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

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2
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1answer
13 views

Solve for a hyperbolic Laplace Transform by expressing as exponents and shiftig on s-axis (5.3-21)

I cannot get past a certain point on this problem as shall be shown. I need guidance in order to complete the problem. The exercise as stated in the text: Represent the hyperbolic function in terms ...
0
votes
0answers
19 views

Volumes of revolutions question

The point $P(a,b)$ lies on the curve $y=arsinh x$. $R$ is the region bounded by the curve, the $x$- and $y$-axes and the line $x=a$. When $R$ is rotated 2$π$ radians about the $x$-axes the solid ...
0
votes
3answers
30 views

How can I calculte the probability of $X$ with a Generlized Hyperbolic Distribution?

I would like to know how to calculate the probability of $X$ when I have fitted a Generalized Hyperbolic Distribution to my data set. The depth of my knowledge is basic t-tests and z-tests. I am ...
2
votes
1answer
26 views

Stuck on an integration question…

$$\int x^{-\frac{1}{2}}\cosh^{-1}(\frac{x}{2}+1)dx$$ The answer I should get is $$2x^{\frac{1}{2}}\cosh^{-1}(\frac{x}{2}+1)-4(x+4)^{\frac{1}{2}}$$ but I keep going wrong. Can someone show me how to ...
1
vote
0answers
24 views

Complex Mapping of $\mathrm{cosh}(w)=z$

Mapping in complex analysis has not been very easy for me unfortunately. I am having difficult trying to find the mapping between the z and w plane. I attempted to simply write that ...
0
votes
1answer
17 views

Generalizing 1D function to higher dimensions

I have a function in 1D given by $f(x) = \tanh(x-x_1) + \tanh(x-x_2)$. I want to generalize this to two dimensions, such that it describes a circle. The function $f(x,y)$ has to have a form such that ...
1
vote
1answer
24 views

A level Integration question.

1a) Prove that $$e^x\operatorname{sech} x\equiv\frac{2e^{2x}}{e^{2x}+1}$$ b) Find $$\frac{d}{dx}[\arcsin(\tanh x)]$$ Simplify your answer as far as possible. c) Hence, or otherwise, solve $$\int ...
7
votes
2answers
249 views
+50

An alternating series identity with a hidden hyperbolic tangent

How does one use the inverse Mellin transform to prove that the following identity holds? $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n(e^{n\pi} + 1)} = \frac{1}{8}(\pi - 5\log(2))$$ The identity follows ...
0
votes
2answers
30 views

Simplifying a hyperbolic trigonometric expression [closed]

How can I rewrite $$\tanh(x)\left(-\frac12 \mathrm{sech}^4(x) + \frac12 \mathrm{sech}^2(x)\right)$$ as $$\frac12 \mathrm{sech}^2(x)\tanh^3(x)?$$
1
vote
0answers
18 views

Hyperbolic trigonometric inequality

Is the following hyperbolic trigonometric inequality correct and if so, is there a simple derivation? $$\tanh A- \tanh B \geq (A-B)(\operatorname{sech}{^2}(A)), \qquad \forall A\ge B\ge 0.$$
0
votes
1answer
24 views

Trouble finding the equation of the inverse of the hyperbolic tangent

I'm trying to find the equation of the inverse of the hyperbolic tangent as follows: Take an $x \in ]-1,1[$ and define $y:=\text{arctanh}(x)$ so that $\tanh y=x$. This means that ...
2
votes
1answer
41 views

Solve $x \tanh(x) = constant$

Does the following equation admit a real solution: $x\cdot \tanh(x) = C$ with $C$ a constant. While I was not able to find a specific answer with symbolic calculations, this solutions seems to ...
1
vote
2answers
49 views

Integral question showing the primitive functions differ only by a constant?

$$\int \frac{dx}{\sqrt{x^2-6x+13}}$$ $$\int \frac{dx}{\sqrt{\left(x-3\right)^2+4}}$$ It can be solved by Method 1 Let $$x-3=2\tan u$$ $$dx=2\sec^2 u\,du$$ Therefore, using the trigonometric ...
2
votes
2answers
51 views

hyperbolic tangent vs tangent

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side (in a right triangle). Is there a similar definition for the hyperbolic tangent? The reason ...
5
votes
2answers
117 views

Asymptotics of $\int_{0}^{+\infty}\!\!\frac{dx}{\sinh^2(\epsilon \sqrt{x^2+1}) } $ for $\epsilon$ near $0$

How to find an asymptotic expansion, for $\epsilon$ near $0$, of the following integral $$ I(\epsilon):=\int_{0}^{+\infty}\frac 1{\sinh^2 (\epsilon \sqrt{x^2+1}) } {\rm d}x. $$ As $\epsilon ...
1
vote
1answer
50 views

Hard integral of root function and hyperbolic function

I need to calculate this integral: $$\int^A_B\frac{\sqrt{x-B}}{\cosh^2x}dx$$ Is there any way to do this?
19
votes
6answers
239 views

What is the importance of $\sinh(x)$?

