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

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0answers
15 views

Solving for hyperquadratic?

I'm working on defining a decision plane for two different classes but am running into trouble. For those familiar with stats I'm working with two classes that have arbitrary distribution, mean and ...
0
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1answer
37 views

Find a linear combination of $u_n$'s satisfying $u(x,1) = \sin(2\pi x) -\sin(3\pi x)$

I have the following problem: $$u_n(x,y) = \sin(n\pi x)\sinh(n\pi y), \;\;\;n = 1, 2, 3, ...$$ Find a linear combination of the $u_n$'s that satisfies: $$u(x,1) = \sin(2\pi x) -\sin(3\pi x)$$ Any ...
3
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4answers
352 views

If $f$ is holomorphic and $\,f'' = f$, then $f(z) = A \cosh z + B \sinh z$

Suppose $f$ is holomorphic in a disk centered at the origin and $f$ satisfies the differential equation $$f'' = f.$$ Show that $f$ is of the form $$f(z)=A \sinh z + B \cosh z,$$ for suitable constants ...
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3answers
65 views

Showing that $\sinh(\mathrm{e}^z)$ is entire

I am attempting to show that $\sinh(\mathrm{e}^z)$, where $z$ is a complex number, is entire. The instructions of the problem tell me to write the real component of this function as a function of $x$ ...
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1answer
65 views

How to find the angle subtended to the origin by the unit hyperbola through the point (1,0)?

I'm trying to find the angle subtended by the unit hyperbola through the point (1,0). I think that I should be integrating something, but I'm not sure how to set it up. I've been trying to think of ...
1
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3answers
126 views

Derivatives of $\sinh x$ and $\cosh x$

Can someone give me an intuitive explanation about the derivatives of $\sinh x$ and $\cosh x$? Something similar to: Intuitive understanding of the derivatives of $\sin x$ and $\cos x$ Thanks!
2
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4answers
98 views

Inverse of $(e^x - e^{-x})/2$

What is the inverse of the function $f(x)=\frac{e^x - e^{-x}}2$? I tried replacing $e^x$ by a variable but I still can't get it.
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0answers
26 views

How are the sine functions along with the hyperbolic functions visualized with imaginary rotations?

Since we know that: cos(t)=cosh(it) and isin(t)=sinh(it) I've been thinking about this, and obviously this is referring to how if you move at a right angle from a circle on a conic section, you end ...
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0answers
57 views

Why are hyperbolic trigonometric functions avoided in (my) high school and early post-secondary school?

I remember seeing hyperbolic trigonometric functions (sinh, cosh, tanh, etc.) in my precalculus textbook back in high school and see them today in my calculus textbook. However, I have not had a ...
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1answer
104 views

Nonlinear Differential Equation

During my study of calculus of variations I came across a nonlinear DE. I know that its solution is hyperbolic cosine with some constants yet do not know how to proceed since the function of interest ...
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2answers
78 views

Differentiate $y=\cosh^{3} 4x$.

Differentiate $y=\cosh^{3} 4x$. $$\frac{dy}{dx} = 3 \cosh^{2} (4x) \sinh (4x)\cdot 4$$ These are the parts that I don't quite understand: \begin{align*} \frac {dy}{dx} &=12 \cosh^{2} ...
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0answers
55 views

$\lim\limits_{n\to\infty}\operatorname{arsinh}(1 + \operatorname{arsinh}(2 + \operatorname{arsinh}(3 + \dots\operatorname{arsinh}(n+\dots)\dots)))=?$

Does the limit $$\lim\limits_{n\to\infty}\operatorname{arsinh}(1 + \operatorname{arsinh}(2 + \operatorname{arsinh}(3 + \operatorname{arsinh}(4+\dots\operatorname{arsinh}(n+\dots)\dots))))$$ exist ...
3
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0answers
75 views

Closed form for $\int^1_0 \mathrm{arccsch}\left( \frac{x^2-x-1}{x^2+x+1}\right)\;\mathrm dx$

Find a closed form for $$\int^1_0 \mathrm{arccsch}\left( \frac{x^2-x-1}{x^2+x+1}\right)\;\mathrm dx$$ What I have tried Expanding the $\mathrm{arccsch}$ into its logarithmic form, however I ...
3
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3answers
161 views

Find the limit of $S_n=\sum_{i=1}^n \big\{ \cosh\big(\!\!\frac{1}{\sqrt{n+i}}\!\big) -n\big\}$, as $n\to\infty$?

$S_n=\sum_{i=1}^n\big\{ \cosh\big(\frac{1}{\sqrt{n+i}}\!\big) -n\big\}$ as $n\to\infty$ I stumbled on this question as an reading about Riemannian sums as in $$ \int_a^b f(x)\,dx =\lim_{x\to ...
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1answer
38 views

Finding the derivative of $\;\operatorname{arccoth}(\sin x)$

I have tried to solve it but I don't why it's wrong. I need to take the derivative of $\;\operatorname{arccoth}(\sin x)$: By using chain rule, I get: $$\dfrac 1{1 - \sin^2 x}\cdot \cos x = \dfrac ...
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2answers
79 views

How to integrate this $\frac{1}{(1-e^{2x})^{1/2}}$?

