The hyperbolic-functions tag has no wiki summary.
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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 $$ \phi(x)= ...
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2answers
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Hyperbolic Functions
Hey everyone, I need help with questions on hyperbolic functions.
I was able to do part (a).
I proved for $\sinh(3y)$ by doing this:
\begin{align*}
\sinh(3y) &= \sinh(2y +y)\\
&= ...
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1answer
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Evaluation of an integral involving hyperbolic sine and exponential
I am wondering if the following integral can be reduced to either a closed form involving elementary functions, or well-known special functions (such as $\operatorname{erf}$, Bessel functions, etc.):
...
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Find area of a curvilinear triangle that includes hyperbolic functions
We were given this question in class and I tried to compute it and it looks to e pretty crazy. Can anyone take a look and let me know if I did it correctly... I would really appreciate it.
...
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2answers
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Series $\sum\limits_{n=1}^\infty \frac{1}{\cosh(\pi n)}= \frac{1}{2} \left(\frac{\sqrt{\pi}}{\Gamma^2 \left( \frac{3}{4}\right)}-1\right)$
I was playing around with Mathematica and found that
$$\sum_{n=1}^\infty\frac1{\cosh(\pi n)} = \frac12\left(\frac{\sqrt{\pi}}{\Gamma \left(\tfrac34\right)^2}-1\right)$$
Does anybody know how to ...
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0answers
50 views
Solving system of equations with hyperbolic functions
Do you maybe know how to solve a system of equations with hyperbolic functions? Imagine the problem of the form:
$$
x=\textrm{sech}(x^2+y^2) \\
y=1-\textrm{sech}^2 (x+y)
$$
Any ideas how to solve it ...
3
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4answers
58 views
Hyperbolic cosine
I have an A level exam question I'm not too sure how to approach:
a) Show $1+\frac{1}{2}x^2>x, \forall x \in \mathbb{R}$
b) Deduce $ \cosh x > x$
c) Find the point P such that it lies on ...
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0answers
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Geometric definitions of hyperbolic functions
I've learned in school that all the trigonometric functions can be constructed geometrically in terms of a unit circle:
Can the hyperbolic functions be constructed geometrically as well? I know ...
6
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3answers
216 views
Show $\sum_{n=1}^{\infty}\frac{\sinh\pi}{\cosh(2n\pi)-\cosh\pi}=\frac1{\text{e}^{\pi}-1}$ and another
Show that :
$$\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}$$
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1answer
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Hyperbolic function transformation in neural network
While I was trying to do a linear transform in neural, I met the following problem:
$$\tanh(ax+b)=\frac{(\tanh(\frac{x}{2})+1)}{2}$$
What is the appropriate $a$ and $b$?
Thanks
the original ...
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1answer
79 views
Integration by parts
Question 2)b) part (ii) is the section that I'm having trouble with:
I don't understand the method used in the solutions; how would you deduce the first line or is that something you should know?
...
3
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1answer
63 views
Evaluation of integral involving $ \tanh(ax) $
Is it possible to evaluate in closed form the integral
$$ \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$$
here $a$ is a ...
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1answer
37 views
For the previous question on hypergeo function
For $\displaystyle\int_0^{\infty}x^n\frac{\sinh ax}{\cosh bx}\text{d}x$, $\left|a\right|<b$
I think we can evaluate $\displaystyle\int_{0}^{\infty }{{{\text{e}}^{cx}}\frac{\sinh ax}{\cosh ...
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1answer
118 views
A hard integral with hyperbolic function
I was self studying integral. I meet a difficult problem here:
$$\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 ...
