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

learn more… | top users | synonyms

3
votes
2answers
42 views

Does the identity ${|\cosh z|}^2={\cos}^2x+{\sinh}^2y$ given in my text hold?

In my text book I saw that ${|\cosh z|}^2={\cos}^2x+{\sinh}^2y$ But when I tried deriving it myself I got this: $${|\cosh z|}^2={\cos}^2y+{\sinh}^2x$$ See my working below: $$\cosh ...
0
votes
1answer
32 views

Having trouble solving a problem involving hyperbolic trignometric functions

We have to find the value of $$ \tanh^{2}a * \cosh^{2} b - \cos^ {2} c \, $$ if $$\sin(a+ib) * \sin(c+id) = 1.$$ Can anyone solve this? Pls share the solution
4
votes
1answer
43 views

What has “$\delta$ Gromov hyperbolic space” got to do with $\mathbb{H}_n$?

The start of section $2$ of this paper, http://homes.cs.washington.edu/~jrl/papers/kl06-neg.pdf defines something called a ``$\delta$ Gromov hyperbolic space". Can someone explain what has this at ...
1
vote
2answers
43 views

If $t=\tanh\frac{x}{2}$, prove that $\sinh x = \frac{2t}{1-t^2}$ and $\cosh x = \frac{1+t^2}{1-t^2}$.

If $t=\tanh\frac{x}{2}$, prove that $\sinh x = \frac{2t}{1-t^2}$ and $\cosh x = \frac{1+t^2}{1-t^2}$. Hence solve the equation $7\sinh x + 20 \cosh x = 24$. I have tried starting by writing out ...
1
vote
0answers
31 views

Elliptic formulation of sum with cosine and hyperbolic cosine

Can anyone check my reformulation below?$$\sum_{k\in \mathbb{Z}}\frac{1}{\cosh{(k+z)\pi}+\cos{(k+z)\pi}}=\frac 2\pi \csc{\pi ...
8
votes
2answers
128 views

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 $${\LARGE\prod_{\Large n\ge0}}~\frac{\cosh\left(n^2+n+\dfrac12\right)+i\sinh\left(n+\dfrac12\right)}{\cosh\left(n^2+n+\dfrac12\right)-i\sinh\left(n+\dfrac12\right)}~=~i$$ ...
1
vote
3answers
40 views

hyperbolic functions proofs

I need to show that $\cosh^4(x)-\sinh^4(x) = \cosh(2x)$ First I found myself going in circles.. $$\cosh (2 x)=\frac{1}{2} \left(e^{-2 x}+e^{2 x}\right)= \sinh(2x)$$ Now I'm trying to get somewhere ...
1
vote
3answers
46 views

Express $\cosh 2x$ and $\sinh 2x$ in exponential form and hence solve for real values of $x$ the equation:$2 \cosh 2x - \sinh 2x =2$

Express $\cosh 2x$ and $\sinh 2x$ in exponential form and hence solve for real values of $x$ the equation: $2 \cosh 2x - \sinh 2x =2$ Here is my idea: $$2 \cosh 2x- \sinh 2x = ...
0
votes
1answer
37 views

Laurent Series of $f(z) = \frac{z}{\sinh(z)}$ in the region $ 4 < |z| <5 $

Determine all coefficients, belonging to $ z^n $ with $ n<5 $, of the Laurent series of the function $f(z)=\frac{z}{\sinh(z)}$ in the region $4 < |z| <5 $. Could someone help me to find the ...
1
vote
1answer
34 views

Finding fomulas for hyperbolic functions

I'm trying to find formulas for hyperbolic functions, starting with this image Knowing that the area between the origin, vertex and a point on hyperbola (enclosed by x-axis and hyperbola itself) is ...
1
vote
1answer
36 views

Hyperbolic Functions Inverse

Why don't we take negative values of the argument in $\cosh^{-1}=x\pm \sqrt{x^2+1}$? and write only $$\cosh^{-1}x = \ln(x+\sqrt{x^2+1})$$
1
vote
2answers
29 views

Inverse Hyperbolic function

For real no x it is fine that $$\sinh^{-1}x=\ln\left(x+\sqrt{x^2+1}\right)$$ But for complex number $z$ Since there is no order on complex numbers Is it same and why?
2
votes
0answers
71 views

