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

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1answer
39 views

A improper integral with integrable singularities

Let $\alpha$ and $v_0$ be both positive. Consider a following integral: \begin{equation} {\mathcal J}^\alpha_{1/2}(v_0) := \int\limits_0^\infty v^{\alpha-1} e^{-v} \frac{1}{\sqrt{v_0-v}} d v ...
1
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1answer
27 views

How to simplify expressions like $\sinh(4\,\text{arcsinh}(x))$?

I understand that expressions like $\sinh(\text{arcsinh}(x))$ simplify immediately and expressions like $\sinh(\text{arccosh}(x))$ tend to simplify after some algebra. However I cannot work out how ...
1
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1answer
95 views

Suppose $z=re^{iθ}$, prove : $|e^{iz}|=e^{-r\sinθ}$.

Suppose $z=re^{iθ}$, prove $|e^{iz}|=e^{-r\sin θ}$. I tried but the result was not as expected: $$e^{iz}=\cosh z+i(\sinh z)\\|e^{iz}|=(\cosh z)^2+[i(\sinh z)]^2=\cosh(2z)$$
-3
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1answer
47 views

Exponential Proof

Let $c(x)=\dfrac{3^x+3^{-x}}{2}$ and $s(x)=\dfrac{3^x-3^{-x}}{2}$. Show that $(c(x))^2=\frac{1}{2}(c(2x)+1)$. How does one go about solving this? I have honesty tried substituting in ...
1
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0answers
53 views

Law of Cosines with imaginary arguments?

Does the law of cosines: $c^2 = a^2 + b^2 - 2 a b \cos \theta$ work with imaginary angles? to get something like: $c^2 = a^2 + b^2 - 2 a b \cosh \theta$ Alternatively, is there a hyperbolic ...
3
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0answers
46 views

Evaluate $\int\frac{1}{x}\coth(ax)\sin^2(\frac{xt}{2})dx$

I am trying to integrate this function: $\int_0^\infty\frac{1}{x}\coth(\frac{\hbar x}{2kT})\sin^2(\frac{xt}{2})dx$ which Wolframalpha (for me) returns nothing, just a blank screen. I thought that it ...
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3answers
27 views

Lateral limits of function involving hyperbolic trignometric functions

I am not being able to calculate the lateral limits at 0 of the following function $f(x) = \frac{\sinh(x)}{2\sqrt{\cosh(x) - 1}}$ I have tried both direct substitution (yields 0/0) and L'Hospital's ...
0
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0answers
8 views

Is the follow equation representative of a Hyperbolic One dimensional conservation law?

For $y : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ and let $j : \mathbb{R} \to \mathbb{R}$ $$\frac{\partial y}{\partial x_2} + \frac{\partial (j \circ y)}{\partial x_1} = 0$$
2
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0answers
32 views

Evaluating Hyperbolic Cotangent (coth) Integral

I am working on some simulation, and the paper that I am basing some of the work off of involves several complex integrals. In particular, the one I am trying to solve is $\int_0^\infty ...
5
votes
2answers
120 views

Explicit Form for Coefficients of Extended Hyperbolic Secant Function

Consider the function: $$\frac{3}{e^x+e^{{w_3}x}+e^{{w_3^2}x}}=\sum_{n=0}^\infty{E_{3,n}\frac{x^n}{n!}}$$ Note here that $w_3=e^{\frac{2i\pi}{3}}$ I am trying to get an explicit formula for the ...
0
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1answer
19 views

Integration of complicated hyperbolic functions

I have a complicated integral as below. I'd be appreciated if anyone could help me to find the answer. $$ I=\int_{0}^{U}du [(\partial_uq)^2+w^2q^2] $$ and the $q(u)$ is defined as $$ ...
5
votes
1answer
76 views

Closed form for finite sum of ${\rm csch}^2$

In a recent problem I was attempting to solve, I hit a road block when I reduced the problem to that of finding a closed form for the following sum $$ \mathcal{S}_n(x)\equiv\sum_{k=1}^n{\rm ...
6
votes
2answers
409 views

What are all functions of the form $\frac{\cosh(\alpha x)}{\cosh x+c}$ self-reciprocal under Fourier transform?

There are some functions that are self reciprocal under cosine Fourier transform: \begin{equation} \frac{1}{\cosh x}, \frac{\cosh x}{\cosh 2x},\frac{1}{1+2\cosh x} \end{equation} It seems that they ...
0
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0answers
17 views

Help me find the values of A,B,C if $Ax^2+By^2+Cz^2=1$ is the equation of a hyperboloid of one sheet that goes through the point $(-1,-4,-3)$?

What are the values of A,B,C if $Ax^2+By^2+Cz^2=1$ is the equation of a hyperboloid of one sheet that goes through the point $(-1,-4,-3)$ Here is what I did: $$Ax^2+By^2+Cz^2=1$$ ...
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0answers
66 views

Examples of integrals solved using hyperbolic functions.

