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

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1answer
50 views

Evaluate the indefinite integral $\int \frac{t\sin at}{b^2+t^2}dt$

It is known DLMF (25.2.8) that for $\Re s>0$ and for integers $N\geq 1$ $$\zeta(s)=\sum_{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-s\int_{N}^\infty \frac{x-\lfloor x \rfloor}{x^{s+1}} dx,$$ where ...
2
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0answers
26 views

Distance between points lying on a hyperbola?

The question is rather simple but I can't find the answer I'm looking for anywhere. On an ordinary 1-dimensional hyperbola, given two points on the hyperbola, what is the length of the path between ...
0
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1answer
23 views

Hyperbolic sin derivation

https://www.youtube.com/watch?v=zd3RyRk6wYI On Khan Academy, Sal derives the hyperbolic function of sin in terms of $i\theta$. My question is, how did he get rid of the $i$ in the denominator? I know ...
26
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1answer
555 views

Curious about an empirically found continued fraction for tanh

First of all, and since this is my first question in this forum, I would like to specify that I am not a professional mathematician (but a philosophy teacher); I apologize by advance if something is ...
0
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0answers
24 views

Complex conjugate of the logarithm of the hyperbolic tangent

Given the Schwarz reflection principle, I would aytomaticaly write down that the complex conjugate of the following function: $$ ln[tanh(z)] $$, where z is a complex number, is: $$ln[tanh(\bar{z})] ...
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2answers
33 views

Let $s(x)=\sum_{n=0}^\infty \frac1{(2n+1)!}x^{2n+1}$, $c(x)=\sum_{n=0}^\infty \frac1{(2n)!}x^{2n}$ prove that: $c(x)^2-s(x)^2=1$

Let $$s(x)=\sum_{n=0}^\infty \frac1{(2n+1)!}x^{2n+1}$$ $$c(x)=\sum_{n=0}^\infty \frac1{(2n)!}x^{2n}$$ prove that $$c(x)^2-s(x)^2=1$$ I know that the following series are representations of the cosh ...
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1answer
18 views

Deriving the logarithmic form of inverse hyperbolic cosecant

I am having trouble finding my mistake in deriving the logarithmic form of inverse hyperbolic cosecant function. Here is my work: $$ y= \mathrm{csch} ^{-1} x \implies \mathrm {csch} \ y= x $$ $$ ...
2
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2answers
127 views

Evaluating: $ \int\sqrt{\tanh(\ln(\sqrt{x}))} dx$ ; $ \int \ln\left(\sqrt{\tanh(\ln(\sqrt{x}))}\right) dx$

I don't have much experience with hyperbolic trig functions... So I don't know how to start solving this. How do I evaluate the following integrals? Any advice, hint or well-thought solution will be ...
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1answer
30 views

Quick question on hyperbolic functions

$\DeclareMathOperator{\arcsinh}{arcsinh}$I have seen that $$ \arcsinh(x) = \ln(x + \sqrt{x^2 + 1}) \tag{1} $$ and also that $$ \arcsinh(x/a) = \ln(x + \sqrt{x^2 + a^2}). \tag{2} $$ I have to ...
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1answer
64 views

can someone explain this simplification for me?? [closed]

Can someone tell me how $$−56−173\,\ln(11)+366\,\ln(13)−\left(\frac{105}2+20\,\ln(2)+366\,\ln(3)\right)$$ simplifies to $$\frac{-217}2−20\,\ln(2)−173\,\ln(11)+732\,{\rm arctanh}\left(\frac58\right)?$$ ...
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0answers
36 views

Simplifying $\frac{2\sinh^2 x}{(\cosh x+1)^3}-\frac{\cosh x}{(\cosh x+1)^2}$

Could you explain me how we simplified this trigonometric expression? $$\frac{2\sinh^2 x}{(\cosh x+1)^3}-\frac{\cosh x}{(\cosh x+1)^2}\qquad\to\qquad\frac{\cosh x -2}{(\cosh x +1)^2}$$ Thanks.
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0answers
91 views
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0answers
28 views

Conformal Map from Domain $D$(where $D$ is bounded by $\{z=x+iy,y=1/x,x>0\}$, $2+2i\in D$) to upper Half Plane

I am trying to find the Conformal Map from Domain $D$(where $D$ is bounded by $\{z=x+iy,y=1/x,x>0\}$, $2+2i\in D$) to upper Half Plane $\mathbb{H}:=\{w:Im(w)>0\}$, I am using $z+1/z$ map but ...
1
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1answer
59 views

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) = ...
11
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3answers
378 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 ...
1
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0answers
31 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
47 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
35 views
0
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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
27 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
22 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
25 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 ...
2
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2answers
59 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 ...
3
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1answer
36 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: $$k^2-2xk+1=0$$ $$k=e^{y}=x\pm\sqrt{x^2-1}$$ ...
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? ...
0
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2answers
41 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 ...
0
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1answer
34 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 ...
3
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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 ...
2
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5answers
65 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 ...
1
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1answer
26 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 ...
3
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2answers
67 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
80 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
46 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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0answers
85 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
106 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 ...
2
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0answers
42 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
43 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 ...
3
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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 ...
1
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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
62 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
votes
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
votes
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? $$\frac{(e^x+e^{-x})}{2} \leq e^\frac{x^2}{2}$$
1
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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 ...