Intriguing Poisson sum with hyperbolic function

I've been playing with lots of Poisson sums lately, and I thought this one to be interesting: $$\sum_{k\in\mathbb{Z}}\left(\frac{1}{(k+x)\sinh{(k+x)\pi q}}-\frac{1}{\pi q (k+x)^2}\right)$$I want to ...
0
votes
6answers
105 views

Is it true that $\frac{1}{\cosh(x) - \sinh(x)} = e^{x}$? [closed]

Is it true that: $$\frac{1}{\cosh(x) - \sinh(x)} = e^{x}$$ Graphically, it seems to be true, but I am not sure if it is true for all $x$. Also, if it is, is it a known result?
8
votes
2answers
195 views

Closed-form of $\int_0^\infty \frac{1}{\left(a+\cosh x\right)^{1/n}} \, dx$ for $a=0,1$

While I was working on this question by @Vladimir Reshetnikov, I've conjectured the following closed-forms. $$ I_0(n)=\int_0^\infty \frac{1}{\left(\cosh x\right)^{1/n}} \, dx \stackrel{?}{=} ...
0
votes
1answer
25 views

Showing that hyperbolic trigonometric functions parameterize the unit hyperbola

I know that the same way circular trigonometry is defined over the circle $ x^2 + y^2 = 1 $, hyperbolic trigonometry is defined over the hyperbola $ x^2 - y^2 = 1 $. What I don't know is how deduced ...
4
votes
2answers
158 views

Evaluating $\sum_{n=0}^{\infty } 2^{-n} \tanh (2^{-n})$

Reading in some tables pages I found $$\sum _{n=0}^{\infty } 2^{-n} \tanh \left(2^{-n}\right)=\tanh (1) \left(1+\coth ^2(1)-\coth (1)\right)$$ I try to split in two sum using the roots of the ...
2
votes
3answers
41 views

Simplification of $\sqrt{2\cosh(x)+2}$

I do not understand the simplification $$ \sqrt{2\cosh(x)+2}=2\cosh(x/2) $$ More generally, I do not understand why $ \sqrt{a\cosh(x)+a}=b\cosh(x/2)$ What is the relationship between $a$ and $b$? ...
1
vote
1answer
62 views

Limit of hyperbolic function

I'm doing some proofs establishing the derivatives for some complex functions, and I was asked to show $(\sinh (z))' = \cosh (z)$. Now I know how to do this with the difference quotients ad such, but ...
1
vote
1answer
50 views

How does the steepness of lines through a hyperboloid change the further away they are from the apex?

I'm a geoscientist and am trying to figure out how the slopes of the flanks of a hyperboloid change for straight lines that cross them. The line of reference is through the apex (red). Now take any ...
0
votes
1answer
58 views

Newton-Raphson For Integer Factorization

Per my earlier question on Naive Grouping for factorization here, below is the modified Newton-Raphson method (integers only) for the polynomial $N -x^2 - yx - x = 0$. ...
1
vote
1answer
76 views

Separate real and imaginary part of $\arccos(z)$

Beginning with $$i \cos \left[ \frac{1}{n} \arccos \left( \frac{i}{\epsilon} \right) + \frac{m \pi}{n} \right]$$ where $m,n \in \mathbf{Z}$, $\epsilon >0$, $\epsilon \in \mathbf{R}$ and $i$ is ...
1
vote
1answer
42 views

Is that hyperbolic identity correct?

Given the expression: \begin{equation} |x|\cosh(kx)+x\sinh(kx), \;\; k>0 \end{equation} By taking cases for $x$, we have: \begin{equation} \bullet \quad x>0: x(\cosh(kx)+\sinh(kx))=x\left( ...
3
votes
0answers
40 views

Representation of hyperbolas.

