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

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0answers
20 views

Newton's method, Hyperbola [on hold]

A cable is freely suspended between two poles 100m apart on a flat ground and the height of each pole is 10m. The overhang required is 6.5m to ensure safety of traffic passing below the cable. The ...
-1
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0answers
17 views

Catenary calculation

I have a catenary wire with the following known factors: P: horizontal force in wire u: weight/mtr s: catenary length v: height difference between y1, y2 From statics follows: y= a cosh (x/a) ...
-6
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0answers
34 views

Non hyperbolic formulas check and proof. [closed]

!I want this non hyperbolic formula to be solved. Both proof and check. thank you
0
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1answer
19 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
146 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 ...
-1
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0answers
24 views

Hyperbolic distance with natural log

We have been learning a lot about hyperbolic lines, midpoints and distances. Would someone be able to assist me how to solve for this problem? The hyperbolic line consists of positive real numbers ...
2
votes
3answers
40 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
60 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
43 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 $N -x^2 - yx - x = 0$. ...
2
votes
1answer
64 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
41 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
38 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
54 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
20 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; ...
21
votes
2answers
388 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
36 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
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0answers
40 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
81 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
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1answer
54 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
51 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
22 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
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0answers
30 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
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4answers
238 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
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0answers
42 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
23 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
1answer
61 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
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1answer
22 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
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1answer
82 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
315 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
38 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]

$$\sqrt{\cos (a)} \sinh \left(\ln (2) a^{\frac{1}{2} \left(e^{-ia }+e^{ia}\right)}\right)+\sqrt{\cos (a)} \cosh \left(\ln (2) a^{\frac{1}{2} \left(e^{-ia }+e^{ia }\right)}\right)=$$
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1answer
38 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
29 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
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1answer
42 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
32 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 ...
1
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1answer
235 views

Proof of Hyperbolic Functions

Find the proof:  (a) Use the definitions cosh(x)= 1/2(ex +e^−x) , sinh(x)= 1/2(e^x − e^−x) to express sinh(x + y) and cosh(x + y) in terms of cosh(x), sinh(x), cosh(y) and sinh(y). (b) Using the ...
6
votes
2answers
220 views

Why is $\sinh$ often pronounced “shine”?

I talked to some guys from the UK and they told me that they would pronounce $\sinh$ as "shine". I am not a native English speaker so I don't know, but in my country we call this function "sintsh" ...
1
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0answers
24 views

Four simultaneous equations

General form of the function: $$y=d\sinh^{-1}\left(\frac{ax+b}2\right)+c$$ I want the function to pass three points, $(0,0)$, $\left(\frac{t}2,\frac{g}2\right)$ and $(t,g)$, and I want the function to ...
2
votes
1answer
531 views

Partials sums of cosh(x) and sinh(x)

Ok, i asked this question yesterday but then hit a snag again. Using these identities $\sinh(x+1) - \sinh (x) = (-1+\cosh(1))\sinh(x) + \sinh(1)\cosh(x)$ $\cosh (x+1) - \cosh (x) = ...
0
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1answer
217 views

Sum of hyperbolic functions, having problems expressing $\sinh(1)$

Using these identities $\sinh(x+1) - \sinh (x) = (-1+\cosh(1))\sinh(x) + \sinh(1)\cosh(x)$ $\cosh (x+1) - \cosh (x) = (-1+\cosh(1))\cosh(x) + \sinh(1)\sinh(x)$ Express the series $C = \cosh 0 + ...
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0answers
14 views

Simplifying equation using hyperbolic transcendental functions… Ahhh!!!!! [duplicate]

I am having real difficulty in seeing how the two equations seen below relate to one another. Can anybody help?? $$\Omega(\theta)=-b.\coth(\operatorname{arsinh}(\exp a\theta . \sinh(c_0)))$$ ...
0
votes
0answers
39 views

Prove that $2\cosh \left(\frac{1}{2} a\right) \sin \left(\frac {1}{2} \alpha\right) = 1.$

Suppose a hyperbolic equilateral triangle has side $a$ and angle $\alpha$.Prove that $$2\cosh \left(\frac{1}{2} a\right) \sin \left(\frac {1}{2} \alpha\right) = 1.$$
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0answers
12 views

Linearization of hyperbolic function with unknown exponent

I have a graph that is clearly some inverse function of the form $y=Ax^n$ where n is a negative. I want to linearize this graph to give me the values of A and n without merely approximating the ...
1
vote
2answers
39 views

$Im(\cosh z)=\sinh x\sinh y$ and $|\sinh z|^2=\sinh^2(z)+\cosh^2(z)$

I have problems in two issues of complex variables ... 1) Prove that $Im(\cosh z)=\sinh x\cosh y$, if $z=x+iy$. I tried to expand $\cosh z= ...
2
votes
0answers
45 views

Prove that $\coth ^2(x)-1\equiv \mathrm{cosech}^2 (x)$

Prove that $\coth ^2(x)-1\equiv \mathrm{cosech}^2 (x)$ My attempts, $\coth^2 (x)-1\equiv(\frac{e^x+e^{-x}}{e^x-e^{-x}})^2-1$ $\equiv \frac{e^{2x}+e^{-2x}+2}{e^{2x}+e^{-2x}-2}-1$ $\equiv ...
0
votes
0answers
82 views

Derivatives of hyperbolic functions and Osborne's rule.

I am slightly confused when it comes to Osborne's rule when you take derivatives of hyperbolic functions. For example. The derivative of cotx is -cosec^2x, so there is a product of sines. So should ...
1
vote
0answers
47 views

Difference between hyperbolic sector, hyperbolic angle and hyperbolic argument

I've been working with hyperbolic functions and am completely confused by the Wikipedia definitions of hyperbolic sectors and angles. Are they the same thing? Based on my trial calculations, they seem ...