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

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5
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1answer
46 views

Curious integrals for Jacobi Theta Functions $\int_0^1 \vartheta_n(0,q)dq$

There are various identities for the Jacobi Theta Functions $\vartheta_n(z,q)$ on the MathWorld page and on the Wikipedia page. But I found no integral identities for these functions. Meanwhile, ...
7
votes
3answers
488 views

Alternative definition of hyperbolic cosine without relying on exponential function

Ordinary trigonometric functions are defined independently of exponential function, and then shown to be related to it by Euler's formula. Can one define hyperbolic cosine so that the formula $$\...
-2
votes
0answers
57 views

++Hyperbolic Sine and Cosine [closed]

Can anyone show that $\sinh x =\frac{ e^x- e^{-x}}{2}$ and $\cosh = \frac{e^x +e^{-x}}{2}$? I know that by separating the RS of both equations of sinh and cosh into two terms gives the 2 asymptotic ...
1
vote
1answer
59 views

Value of $\int_{-\infty}^\infty\frac{\cos x}{1+x^2} \, dx$

I am a bit puzzled by the expression $\displaystyle I=\int_{-\infty}^\infty\frac{\cos x}{1+x^2}\,dx$. If I try solving it using Cauchy's formula, I arrive to $I=2\pi i \frac{\cos i}{2i} = \pi\cos i$. ...
2
votes
1answer
94 views

Can you help me to evaluate $\int_{-\infty}^\infty\frac{x^2}{-1+\cosh (2x)}dx$ as $\pi^2/6$? And do you find a similar integral for $\zeta(4)$?

I was inspired in the shape of the integrals for $\zeta(2)$ in A. Córdoba, Encounters at the interface between Number Theory and Harmonic Analysis, Proceedings of the Segundas Jornadas de Teoría de ...
0
votes
1answer
38 views

Hyperbolic versus circular trigonometry [closed]

The circular trig function can be used to solve triangles in the Euclidean plane. Can the hyperbolic trig functions be used in any similar way?
1
vote
1answer
28 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 ...
4
votes
3answers
143 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 ...
0
votes
2answers
107 views

I cannot find the following integral in an integral table.

In the appendix A of this paper there is an integral that the author says can be solved using any good integral table. However I cannot seem to find it on any integral table (ex: gradshteyn and ryzhik)...
0
votes
2answers
65 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
votes
1answer
70 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 ...
2
votes
2answers
1k views

Hyperbolic Functions

Hey everyone, I need help with questions on hyperbolic functions. I was able to do part (a). I proved for $\sinh(3y)$ by doing this: \begin{align*} \sinh(3y) &= \sinh(2y +y)\\ &= \sinh(2y)\...
1
vote
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 ...
0
votes
1answer
32 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}(...
1
vote
1answer
59 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
votes
0answers
36 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
votes
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
votes
1answer
567 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
votes
0answers
27 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: $$ln[tanh(\bar{z})] $$....
1
vote
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
votes
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{...
2
votes
2answers
136 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 ...
0
votes
1answer
31 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 ...
0
votes
1answer
68 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 $$\frac{-217}2−20\,\ln(2)−173\,\ln(11)+732\,{\rm arctanh}\left(\frac58\right)?$$ ...
0
votes
0answers
41 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
votes
0answers
94 views

Two complementary continued fractions that are algebraic numbers

Define the two similar continued fractions, $$x=\cfrac{1}{km\color{blue}+\cfrac{(m-1)(m+1)} {3km\color{blue}+\cfrac{(2m-1)(2m+1)}{5km\color{blue}+\cfrac{(3m-1)(3m+1)}{7km\color{blue}+\ddots}}}}\tag1$$...
1
vote
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
vote
1answer
66 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^\...
11
votes
3answers
461 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
vote
0answers
38 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: $...
0
votes
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 $
0
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3answers
40 views
0
votes
1answer
34 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 ...
0
votes
3answers
31 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: $$\lim _{ n\rightarrow { 0 }^{ + } }{ \frac { \cosh { (nx) } -1 }{ \cosh { n } -1 } } = { x }^{ 2 }$$ ...
0
votes
1answer
25 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 ...
0
votes
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 ...
2
votes
2answers
76 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
votes
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\...
1
vote
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 ...
0
votes
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
votes
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{...
0
votes
1answer
43 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
votes
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
votes
5answers
68 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
vote
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
votes
2answers
73 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 ...
1
vote
6answers
98 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} ...
0
votes
1answer
22 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 ...
0
votes
0answers
30 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}...
1
vote
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} + \...