The hyperbolic-functions tag has no wiki summary.
8
votes
2answers
73 views
$\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx$
Is there any closed-form representation for the following integral?
$$\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx,$$
where $\mathrm{sech}\,x$ is the hyperbolic secant, ...
18
votes
1answer
147 views
Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$
Please help me to find a closed form for the infinite product
$$\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$$
where $\tanh(z)=\frac{e^z-e^{-z}}{e^z+e^{-z}}$ is the hyperbolic tangent.
0
votes
2answers
36 views
Trigonometric hyperbolic function - sinhx
It is given that $\sinh x = \frac{e^x-e^{-x}}{2}$ which is given in various sources, however it has not been explained diagramatically or I am unable to get the derivation of these functions.
So, I ...
2
votes
1answer
57 views
definite Integration
A integration is given,
$$M = \int_{- \infty}^{\infty} \left[\frac{1}{2} \left(\frac{d\phi}{dx}\right)^2 + \frac{\lambda}{4}(\phi^2-v^2)^2\right] dx,$$ where $$m=v\sqrt\lambda$$ and $$ \phi(x)= ...
1
vote
2answers
66 views
Hyperbolic Functions
Hey everyone, I need help with questions on hyperbolic functions.
I was able to do part (a).
I proved for $\sinh(3y)$ by doing this:
\begin{align*}
\sinh(3y) &= \sinh(2y +y)\\
&= ...
2
votes
1answer
36 views
Evaluation of an integral involving hyperbolic sine and exponential
I am wondering if the following integral can be reduced to either a closed form involving elementary functions, or well-known special functions (such as $\operatorname{erf}$, Bessel functions, etc.):
...
0
votes
0answers
51 views
Find area of a curvilinear triangle that includes hyperbolic functions
We were given this question in class and I tried to compute it and it looks to e pretty crazy. Can anyone take a look and let me know if I did it correctly... I would really appreciate it.
...
8
votes
2answers
154 views
Series $\sum\limits_{n=1}^\infty \frac{1}{\cosh(\pi n)}= \frac{1}{2} \left(\frac{\sqrt{\pi}}{\Gamma^2 \left( \frac{3}{4}\right)}-1\right)$
I was playing around with Mathematica and found that
$$\sum_{n=1}^\infty\frac1{\cosh(\pi n)} = \frac12\left(\frac{\sqrt{\pi}}{\Gamma \left(\tfrac34\right)^2}-1\right)$$
Does anybody know how to ...
0
votes
0answers
56 views
Solving system of equations with hyperbolic functions
Do you maybe know how to solve a system of equations with hyperbolic functions? Imagine the problem of the form:
$$
x=\textrm{sech}(x^2+y^2) \\
y=1-\textrm{sech}^2 (x+y)
$$
Any ideas how to solve it ...
3
votes
4answers
58 views
Hyperbolic cosine
I have an A level exam question I'm not too sure how to approach:
a) Show $1+\frac{1}{2}x^2>x, \forall x \in \mathbb{R}$
b) Deduce $ \cosh x > x$
c) Find the point P such that it lies on ...
1
vote
0answers
37 views
4
votes
0answers
84 views
Geometric definitions of hyperbolic functions
I've learned in school that all the trigonometric functions can be constructed geometrically in terms of a unit circle:
Can the hyperbolic functions be constructed geometrically as well? I know ...
6
votes
3answers
221 views
Show $\sum_{n=1}^{\infty}\frac{\sinh\pi}{\cosh(2n\pi)-\cosh\pi}=\frac1{\text{e}^{\pi}-1}$ and another
Show that :
$$\sum_{n=1}^{\infty}\frac{\cosh(2nx)}{\cosh(4nx)-\cosh(2x)}=\frac1{4\sinh^2(x)}$$
$$\sum_{n=1}^{\infty}\frac{\sinh\pi}{\cosh(2n\pi)-\cosh\pi}=\frac1{\text{e}^{\pi}-1}$$
0
votes
1answer
36 views
Hyperbolic function transformation in neural network
While I was trying to do a linear transform in neural, I met the following problem:
$$\tanh(ax+b)=\frac{(\tanh(\frac{x}{2})+1)}{2}$$
What is the appropriate $a$ and $b$?
Thanks
the original ...
0
votes
1answer
83 views
Integration by parts
Question 2)b) part (ii) is the section that I'm having trouble with:
I don't understand the method used in the solutions; how would you deduce the first line or is that something you should know?
...
3
votes
1answer
69 views
Evaluation of integral involving $ \tanh(ax) $
Is it possible to evaluate in closed form the integral
$$ \int_{-\sqrt{x}}^{\sqrt{x}}\frac{r\tanh(ar)}{\sqrt{x-r^{2}}}dr=2\int_{0}^{\sqrt{x}}\frac{r\tanh(ar)}{\sqrt{x-r^{2}}}dr$$
here $a$ is a ...
0
votes
1answer
37 views
For the previous question on hypergeo function
For $\displaystyle\int_0^{\infty}x^n\frac{\sinh ax}{\cosh bx}\text{d}x$, $\left|a\right|<b$
I think we can evaluate $\displaystyle\int_{0}^{\infty }{{{\text{e}}^{cx}}\frac{\sinh ax}{\cosh ...