I've read in some questions here that various types of integrals usually solved by involving $\tan$ and $\sec$ into the mix can sometimes be solved in an easier manner using hyperbolic functions, as ...
4
votes
1answer
52 views

Hyperbolic Trig Inequality

The following hyperbolic trig inequality came up. $$0 \leq y \leq x \leq 2 \implies \sinh(x)-\sinh(y) \leq \sinh(x-y)\cdot e^{xy/2}.$$ I spent many hours trying to prove it. The first few terms of ...
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3answers
56 views

What is the range of this function: (sin(x)+tanh(x))/(1+sin(x)tanh(x))?

I don't really have any idea how to solve this problem. It appears to have horizontal asymptotes at values of $x=1$ and $x=-1$ but I have no way of proving these. I have taken the derivative (an ...
0
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4answers
100 views

When can sinh(x) and cosh(x) be equal?

I know that for large positive numbers cosh(x) and sinh(x) would almost be equal to $e^x/2$ as $e^{-x}/2$ would become negligible given the magnitude of x in both cases. And so for a number like ...
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3answers
48 views

Absolute value of hyperbolic function [closed]

Is this statement true, if yes, can anyone show me why? $$ \cosh(z)\cosh(z^*) = |\cosh(z)|^2 $$
1
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1answer
29 views

Integration involving hyperbolic functions

As part of a physics problem involving a particle of mass m that slid down an inclined plane of angle $\theta$ and experienced a frictional/retarding force of $f = kmv^2$, I reduced the problem to the ...
1
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1answer
30 views

Derivative of hyperbolic function: $\;\displaystyle f(x) = \sinh \left(\cosh \left(x^9\right)\right) \,$

Derivative of $\;\displaystyle f(x) = \sinh \left(\cosh \left(x^9\right)\right) \,$ ? Okay, so I tried u substitution: $ u = \cosh(x^9)$ $ du = \sinh(x^9) \cdot 9x^8 dx$ -> $$ \sinh(u) ...
1
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1answer
61 views

Can hyperbolic functions be interpreted in terms of a circle? [closed]

Trigonometric functions $\cos$ and $\sin$ are related to points on the unit circle: they are the $x$ and $y$ coordinates. Is there such a relation for $\cosh$ and $\sinh$, hyperbolic functions? What ...
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0answers
20 views

How can I get a and b out of these equations involving $tanh$?

For every value $x$, we know: $$\dfrac{tanh(xa+b)}{tanh(a+b)}$$ Is it now possible to know $\dfrac{a}{b}$? The only thing I think I was able to show, was that: ...
0
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1answer
91 views

Integral involving the hyperbolic tangent

Find the integral of $ \int \limits _{-\infty} ^{\infty} e^{\Bbb i x \frac h k} \tanh x \ \Bbb dx$. I tried to expand this but that didn't help.
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1answer
550 views

Hyperbolic functions and sequence and series [closed]

Using these identities: sinh(mx+x)=cosh(mx)sinh(x)+sinh(mx)cosh(x) cosh(mx+x)=cosh(mx)cosh(x)+sinh(mx)sinh(x) Express the following sums in terms of just cosh((n+1)x), sinh((n+1)x), cosh(x) and ...
2
votes
3answers
51 views

Evaluate the integrals in terms of inverse hyperbolic functions & natural logs

Having trouble with my homework. It is asking evaluate the integral in terms of 1) inverse hyperbolic function and 2) natural logarithm The integral is $$ \int_0^{2\sqrt3} \frac{dx}{\sqrt{4+x^2}} $$ ...
1
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1answer
20 views

Find derivate of y with respect to v (hyperbolic functions)

Need to find derivative of $ y = \ln \operatorname{cosh}v - \frac 1 2 \operatorname{tanh}^2 v $ Any help would be appreciated. The answer is $ \operatorname{tanh}^3v $ but I have no clue how to get ...
0
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1answer
32 views

Finding values of hyperbolic functions

Struggling in Calc2, the question gives a value of sinh x = -3/4 and asking me to find the values of the remaining five hyperbolic functions. Can anybody help me as to how to approach this problem? I ...
3
votes
2answers
68 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
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1answer
41 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
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1answer
56 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 ...
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3answers
98 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 ...
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0answers
49 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
157 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
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3answers
56 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
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3answers
283 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
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1answer
58 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
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1answer
41 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
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1answer
40 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 $$\cosh^{-1}x = \ln(x+\sqrt{x^2+1})$$
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2answers
38 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
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0answers
123 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
114 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?
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2answers
263 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
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1answer
56 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
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2answers
189 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
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3answers
45 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
71 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
57 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
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1answer
92 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
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1answer
117 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 ...