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

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4
votes
1answer
58 views

solving $2\cosh2x = 13\cosh x - 12$

I've been asked to solve: $2\cosh2x = 13\cosh x - 12$ I showed earlier in the question that $\cosh2x = 2\cosh^2x -1$ So I can say that: $2(2\cosh^2x -1) = 13\cosh x - 12$ $\therefore 4\cosh^2x ...
0
votes
1answer
47 views

find $\int{artanh(x)} dx$

I've been asked to find $\int{artanh(x)} dx$ The first thing I did was I said: let $y = artanh(x)$ $\therefore \dfrac{dy}{dx} = \dfrac{1}{1-x^2}$ $\therefore dx = (1-x^2)dy$ Also, from $y = ...
0
votes
1answer
138 views

Need help with a Fourier Transform Question

I need an way to solve this Fourier transform problem. $$ f(t)= \begin{cases} \cosh(t) & \text{ For } |t|<1\\ 0 & \text{ For }|t|>1 \end{cases} $$ The given answer for the ...
0
votes
2answers
61 views

Expressing $\mathrm{B}(\sinh(x), \cosh(x))$ in terms of elementary functions

Is it possible to express: $\mathrm{B}(\sinh(x), \cosh(x))$ (where $\mathrm{B}$ is the beta function) In closed form, in terms of elementary functions?
0
votes
2answers
40 views

Questions about hyperbolas and integration

I have a couple of questions regarding hyperbolas and their integrals. If it's too much, don't feel like you have to answer all 3 questions. My first question: The integral of a function like 1/x^2 ...
0
votes
0answers
16 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
votes
1answer
38 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
votes
4answers
376 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 ...
1
vote
3answers
68 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 $x$ ...
0
votes
1answer
66 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
vote
3answers
132 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
votes
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.
0
votes
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 ...
1
vote
0answers
58 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 ...
1
vote
1answer
106 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 ...
0
votes
2answers
81 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} ...
1
vote
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
votes
0answers
78 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
votes
3answers
163 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 ...
1
vote
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 ...
0
votes
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
votes
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
votes
1answer
94 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 ...
0
votes
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
votes
1answer
83 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
votes
1answer
46 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)= ...
1
vote
2answers
130 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: ...
1
vote
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
votes
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: ...
1
vote
1answer
218 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
votes
1answer
99 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
votes
0answers
216 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 ...
47
votes
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
votes
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 ...
6
votes
1answer
183 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 ...
1
vote
3answers
63 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
votes
1answer
921 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
votes
1answer
54 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
votes
1answer
299 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
votes
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
197 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
283 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 ...
1
vote
2answers
204 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
votes
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 ...
1
vote
1answer
119 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 ...
1
vote
0answers
84 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
votes
1answer
151 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 ...
0
votes
1answer
48 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 ...
-3
votes
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
votes
0answers
235 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 ...