I've tried $\int \frac {sinh(lnx)} {2x} dx = \int \frac {e^{lnx}-e^{-ln{x}}} {2x} dx = \int \frac {x-e^{-ln{x}}} {2x} dx $
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3answers
34 views
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1answer
32 views

Why is there a factor of 1.7159 with the tanh function used in neural network activation?

I was reading about neural networks when I came across the line : Recommended f (x) = 1.7519 tanh (2/3 * x). How do we arrive at these values (we can fix the other once the other is obtained using ...
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3answers
26 views

Why does this limit of hyperbolic cosines equate to a parabola?

I bolded my main question below, and I would like to understand why the following limit is true: $$\lim _{ n\rightarrow { 0 }^{ + } }{ \frac { \cosh { (nx) } -1 }{ \cosh { n } -1 } } = { x }^{ 2 }$$ ...
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1answer
20 views

Deduction of 2nd order ODE general solutions

When I have some ODE, for example: $$ u''(t) + 5u(t) = 0 $$ I put together a characteristic equation: $$ \lambda ^2 + 5 = 0 $$ Then I compute its roots $r_1$ and $r_2$. And now there are some ...
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2answers
24 views

Clarification Needed Regarding $\sinh^{-1}(-3)$

As the definition of $\sinh^{-1}(x)$ goes : $\sinh^{-1}(x)=\ln\left(x+\sqrt{x^{2}+1}\right)$ So what I expect to get is $\sinh^{-1}(-3)=\ln\left(-3+\sqrt{10}\right)$ The value inside of the ...
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2answers
51 views

Show that $\int_{0}^{\infty} \frac{\cosh(ax)}{\cosh(\pi x)} dx=\frac{1}{2}\sec(\frac{a}{2})$ using Residue Calculus

Show that the following expression is true $$\int_{0}^{\infty} \frac{\cosh(ax)}{\cosh(\pi x)} dx=\frac{1}{2}\sec(\frac{a}{2})$$ Edit: I forgot to mention that $|a|<\pi$ Specifically, using ...
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1answer
35 views

Inverse of cosh(x)

My goal is to find the inverse of $y=\cosh(x)$ Therefore: $$x=\cosh(y)=\frac{e^y+e^{-y}}{2}=\frac{e^{2y}+1}{2e^y}$$ If we define $k=e^y$ then: $$k^2-2xk+1=0$$ $$k=e^{y}=x\pm\sqrt{x^2-1}$$ ...
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1answer
25 views

If $I_n$ is defined as $\int^1_0\sinh^nx$ show that $nI_n+(n-1)I_{n-2}=\cosh1\sinh^{n-1}1$

I've tried evaluating the first three terms, so I have the results: $I_1=\cosh1-1$ $I_2=\frac{1}{4}\sinh2-\frac{1}{2}$ $I_3=\frac{1}{12}\cosh3-\frac{3}{4}\cosh1+\frac{2}{3}$ These do satisfy the ...
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1answer
55 views

Integrate $\int\frac{dx}{a^2-x^2}$

For what values of $x$ is this valid? $$\int\frac{dx}{a^2-x^2}=\frac{1}{a}\tanh^{-1}\frac{x}{a}+C$$ I think the anwer should be $-a<x<a$ because of the domain of $tanh^{-1}$. Is this correct? ...
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2answers
31 views

system of equation with cosh and sinh

is there simple a way solve this system to find the unknown x and y $$cosh\frac{a+x}y=\frac{b}{y}$$ $$sinh\frac{a+x}y=tanθ$$ My attemp: dividing these equations we get ...
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1answer
32 views

$\tanh(x)$ is bijective, where to get continuity?

I'm trying to show $\tanh(x)$ is bijective using the intermediate value theorem. It works by noting $\tanh(x)$ as strictly increasing by differentiating $\tanh(x)$ and then surjective using limits to ...
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1answer
35 views

Applied Hyperbolic sinh(x) question - Getting Started

I'm REALLY stuck on this question as I don't really know how to begin, I understand that it has something to do with: $$\sinh⁡ x=\frac{e^x-e^{-x}}{2}$$ I'm definitely not asking for someone to do it ...
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5answers
64 views

Prove $\sinh x > x$ for all $x >0$

I did a proof for $\sinh x > x$ for all $x > 0$. But I am not sure if the proof is mathematically valid. I started by showing that $\frac{d}{dx} \sinh x = \cosh x$ and that the limit of $\cosh ...
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1answer
25 views

On convergence/divergence of improper integral with hyperbolic function

I am trying to determine whether $\int_0^{\infty}{(\frac{1}{xsinh(x)}-\frac{1}{x})dx}$ converges or diverges. It seems like inevitably divergent in 0 point. But how to show it? Maybe it should be ...
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2answers
65 views

