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

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1answer
24 views

Geodesic sphere in $\mathbb H^2$

I saw the definition of a geodesic sphere, and I think I'm not able to "see" how do they look like. For example, it's obvious that in $\mathbb R^n$ geodesic spheres are simply normal spheres, and that ...
0
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2answers
62 views

Sum involving $\cosh$ and $\sinh$

I would like to prove the equation $$\frac{\sinh\left(\left (1-\frac{1}{2m} \right)x\right)}{\sinh(x/2m)}=1+ \sum\limits_{n=1}^{m-1}2\cdot \cosh\left(\left( 1-n/m \right)x\right),\quad \forall x > ...
3
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1answer
67 views

Find real parts of the complex roots of this $9^{th}$ order polynomial in explicit form

I have a following polynomial. (See WolframAlpha ): $$x^9-6x^8+14x^7-16x^6+36x^5-56x^4+ 24x^3-320x+\frac{640}{9}=0 \tag{1}$$ Wolfram says that $(1)$ has three real roots and three pairs of complex ...
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3answers
132 views

Position of Object Suspended on a String (Need Another Answer)

I'm going to try to make as few errors in typing this as possible, so please bear with me and ask me to clarify/correct whatever needed. Q: If an object is suspended on a string hung between two ...
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0answers
24 views

When to substitute with trigo-/hyperbolic-function [closed]

I try to figure out, what indicators could look like to decide if substitution with trigonometrical function or substitution with hyperbolic function works/ works better to integrate a function. Is ...
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1answer
31 views

Is there an easy way to simplify $\tanh(2\operatorname{arctanh}(x))$ and the like?

Is there an easy way to generally simplify any hyperbolic functions of inverse hyperbolic functions, with examples shown below? $$\tanh(2\operatorname{arctanh}(x))$$ $$\coth(\operatorname{arccosech}(...
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1answer
58 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
30 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 ...
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0answers
26 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
35 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 ...
0
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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: $$ y= \mathrm{csch} ^{-1} x \implies \mathrm {csch} \ y= x $$ $$ \frac{...
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1answer
67 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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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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0answers
39 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.
3
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0answers
93 views

Two complementary continued fractions that are algebraic numbers

Define the two similar continued fractions, $$x=\cfrac{1}{km\color{blue}+\cfrac{(m-1)(m+1)} {3km\color{blue}+\cfrac{(2m-1)(2m+1)}{5km\color{blue}+\cfrac{(3m-1)(3m+1)}{7km\color{blue}+\ddots}}}}\tag1$$...
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0answers
29 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
64 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) = \int_0^\...
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0answers
34 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
48 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
40 views

Is there an inequality for $\sinh(x)$ which is similar to this inequality $\cosh(x)\leq e^{x^2/2}$

Is there an inequality for $\sinh(x)$ which is similar to this cosh x inequality?
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1answer
33 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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2answers
26 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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1answer
24 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
72 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
40 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: $$k^2-2xk+1=0$$ $$k=e^{y}=x\pm\sqrt{x^2-1}$$ $$y=\ln(x\...
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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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1answer
35 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 ...
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5answers
67 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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2answers
42 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 $$tanh\frac{a+x}y=y\frac{...
1
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1answer
27 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
71 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
93 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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0answers
29 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} \sum_{k=0}^{\infty}(2k-a)g_{0,k}...
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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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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
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 ...
3
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1answer
51 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 ...
4
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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 \sum_{k=0}^{...
3
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2answers
115 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
43 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 + ...
1
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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
votes
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 ...
3
votes
3answers
48 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
93 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
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 ...