I am well aware of the matrix representation for rotation of points on a circle with reals $$M_{R(\theta)} = \left(\begin{array}{rr} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & ...
3
votes
0answers
55 views

Looking for a bound on a function involving $\sinh$

Fix $T > 0$ and let $t \in (0,T)$ let $c > 1$ be a constant (which may be bigger than $T$). Consider the function $$f(c,t,T) = \frac{\sinh ((T-t)c)}{\sinh (Tc)}.$$ I am looking for a bound of ...
1
vote
1answer
24 views

Separate real and imaginary part of $j \cos (z)$

Given the following expression $$w = j \cos \left[ \displaystyle \frac{1}{n} \arccos \left( \frac{j}{\epsilon} \right) + \frac{m \pi}{n} \right] = j \cos (z)$$ (which is related to this question; ...
25
votes
4answers
547 views

Integral ${\large\int}_0^\infty\frac{dx}{\sqrt[4]{7+\cosh x}}$

How to prove the following conjectured identity? $$\int_0^\infty\frac{dx}{\sqrt[4]{7+\cosh x}}\stackrel{\color{#a0a0a0}?}=\frac{\sqrt[4]6}{3\sqrt\pi}\Gamma^2\big(\tfrac14\big)\tag1$$ It holds ...
-1
votes
1answer
37 views

Need Help Evaluating This Indefinite Integral

I would appreciate any help finding a possible closed form solution of this integral. $$\int\sqrt{\cosh(u)-\cos(v)}\cdot e^\frac{u}{2}~du$$ Any help would be greatly appreciated! The solution for ...
0
votes
0answers
43 views

Transform complex trigonometric expression with $\arccos$

In the proof that the poles of a Chebyshev filter lie on an ellipse, there is the following transformation, for the $s$ values correspondant to the poles. From (1) $$s_{pm} = j \cos \left[ ...
0
votes
0answers
87 views

Naive Grouping for Factorization

I have a naive grouping method for factorization. I am curious as to its novelty and aspects of the code below that will increase its efficiency. The method is best described with an example: For n ...
0
votes
2answers
38 views

Simplification of integrals.

When dealing with hyperbolic functions, simplifying: $$ 2\pi\int_{-a}^a a \cdot \cosh^2\left(\frac{x}{a}\right)dx $$ Yields: $$ 2\pi a\cdot \frac{a}{2}(2+\sinh 2) $$ How is this possible?
2
votes
1answer
55 views

value of $\arctan (\cosh u)$ as $u \to -\infty $

I am interested in the value of $\arctan (\cosh u)$ as $u\to -\infty $ $$\arctan (\cosh u)= \dfrac i 2 \log \left| \dfrac {1-i\cosh u}{1+i\cosh u} \right|$$ and since $$\cosh u= \dfrac ...
3
votes
1answer
59 views

How do I derive the Maclaurin series for $\tanh(x)$?

I've thought of doing it by writing $\tanh(x)$ as $(1-e^{-2x})/(1+e^{-2x})$ and then using the Maclaurin series for $e^{x}$ or just as $\sinh(x)/\cosh(x)$ and using the Maclaurin series for $\sinh(x)$ ...
1
vote
2answers
23 views

How to show that $\sin(iy)=i\sinh y$

I know that $$\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$$ Then substituting $x=iy$: $$\sin(iy)=\frac{e^{i(iy)}-e^{-i(iy)}}{2i}=\frac{e^{-y}-e^{y}}{2i}$$ Then, according to my lecture handout (this step is the ...
1
vote
0answers
33 views

Exponential and Hyperbolic Functions Before Power Series

Is there a textbook that covers the exponential function and the possibly the hyperbolic functions from a precalculus geometrical viewpoint? I'm essentially asking for something like a trigonometry ...
13
votes
4answers
248 views

Prove that $\sinh(\cosh(x)) \geq \cosh(\sinh(x))$

Prove that $$\sinh(\cosh(x)) \geq \cosh(\sinh(x))$$ I tried to tackle this problem by integrating both lhs and rhs, in order to get two functions who show clearly that inequality holds. I've ...
0
votes
0answers
47 views

Circumference of hyperbolic circle is $2\pi \sinh r$

I'm looking for a proof that in the Poincare disk model the circumference of a circle of radius $r$ is $2\pi \sinh r$. I have seen this result in many places but I haven't been able to find a proof. ...
1
vote
1answer
26 views

Distance in the $y$-axis of the hyperbolic plane

I'm reading Stillwell's Geometry of Surfaces but I'm having a little bit of trouble because my background in calculus isn't great, I'm struggling with these problems: In the upper half-plane model, ...
1
vote
2answers
88 views

Solving the integral $\int_{-\infty}^{\infty} (1+x^2)^{-3/2}$ with $\sinh$, $\cosh$?

