Questions tagged [hyperbolic-functions]

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

The hyperbolic functions are analogs of the usual trigonometric functions; we may define such functions as the hyperbolic sine

$$\sinh{x} = \frac{e^x - e^{-x}}{2}$$

and the hyperbolic cosine

$$\cosh{x} = \frac{e^x + e^{-x}}{2}$$

as well as the hyperbolic tangent

$$\tanh{x} = \frac{\sinh{x}}{\cosh{x}}=\frac{e^x - e^{-x}}{e^x + e^{-x}}$$

These functions are differentiable. More precisely, $\cosh'=\sinh$, $\sinh'=\cosh$, and $\tanh'=1-\tanh^2$.

Just as the point $(\cos{t}, \sin{t})$ describes a point on the unit circle $x^2 + y^2 = 1$, the point $(\cosh{t}, \sinh{t})$ defines a point on the unit hyperbola $x^2 - y^2 = 1$.

Reference: Hyperbolic function.

1108 questions
13
votes
2 answers

An amazing property of the Catenary

I discovered that if we want an arc of catenary in the interval $[a,b]$ we solve $$\int_a^b \sqrt{\cosh '(x)^2+1} \, dx=\int_a^b \cosh x \, dx$$ which means that the "result" of the length is equal to the result of the area in the same interval,…
12
votes
3 answers

Show $\sum_{n=1}^{\infty}\frac{\sinh\pi}{\cosh(2n\pi)-\cosh\pi}=\frac1{\text{e}^{\pi}-1}$ and another

Show that : $$\sum_{n=1}^{\infty}\frac{\cosh(2nx)}{\cosh(4nx)-\cosh(2x)}=\frac1{4\sinh^2(x)}$$ $$\sum_{n=1}^{\infty}\frac{\sinh\pi}{\cosh(2n\pi)-\cosh\pi}=\frac1{\text{e}^{\pi}-1}$$
Ryan
  • 3,775
  • 1
  • 15
  • 31
12
votes
1 answer

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)}\\ &=& \frac{\sin(x)\cosh(y)+i\cos(x)\sinh(y)}{x+iy}\\ &=&…
Bartek
  • 5,965
  • 4
  • 25
  • 64
11
votes
3 answers

Number of zeros of $f(x)= \frac{1}{2} E\left[ \tanh \left( \frac{x+Z}{2} \right) \right]-\tanh(x)+\frac{x}{2}$ where $Z$ is standard normal

Consider the following function: \begin{align} f(x)= \frac{1}{2} E\left[ \tanh \left( \frac{x+Z}{2} \right) \right]-\tanh(x)+\frac{x}{2}, \end{align} where $Z$ is standard normal. Question: How to show that this function has only three zeros? Note,…
Boby
  • 5,407
  • 3
  • 19
  • 53
11
votes
1 answer

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, there are beautiful identities for the simple case of…
11
votes
2 answers

Calculate the sum: $\sum_{x=2}^\infty (x^2 \operatorname{arcoth}(x) \operatorname{arccot} (x) -1)$

$${\color\green{\sum_{x=2}^\infty (x^2 \operatorname{arcoth} (x) \operatorname{arccot} (x) -1)}}$$ This is an impressive sum that has bothered me for a while. Here are the major points behind the sum... Believing on a closed form: I believe that…
user266519
11
votes
2 answers

Proving $~\prod~\frac{\cosh\left(n^2+n+\frac12\right)+i\sinh\left(n+\frac12\right)}{\cosh\left(n^2+n+\frac12\right)-i\sinh\left(n+\frac12\right)}~=~i$

How could we prove that $${\LARGE\prod_{\Large n\ge0}}~\frac{\cosh\left(n^2+n+\dfrac12\right)+i\sinh\left(n+\dfrac12\right)}{\cosh\left(n^2+n+\dfrac12\right)-i\sinh\left(n+\dfrac12\right)}~=~i$$ This is problem $A12-1$ from the Harvard College…
Lucian
  • 46,778
  • 1
  • 77
  • 148
10
votes
5 answers