1
vote
1answer
125 views
A hard integral with hyperbolic function
I was self studying integral. I meet a difficult problem here:
$$\int_{0}^{\infty }{{{x}^{n}}\frac{\sinh ax}{\cosh bx}}\text{d}x=\frac{\pi }{2b}\cdot \frac{{{\text{d}}^{n}}}{\text{d}{{a}^{n}}}\tan ...
1
vote
1answer
39 views
Some estimate concerning hyperbolic functions
I want to show that $|\sinh(az)|\leq|\sinh(z)|$ for all $z\in\mathbf{C}$ (or at least for all $z\in\mathbf{H}$, the upper half plane), provided that $0<a<1$ However, I am not even certain ...
1
vote
2answers
64 views
Expressing hyperbolic functions in terms of $e$.
Express $\tanh(-3)$ in terms of $e$, where $\tanh$ is the hyperbolic tangent.
This is what I did:
$$\begin{align}
\tanh(-x)&=\dfrac{e^{-2x}-1}{e^{-2x}+1}\\\\\\
...
5
votes
1answer
81 views
Integral with hyperbolic cosine squared
Does anyone can give me a hint how to integrate the following:
$$\int_0^\infty{\frac{x^2 {\rm d}x}{\mathrm{cosh}^2(x)}}.$$
The answer is $\frac{\pi^2}{12}$ (taken from the book).
I've started with
...
3
votes
2answers
79 views
Why do we get two solutions when inverting $y = \sinh x$?
Using the definition $\sinh x = \dfrac{e^x-e^{-x}}{2},\;$ let's say we want to solve $\;y = \sinh x \;$ for $x$.
It's not hard to show that $\;\sinh x \;$ is bijective, so this should have exactly ...
1
vote
1answer
150 views
Inverse Laplace Transform involving $\cosh$.
While doing an assignment on solving a PDE I stumbled into the following inverse Laplace transform question (involving $\cosh$? I can't believe it). Mathematica gives no solution and I have no idea ...
1
vote
0answers
74 views
limit of a tricky multi-variable function
I am reading a paper on the sine-gordon model. The solution for a two kink solution is given as:
...
0
votes
1answer
91 views
Differentiating power series
Consider the power series
$$\sum_{n=0}^\infty{\frac{x^{2n}}{(2n)!}}$$
From this, it follows that its sum defines an infinitely differentiable function $f$, given by ...
2
votes
1answer
104 views
Definite integral involving hyperbolic cosine
I have had no experience so far with hyperbolic functions so any help will be appreciated. This is on the chapter of complex integration but I would especially appreciate it if you could turn this ...
10
votes
1answer
354 views
Can the real and imaginary parts of $\dfrac{\sin z}z$ be simplified?
I have calculated the real and imaginary parts of $\dfrac{\sin z}z.$ I've obtained
$$\begin{eqnarray}
\frac{\sin z}z&=&\frac{\sin(x+iy)}{(x+iy)}\\
&=& ...
0
votes
2answers
35 views
How do I show this hyperbolic identity?
I am trying to derive
$$y = \exp \left(\dfrac{a+b}2t \right) \left(k_1 \cosh \left(\dfrac{(a-b)t}2 \right) + k_2 \sinh \left(\dfrac{(a-b)t}2 \right) \right)$$
From $$y = c_1 \exp(at) + c_2 ...
0
votes
1answer
48 views
How do I write this as a solution for hyperbolic sine and cosine?
Consider the following PDE. Where $u(x,t) = X(x)T(t)$
$u_{tt}+u_t = u_{xx}$
$u(0,t)=u(\pi,t)=0$
$u(x,0)=0$
$u_t(x,0)=10$
I am having trouble solving for my $T(t)$, it comes down ...
2
votes
3answers
197 views
Use residues to evaluate $\int_0^\infty \frac{\cosh(ax)}{\cosh(x)}\,\mathrm{d}x$, where $|a|<1$
Use residues to evaluate $$
\int_0^\infty \frac{\cosh(ax)}{\cosh(x)}\,\mathrm{d}x
$$
where $|a|<1$.
Try considering the integral of the form
$$
\int_C \frac{\exp(az)}{\cosh(z)}\,\mathrm dz,
$$
...
0
votes
0answers
131 views
Integral Transform with Hyperbolic Functions
I am at it with understanding the nitty-gritty of the integral transform suggested in a previous question of mine:
Length of a Parabolic Curve
To solve this integral, you can use the substitution
...
2
votes
0answers
30 views
hyperbolic group ; showing the existence of a ration function with a certain condition
I'm currently working out of a book called differentialgeometry and minimal surfaces written by Jost-Hinrich Eschenburg und Jürgen Jost.
Right now I'm looking at an exercise (12.5) under the ...
0
votes
2answers
146 views
Hyperbolic angle
I ve been looking in wikipedia and other sites for "hyperbolic angle", but it is not drawn anywhere. Only an area is shaded everywhere. Is it even possible to draw it?
4
votes
4answers
146 views
Evaluating $\int_0^1 x \sinh (x) \ \mathrm{dx}$
I am looking to evaluate
$$\int_0^1 x \sinh (x) \ \mathrm{dx}$$
0
votes
2answers
96 views
Stuck With The Differentiation Of A Inverse Hyperbolic Function
I'am suppose to show that $$\frac{\mathrm{d} }{\mathrm{d} x}[x \operatorname{cosh}^{-1}(\cosh x)] = 2x$$
And this is what i've tried.Upon differentiating the above function wrt $x$ using the product ...