Please how to integrate this $$\frac{1}{(1-e^{2x})^{1/2}}$$ I have tried $u= e^x$ But I think that is wrong So can anyone help me ?
0
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1answer
68 views

Why ${\rm arcosh}(\cosh x) =x $?

I'd like to know why ${\rm arcosh}(\cosh x) =x$. Also I have read that the derivative of ${\rm arcosh}(\cosh x) = \sinh x/|\sinh x|$. Why? Thanks all
0
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1answer
92 views

Find the derivative of y when y= ln (arccosh x)

I want to know how to find the derivative of y when y= ln (arccosh x) I know arccosh x = 1/[x^2 -1]^(1/2) So 1/[(arccosh x)^[2] [x^2 -1]^(1/2)] But the right answer is 1/[(arccosh x)^[2] [x^2 ...
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1answer
63 views

Find $x$ if $\sinh(x)=2$

I want to know how to find $x$ if $\sinh(x)=2$. I already know that $\sinh(x) = \dfrac{e^x -e^{-x}}{2}$. Hence, $$\frac{e^x -e^{-x}}{2} = 2 \implies e^x -e^{-x}=4$$ but I don't know what should I do ...
0
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1answer
81 views

When I choose arctanh or arccoth?

I want to know When I choose $\operatorname{arctanh}$ or $\operatorname{arccoth}$ ? E.x $$\int_0^3 \frac{1}{49-4x^2} dx$$ It will be $(1/14) \operatorname{arctanh} (2x/7)$ or $(1/14) ...
0
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1answer
44 views

limit of a hyperbolic function

How to evaluate this limit without using hopital rule: $$\lim_{c\rightarrow + \infty}{\frac{\text{sinh}\sqrt{c}}{2\sqrt{x}}}$$ Here is what I have done so far: we know that $\text{sinh}(x)= ...
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2answers
124 views

The derivative of $\tanh x$

I'm trying to calculate the derivative of $\displaystyle\tanh h = \frac{e^h-e^{-h}}{e^h+e^{-h}}$. Could someone verify if I got it right or not, if I forgot something etc. Here goes my try: ...
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3answers
51 views

Is there a means of analytically proving the following identity?

Okay, so before I begin, my background is more in the world of applied, rather than pure, mathematics, so this question is motivated by a physics problem I'm looking at just now. Mathematically, it ...
2
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3answers
80 views

What is wrong? Symmetric function

I need some advice here. What is $y(\ln(4))$ if the function $y$ satisfies: $$\frac{dy}{dx} = 1-y^2$$ and is symmetric about the point $(\ln(9),0)$. Solving that equation I end up with: ...
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1answer
205 views

Derivation of parametric equations of a hyperbola

Can somebody please show me how to derive the parametric coordinates of a hyperbola from $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$ without just substituting them in? Thanks
0
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1answer
95 views

Does $e^{i*\theta}$ Relate To Hyperbolic Sine/Cosine?

I would like to understand the relationship betwene $e^{i\cdot \theta}$ and hyperbolic sine and cosine. Here is what I have done so far: Given: $$\sinh(x)+\cosh(x)=e^x $$ ...
4
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0answers
205 views

Geometric interpretation of hyperbolic functions

When proving identities like $$\cosh(2x)=\cosh^2(x)+\sinh^2(x)$$ $$\cosh^2(x)=\sinh^2(x)+1$$ algebraically, I am beset by the feeling that there should be a geometrical interpretation that makes them ...
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2answers
1k views

Integral $\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}dx$

Consider the following integral: $$\mathcal{I}=\int_1^\infty\frac{\operatorname{arccot}\left(1+\frac{2\,\pi}{\operatorname{arcoth}x\,-\,\operatorname{arccsc}x}\right)}{\sqrt{x^2-1}}dx,$$ where ...
2
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0answers
228 views

Parametric Equation of a Hyperbolic Paraboloid

I need to make two trace plots of the hyperbolic paraboloid $z=x^2-y^2$. In the first plot, we set $z$ equal to a constant $k$, $z=k$. How do I find the parametric equation for this representation of ...
5
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1answer
179 views

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 ...
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3answers
60 views

Need help solving equation involving $\cosh$

I am trying to solve this equation for $a$ $$R= (a)\cosh\left(\frac{l}{a}\right)$$ where $R$ and $l$ are real positive constants. I tried breaking $\cosh$ into exponentials but this didnt seem to ...
0
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1answer
894 views

Taylor Series of Hyperbolic Cotangent Coth(x)

Expanding about 0 gets me a divergence on the first term, and the wikipedia article says nothing about how to derive it other than taylor series. It makes me think I'm supposed to use Laurent Series, ...
0
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1answer
52 views