I stumbled across $\sinh(x)$. I am only a calculus uno student, but was wondering when this function comes into play, and what is its purpose? Last, does it have world applications, or is it a ...
7
votes
1answer
109 views

Finding taylor expansion for $\tanh(x)$

I am a high school student and am trying to find the taylor expansion of $\tanh(x)$ in terms of a summation form. I have gotten this far, and am aware it might get complicated very quickly. If someone ...
3
votes
1answer
37 views

How to find cosh(arcsinh(f(x)))?

With the regular trig functions, if I ever end up with something like $\operatorname{trig}_1(\operatorname{arctrig}_2(f(x))$, where $\text{trig}_1$ and $\text{trig}_2$ are two arbitrary trigonometric ...
0
votes
4answers
65 views

Taylor series extension of tanh

I know how to find the taylor expansion of both sinh and cosh, but how would you find the taylor expansion of tanh. It seems you can't just divide both the taylor series of sinh and cosh so how would ...
0
votes
1answer
32 views

Hyperbolic Intuition.

I am working with hyperbolic functions and was wondering where they actually came from. I am under the understanding that Ricotta (and I think Johann Heinrich Lambert also did work in this area), did ...
1
vote
1answer
32 views

Evaluating $\lim_{x\to\infty}\left(\frac{\sinh x}{x}\right)^{{1}/{x^2}}$

$$\lim_{x\to\infty}\left(\frac{\sinh x}{x}\right)^{\dfrac{1}{x^2}}$$
1
vote
3answers
48 views

How to find the inverse function of $f(x)=\frac{\sinh(\ln(\cosh x))}{\sinh x}$

How to find the inverse function of $f(x)=\frac{\sinh(\ln(\cosh x))}{\sinh x}$? I' ve tried the following: $y=\frac{\sinh(\ln(\cosh x))}{\sinh x}$ . Now I should express $x$ in terms of $y$. Then: ...
6
votes
0answers
71 views

Can these two indefinite integrals be evaluated in closed form?

I'm wondering whether any of these two indefinite integrals $$\int \frac{1}{\sqrt{1+\alpha \sinh(x)^{-4/3}}}dx$$ $$\int \frac{\sinh(x)^{-4/3}}{\sqrt{1+\alpha\sinh(x)^{-4/3}}}dx$$ can be evaluated in ...
4
votes
1answer
41 views

Why is $\arg(i\cosh x)=\frac{\pi}{2}$?

I was told $\arg(i\cosh (x))=\frac{\pi}{2}$ and $\arg(\cosh (x))=0$ but I can't figure out why. Could someone explain it to me?
1
vote
1answer
43 views

Solve ${y}' = \cosh^{-1}\left ( x \right ) + \mathrm {Si}(x)$

I am wondering how to find an explicit, closed-form solution for the following first-order differential equation: $${y}' = \cosh^{-1}\left ( x \right ) + \mathrm {Si}(x)$$ Where $\mathrm {Si}(x)$ ...
7
votes
2answers
136 views

Proof $1 -\frac{1}{5} + \frac{1}{9} - \frac{1}{13} + … = \frac{\pi + 2\ln(1+\sqrt2)}{4\sqrt2}$

I'm trying to show that $$1 -\frac{1}{5} + \frac{1}{9} - \frac{1}{13} + \cdots = \frac{\pi + 2\ln(1+\sqrt2)}{4\sqrt2}.$$ I thought of using the power series for $\tanh^{-1}z$ which I found was ...
0
votes
0answers
17 views

inverse hyperbolic function of a complex argument

It is not too difficult to prove that $f(z)=\cosh z$ is a bijection from $$\def\C{{\Bbb C}}D=\{\,z=x+iy\in\C\mid 0<y<\pi\,\}$$ to $$R=\{\,w=u+iv\in\C\mid v\ne0\,\}\cup\{\,w\in\C\mid ...
0
votes
0answers
32 views

$d/dx((\sinh^{-1}(\tan x)))$

I'm just wondering if I did everything correctly for this question, I know the answer is correct but I'm not 100% sure the steps I took to get there are valid: \begin{align} d/dx(\sinh^{-1}(tanx)) ...
0
votes
3answers
50 views

Help with a hyperbolic trig problem

$$\tanh n=\operatorname{csch}n$$ Solve so that $n=\ln(x\pm x^{1/2})$ $%replace "x^{1/2}" with "\sqrt{x}" if you want. - editor$ I need some advice with this problem; I answered a similar one ...
1
vote
1answer
31 views

How to solve transcendental hyperbolic equation

How can I solve the functional relation $$ e^{-af'(x)}\cosh( f(x) ) = bx $$ for $f(x)$? It would suffice to solve for $x>0$, $a>0$ and $b>0$.
1
vote
1answer
35 views

Recognising that $\sum_{n=0}^\infty \frac{a^2-b^2(2n+1)^2}{(a^2+b^2(2n+1)^2)^2}=-\frac{\pi^2\mathrm{sech}^2\left(\frac{a\pi}{2b}\right)}{8b^2}$