Prove that $|\sin z| \geq |\sin x|$ and $|\cos z| \geq |\cos x|$

For any $z=x+iy$, prove the following: $$|\sin z| \geq |\sin x|$$ $$|\cos z| \geq |\cos x|$$ $\epsilon$-$\delta $ proof is not required. I don't really know how to proceed. I know in order to ...
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6answers
76 views

How to remember hyperbolic functions [closed]

I forget them all the time in solving PDE. Can someone provide a way to remember them: $$ \cosh\left(x\right)=\dfrac{e^x+e^{-x}}{2} \qquad \text{ and } \qquad \sinh\left(x\right)=\dfrac{e^x-e^{-x}}{2} ...
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1answer
21 views

Finding limit of hyperpolic expression.

I am having trouble of how to solve this kind of problem. I have to show the limit of the function: $f(x)=\frac{1 - \tanh x}{e^{-2x}}$ $\lim_{x\to\infty} f(x)$ I am to do this without using ...
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0answers
27 views

Infinite sum of a product of hyperbolic functions, help!!

Let $g_{a,b}=\mathrm{csch}(n(a-b))$ when $a$ is different from $b$ and $0$ if $a=b$. $n$ is a positive real. I am trying to compute the following sum \begin{equation} ...
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1answer
17 views

Proof of hyperbolic function limits and values.

I am hoping you guys can help me, since it seems that i'm doing something wrong. This task is to be solved without the use of differential calculus. I have the function: $f(x)= \frac{1}{\cosh x} + ...
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0answers
21 views

Reduction of expression algebraically

I have asked this question before and it helped me get a little further, but not at a solution. I have to algebraically reduce the expression: $\sinh(2 \cdot \sinh^{-1}(y))$ Now i had the idea of ...
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1answer
45 views

Prove cosh(x) and sinh(x) are continuous.

I failed this task at my univiersity and i do not understand why. No feedback was given. I have to prove that cosh(x) and sinh(x) are continious. I proved it for cosh(x) and said the same principles ...
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1answer
21 views

reduction of formula algebraically

I have been working on this one for a couple of hours and i just get stuck on every attempt i make. I have to reduce the formula algebraically: $\sinh(2 \cdot \sinh^{-1}(y))$ And I just can't seem ...
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83 views

How to prove sum related to hyperbolic tangents $\sum_{k=0}^{n-1}\frac{\tanh(…)}{1+\frac{\tanh^2x}{\tan^2(…)}}=\tanh(2nx)$

I have no Idea how to start I think to switch it to definite integral, use complex analysis, or some real analysis tricks and at the end I failed to make any progress. $$ \displaystyle ...
3
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2answers
102 views

Derivation of $\tanh$ solution to $\frac{1}{2}f''=f^3 - f$

I am a mechanical engineering student, and I am trying to solve the following ODE: $$\frac{1}{2}f''=f^3 - f$$ where $f=f(x)$ and the boundary conditions are $f(0)=0$ and $f'(\infty)=0$. On the ...
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0answers
39 views

Summation of series involving $\sinh$ of a square root

In a physics problem that I assigned in one of my classes recently, the following series arises: $$ S = \sum_{\text{odd } n} \sum_{\text{odd } m} \frac{(-1)^{(n+m)/2}}{nm} \frac{\sinh( \pi \sqrt{n^2 + ...
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5answers
41 views

Limit of function of hyperbolic

How can I - without using derivatives - find the limit of the function $f(x)=\frac{1}{\cosh(x)}+\log \left(\frac{\cosh(x)}{1+\cosh(x)} \right)$ as $x \to \infty$ and as $x \to -\infty$? We know ...
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0answers
22 views

Limit of $\lim\limits_{x \to \infty}\frac{1}{\cosh(x)}+\log\left(\frac{\cosh(x)}{1+\cosh(x)}\right)$ without l'hopital [duplicate]

I've got this function and I need to find the limit. I have tried multiple things. I've replaced $\cosh(x)$ with $e^x+e^{-x}$ and calculate on that. But it seems like no matter what I do, I always ...
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3answers
47 views

Find $\lim_{x\to \infty}\frac{1-\tanh(x)}{e^{-2x}}$ without L'Hopital

I have a problem with solving limits of hyperbolic functions. Since I am not allowed to use l'Hopital I know that I have to change the fraction so I don't get a $0/0$ or $\infty/\infty$. But my ...
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2answers
86 views

Coincidence that series of arctan is alternating series of artanh?