Rewriting a hyperbolic equation in standard form

$9x^2-4y^2-72x$ = 0 How would that be done? So far, I got up to $\frac{9(x^2-8x)}1-\frac{4(y-0)^2}1=0$
4
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1answer
291 views

fourier expansion of $\coth$ and justifying an identity

The problem: Justify the following equalities: $$\cot x = i\coth (ix) = i \sum^\infty_{n=-\infty} \frac{ix}{(ix)^2+(n\pi)^2}=\sum^\infty_{n=-\infty}\frac{x}{x^2+(n\pi)^2}$$ I am trying to figure ...
2
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0answers
47 views

Integral with $\cosh$ and $\log$ in the integrand

I am trying to find a good way to simplify (or even solve) the following integral: $$ \int_0^{-\log \varepsilon} \cosh^3r \frac{\partial^2}{\partial r^2} \log (\det E) dr $$ where $r > 0$ is a ...
2
votes
1answer
191 views

How to convert between the hyperboloid model and the Poincare patch for $\mathbb{H}_n$?

This question is motivated by the results in this paper, http://calvino.polito.it/~camporesi/JMP94.pdf In this paper some of its most important results about the asymptotics of symmetric traceless ...
13
votes
3answers
271 views

Closed form for $\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx$

Let $$f(a)=\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx,$$ where $\operatorname{sech}(z)=\frac2{e^z+e^{-z}}$ is the hyperbolic secant. Here are values of $f(a)$ at some ...
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2answers
194 views

Geometric meaning of hyperbolic functions.

Trigonometric functions have strong geometric meanings, which make them useful for dealing with complex number (which can be thought as point on the complex plane). In my point of view, by the ...
2
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0answers
74 views

Solution of nonlinear waves( breathers)

The sine-Gordon equation is known as $$\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} + \sin u = 0,$$ Can you please derive the equation which is known as breather equation ...
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1answer
117 views

Integral of $\operatorname{cosech}^4 x$

$\newcommand{\cosech}{\operatorname{cosech}}$ I ran into this integral when trying to solve a problem, and I could not get my head around it. $$\int \cosech^4(x)dx$$ I tried to split into two ...
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0answers
83 views

Solution by of nonlinear equation

$$\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} + \sin u = 0$$ From the sine-Gordon equation we can easily solve, \begin{equation} \phi(x) = \pm 4 \tan^{-1}\left[e^{\frac{x-t ...
2
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1answer
148 views

Simplifying $\cosh \mathrm{arcsinh} \ x$

How can I simplify the following: $$\cosh \mathrm{arcsinh} \ x$$ I know that an expression of the form $f(g^{-1}(x))$ where $f$ and $g$ are trigonometric functions can be simplified by constructing a ...
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1answer
47 views

Need some algebra explaination for hyperbolic sinus

I am reading a book where i have come across an expression: $$ \frac{e^{x}\left(-a^2+b^2+2iab\right)- e^{-x}\left(-a^2 +b^2-2iab\right)}{2} $$ In the book they state that we can express this ...
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1answer
94 views

How to simplify this equation?

How to simplify this equation? I know that: $\sin^2 p+\cos^2 p = 1$ But how to go further?
3
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0answers
229 views

Integrating a fractional power of a rational function

I am currently working on a project where I stumbled upon the integral $$ \int \frac{\sinh \left(\frac{R}{2}\right)}{(\coth R - 6R \coth\left(\frac{R}{2}\right) + 9)^{1/4}} \,dR $$ where $R$ is a ...
7
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1answer
1k views

How were Hyperbolic functions derived/discovered?

Trig functions are simple ratios, but what does Cosh, Sinh and Tanh compute? How are they related to euler's number anyway?
4
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1answer
498 views

Geometric construction of hyperbolic trigonometric functions

If we have a circle we can geometrically construct the trigonometric functions as shown. The functions all derive from sin and cos. If we say that the circle is a conic section and imagine it on the ...
2
votes
2answers
100 views

proving that the differences of squares of hyperbolic sin/cos is an integer.

The hyperbolic sine and cosine are defined as following: $\sinh(x)=\dfrac{e^x-e^{-x}}{2}$ $\cosh(x)=\dfrac{e^x+e^{-x}}{2}$ How do I show that their differences of squares are always an integer for ...
0
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1answer
50 views

A tough calculation involving hyperbolic contangents.

From here: http://en.wikipedia.org/wiki/Brillouin_function Define $$B_j(x)=\frac{2j+1}{2j} \coth \left( \frac{2j+1}{2j} x \right) - \frac{1}{2j} \coth \left( \frac{1}{2j} x \right)$$ I want to do ...
1
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1answer
54 views

Hyperbolic integration solving

$$ \therefore x-x_0 = \pm \int_{\phi(x_0)}^{\phi(x)} \frac{d \phi}{\sqrt\frac{\lambda}{2}\left( \phi^2-(\frac{m}{\sqrt \lambda})^2\right)} $$ How can we write the above equation to as, $$ \phi(x) = ...