Integral of $\ln(x)\operatorname{sech}(x)$

How can I prove that: $$\int_{0}^{\infty}\ln(x)\,\operatorname{sech}(x)\,dx=\int_{0}^{\infty}\frac{2\ln(x)}{e^x+e^{-x}}\,dx\\=\pi\ln2+\frac{3}{2}\pi\ln(\pi)-2\pi\ln\!\Gamma(1/4)\approx-0.5208856126\!\dots$$ I haven't really tried much of anything…
10
votes
5 answers

Geometric meanings of hyperbolic cosine and sine

In euclidean geometry, $\cos$ and $\sin$ are used for angles in trigonometry. Is there an equivalent for $\cosh$ and $\sinh$ the hyperbolic cosine and sine, and not cosine and sine ?
projetmbc
  • 768
  • 6
  • 14
9
votes
5 answers

Prove that $\lim\limits_{x \to 0} \sinh(x)/x =1$.

Prove that $ \lim\limits_{x \to 0} \frac{\sinh x}{x} =1.$ I am having some trouble proving this without derivative. Some help would be much appreciate!
user157308
  • 606
  • 2
  • 6
  • 15
9
votes
1 answer

A conjecture $\int_{-\infty}^\infty\frac{\arctan e^x}{\cosh x}\cdot\frac{\tanh\frac{x}2}{x}dx\stackrel?=\frac\pi2\ln2$

I need to find a closed form for this integral: $$\mathcal{I}=\int_{-\infty}^\infty\frac{\arctan e^x}{\cosh x}\cdot\frac{\tanh\frac{x}2}{x}dx.$$ A numerical integration results in an approximation $\mathcal{I}\approx1.0887930451518...$, and…
9
votes
0 answers

$\int_0^\infty \log\left(1 - \frac{x}{\sinh x}\right)\,dx$ and generalisations

I'm interested in the value of $$ \theta(2) := -\frac{1}{2\pi^2}\int_0^\infty \log\left(1 - \frac{x}{\sinh x}\right)\,dx. $$ I have some hopes this admits a closed-form solution. Expanding the logarithm yields the integral $I_n :=…
9
votes
2 answers

Why is $\sinh(45°)$ not infinity? How does it ever intersect with the hyperbola, seeing as it goes along the asymptote?

From what I know, the hyperbolic trigonometric functions are almost the same as the circular trigonometric functions ($\sin, \cos, \tan$, et cetera without the $h$ suffix), except they output when a line coming from the centre at the given angle…
Tachytaenius
  • 252
  • 1
  • 9
9
votes
3 answers

Prove that $\cosh^{-1}(1+x)=\sqrt{2x}(1-\frac{1}{12}x+\frac{3}{160}x^2-\frac{5}{896}x^3+....)$

How can we prove the series expansion of $$\cosh^{-1}(1+x)=\sqrt{2x}\left(1-\frac{1}{12}x+\frac{3}{160}x^2-\frac{5}{896}x^3+...\right)$$ I know the formula for $\cosh^{-1}(x)=\ln(x+\sqrt{x^2-1})$ so, $$\cosh^{-1}(1+x)=\ln(1+x+\sqrt{x^2+2x}).$$…
diya
  • 3,541
  • 1
  • 26
  • 75
8
votes
1 answer

A problem of Ramanujan's interest: closed form of $1 + 2\sum_{n=1}^{\infty} \frac{\cosh(n\theta)}{\cosh(n\pi)} $

I am Brian Diaz, and I am new to the math.stackexchange community. I have been struggling with attempting to find a closed form of the following series: $$ \varphi(\theta) = 1 + 2\sum_{n=1}^{\infty} \frac{\cosh(n\theta)}{\cosh(n\pi)} $$ Admittedly,…
1 2
3
73 74