I noticed that the power series for $\arctan$ is the alternating series of that for $\operatorname{arctanh}$. Does it have a special meaning or even some kind of special importance?
2
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1answer
58 views

Find the positive root of the equation $\cosh x+\cos x-3=0$, other than numerically

I know you are able to find the root of the equation by using Newton-Raphson method. But is there any other way? $$\cosh x+\cos x-3=0$$ I thought maybe you could say that $-1\leq \cos x \leq 1$. So ...
2
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1answer
20 views

Find Hyperbola equation from non orthogonal asymptotes

I am looking for an easy way to find the hyperbola that has two non vertical asymptotes $y=m_1x+q_1$ and $y=m_2x+q_2$ and with a vertex located at a distance $r$ from the point where the two ...
2
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4answers
95 views

Why is $\frac{(e^x+e^{-x})}{2}$ less than $e^\frac{x^2}{2}$?

I have read somewhere that this equality holds for all $x \in \mathbb {R}$. Is it true, and if so, why is that? $$\frac{(e^x+e^{-x})}{2} \leq e^\frac{x^2}{2}$$
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1answer
70 views

Another way to calculate $\cos(x)$ and $\cosh (x)$

I usually don't see any expressions for approximate calculation of trigonometric functions aside from their Taylor series. But Taylor series work better for small angles, so in general they are not ...
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2answers
32 views

Can hyperbolic sine and cosine be combined into a single function of shifted argument?

For trigonometric functions we have a nice identity: $$A\cos x+B\sin x=\sqrt{A^2+B^2}\sin(x+\operatorname{atan2}(A,B)).\tag1$$ At the core of it is the well-known identity of ...
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3answers
40 views

Proof of integral involving the inverse hyperbolic secant and cosent

We know that $$ \int \frac{dx}{x \sqrt{a^2 \pm x^2} } = -\frac{1}{a} \ln \frac{a+ \sqrt{a^2 \pm x^2}}{\lvert x\rvert }+C$$ I tried proving this integral setting $x = a \ \mathrm{csch} \ u $ and using ...
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1answer
40 views

Proof of integral involving hyperbolic tangent

We know that $$ \int \frac{dx}{a^2-x^2} = \frac{1}{2a}\ln \left| \frac{a+x}{a-x}\right| +C$$ (That absolute value sign is supposed to be longer. I apologize for ignorance on how to make that longer on ...
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3answers
315 views

Is there a formula for the area under $\tanh(x)$?

I understand trigonometry but I've never used hyperbolic functions before. Is there a formula for the area under $\tanh(x)$? I've looked on Wikipedia and Wolfram but they don't say if there's a ...
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1answer
20 views

How to find $x$ such that $f(x)$ takes a prescribed value [closed]

Find $x$ such that \begin{equation} x\tanh(x\sqrt{2\alpha})=\frac{2}{\sqrt{2\alpha}} \end{equation}
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1answer
47 views

Finding if there is a maximum or minimum on a curve?

My apologies for being very brief with this question, the reason for this is because I don't know where to start. The question is as follows: A curve has the equation $\lambda \cosh(x) + \sinh(x)$, ...
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2answers
306 views

How to I show that $\lim_{x\to0} \frac{1}{x^2}\left(\frac{\sinh x}{x}-1\right) = \frac{1}{6}$

I can do this limit with a symbolic calculator and get the result. $$\lim_{x\to0} \left[ \frac{1}{x^2}\left(\frac{\sinh x}{x} - 1\right) \right] = \frac{1}{6}$$ But how would I do it by hand, and ...
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2answers
28 views

Difficulty finding the sum of a hyperbolic function.

Can someone please point out where I am (If I am) going wrong during the solution process of the following question: I have been presented with the following : $$4sinh(2ln(2))-cosh(ln2)$$ and told ...
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2answers
52 views

What's the relationship between hyperbola, hyperbolic functions and the exponential function?

The hyperbolic functions can be expressed using the exponential function. However how are these related to "hyperbolas"?
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3answers
67 views

How to evaluate these two integrals about hyperbolic functions?

While I was calculating the two integrals below \begin{align*} \mathcal{I}&=\int_{0}^{\infty }\frac{\cos x}{1+\cosh x-\sinh x}\mathrm{d}x\\ \mathcal{J}&=\int_{0}^{\infty }\frac{\sin x}{1+\cosh ...
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1answer
37 views

Length of the arc of a hypercycle

I am still puzzeling to get a nice equation for the arclength of an hypercycle. (I asked a similar question (less developed) about a year ago that was never answered, now i am a bit further, i ...
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3answers
83 views

Evaluating a complex limit

I would love some advice on how to approach the following limit: $$\lim_{z\to \infty} \frac{\sinh(2z)}{\cosh^2(z)}$$ or let $z= \dfrac{1}{t}$ then $$\lim_{t\to 0} ...
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1answer
21 views

Limit of complex hyperbolic tan

For the hyperbolic tan function, we have the property $\displaystyle \lim_{a \to \infty} \tanh(a(z-b)) = \mathrm{signum}(z-b)$ where $z$, $a$ and $b$ are real. But what happens if $b$ is complex ...