I want to solve the following integral: $$\int_{-\infty}^{\infty} (1+x^2)^{-3/2}$$ I thought maybe it's possible with $\sinh$ or $\cosh$ or something similar, but I can't figure it out. Thanks in ...
1
vote
1answer
62 views

Evaluating $ \int_0^\theta \cosh(a\sin x) dx$

The integral below seems quite simple, but I couldn't find anywhere the result. $$ I = \int_0^\theta \cosh(a\sin x) dx$$ I tried to expand it into Taylor expansion series and successfully evaluate the ...
0
votes
1answer
23 views

Transformation of parameter between the hyperbolas $xy=1$ and $x^2-y^2=2$ during rotation?

It is fairly straightforward to see that the hyperbola $xy=1$ is simply the hyperbola $x^2-y^2=2$ rotated by $\pi/4$. All we do is apply the corresponding rotation matrix to the vector ...
0
votes
1answer
84 views

Integral of $\int_{-\infty}^\infty\frac{e^{-y^2/2} \sinh(cy)^2} {\cosh(Mcy) }dy$

Is it difficult to compute or find a good computable lower bound on the integral \begin{align*} \int_{-\infty}^\infty\frac{e^{-y^2/2} \sinh(cy)^2} {\cosh(Mcy) }dy \end{align*} where $c$ and $M$ are ...
6
votes
1answer
322 views

Integral of the ratio of two exponential sums

I am trying to find a lower bound on the following integral \begin{align*} \int_{y=-\infty}^{y=\infty} \frac{ (\sum_{n=[-N..N]/\{0\}}n e^{-\frac{(y-cn)^2}{2}})^2} {\sum_{n=[-N..N]/\{0\}} ...
3
votes
1answer
45 views

a hyperbolic summation

Find the value of $$\lim_{n\to\infty} \sum_{k=1}^{n}\frac{1}{\sinh 2^k}$$ Numerical approximations gives me a value of $\frac{2}{e^2-1}$. I tried to write the sum as ...
0
votes
3answers
93 views

How to simplify if $a > 0$ and $\cos(a) < 0$ [closed]

$$\sqrt{\cos (a)} \sinh \left(\ln (2) a^{\frac{1}{2} \left(e^{-ia }+e^{ia}\right)}\right)+\sqrt{\cos (a)} \cosh \left(\ln (2) a^{\frac{1}{2} \left(e^{-ia }+e^{ia }\right)}\right)=$$
-1
votes
1answer
54 views

hyperbolic isometry

I have a project I have to do. In order to do it I need to investigate this book. In page 94 they defined hyperbolic isometry on a metric s.t it possesses no fixed point in the tree. After that they ...
1
vote
1answer
30 views

Prove this equality about hyperbolic right triangles

If K is the area of a hyperbolic right triangle ABC in which the right angle is at C, prove that $$ \sin K=\frac{ \sinh a \sinh b}{1+\cosh a\cosh b}$$ My attempt at the solution: I basically need ...
1
vote
2answers
108 views

Find: $\int_0^{\infty}\frac{\sinh x}{1+\cosh^2x}dx$

$$\int_0^{\infty}\frac{\sinh x}{1+\cosh^2x}dx$$ Here's what I've attempted: Using the identity $1+\cosh^2x=\sinh^2x$ I got: $$\int_0^{\infty}\frac{\sinh x}{\sinh^2x}dx=\int_0^{\infty}\frac1{\sinh ...
0
votes
1answer
57 views

Finding equation of hyperbola with only foci and asymptote

This is a concept we learned in class today, which I still can't seem to grasp. I have no specific question that necessarily has to be done, so I will use one of the examples my book gives me: Given ...
3
votes
2answers
34 views

Differentiating inverse hyperbolic function

I am trying to differentiate $\tanh^{−1}\left(x/(1 + x^2)\right)$, but am finding it difficult understanding what to do. I think you have to place the differential of the angle of the hyperbolic ...