So I know from Mathematica that: $$\sum_{n=0}^\infty \frac{a^2-b^2(2n+1)^2}{(a^2+b^2(2n+1)^2)^2}=-\frac{\pi^2\mathrm{sech}^2\left(\frac{a\pi}{2b}\right)}{8b^2}$$ I am wondering how someone could ...
1
vote
1answer
51 views

How do I find the domain and range of $\tanh(x)$

I used the formula $$\frac{e^x - e^{-x}}{e^x + e^{-x}}$$ I calculated the inverse to find the range, but I got the incorrect answer. Please help me find the domain and range of $\tanh(x)$.
1
vote
2answers
59 views

Taking limit with hyperbolic functions

I have a problem with evaluating $$\sinh^{-1}(C \sinh (ax))\bigg|_{-\infty}^{+\infty}$$ where $C$ and $a$ are real positive constants.
0
votes
2answers
25 views

Rewriting solution in terms of hyperbolic trigs

I have to find the inverse laplace transform of: $\mathcal{L}^{-1}(\frac{s}{-8+2s+s^2})$ I found it was $\frac{2}{3}e^{-4t}+\frac{1}{3}e^{2t}$ But the question I'm asked is, determine $A,B,C,D$ ...
3
votes
1answer
47 views

Complicated integral, where $\int\coth(x)dx$ is somehow written in terms of $\int |x|e^{ix}dx$

In Gardiner's Quantum Noise the following integral equality is used (eq 3.3.10, 3.3.14): $$\int_0^{\infty}d\omega ...
0
votes
1answer
51 views

Simplify a parametric equation with hyperbolic trigonometric functions

I've the following parametric equations for a curve: $$\begin{cases}x(t)=a\cdot \operatorname{sech} (t) \\ y(t)=a\cdot(t-\tanh(t))\end{cases}$$ Now let $\theta(t)=-\arctan(\sinh(t))$ how does the ...
0
votes
2answers
30 views

$x/|x|$ question about division

What is $\frac{x}{|x|}$ can it be simplified? Because look at this. $\frac{r\cosh(x)}{\sqrt{\cosh^2(x)}} = \frac{r\cosh(x)}{|\cosh(x)|}$ How do you do this?
1
vote
0answers
16 views

matrix in normal coordinates

Writing the matrix $ \begin{pmatrix} -\frac{k}{\gamma} & \frac{k}{\gamma}&0&0&0&\cdots&0&0&0&0 \\ \frac{k}{\gamma} &-2\frac{k}{\gamma}& ...
0
votes
1answer
15 views

Reverse map for an equation .

I don't know this is actually reverse mapping or what but i have following equation. $$x = \tanh(a \cdot b ) + c $$ How do I solve for $a$? Does it has anything to do with inverse hyperbolic ...
0
votes
3answers
23 views

The relation between hyperbolic sine and hyperbolic cotangent

I was wondering if someone can verify (or not) the correctness of the following function? $$\frac{1}{\sinh^2X}=\coth^2X-1$$ I saw it in a paper but I am weak in math, so I am unsure if it is correct ...
2
votes
2answers
61 views

help with hyperbolic functions like sinh and tanh

Show that $\sin^{-1}(\tanh x)=\tan^{-1}(\sinh x)$. Got a hint that $\sin\theta=\tanh x$ but I still don't know how to proceed...
0
votes
1answer
76 views

Prove that the function f(x) = cosh(x)+ cos(x) is strictly increasing for non-negative x

I know that using the mean value theorem I should get $f'(x) =$ sinh$(x)$ - sin$(x)$, but from that on I have no ideas on how to show that $f'(x) > 0$ in the specified interval. Basic trigonometric ...
0
votes
2answers
37 views

Stuck on an indefinite integral probably using hyperbolic substitution.

First off, please don't give the answer. I'm really after a starting point. I'm trying to solve the integral $$\int \frac{1}{25e^x+9}~dx$$ I have done a few others where I have an $x$ instead of an ...
0
votes
2answers
30 views

Proving hyperbolic identities for $coth^2x-1 \equiv cosech^2x$

I've been working on hyperbolic functions lately. All is well, however I seem to come across a couple difficulties here and there when it comes to actually proving hyperbolic identities, thus I'm ...
0
votes
4answers
34 views

Evaluate coshX given that tanhX

Whilst working out some hyperbolic evaluation questions, I've come across this particular one. So far with any question I've come across I've simply tackled it step by step using hyperbolic ...
6
votes
0answers
98 views

Solving a problem of Ramanujan's interest

I am Brian Diaz, and I am new to the math.stackexchange community. I have been struggling with attempting to find a closed form of the following series: $$ \varphi(\theta) = 1 + 2\sum_{n=1}^{\infty} ...
0
votes
0answers
38 views

How to find z of cosh(z) = -2 & choosing value

$\cosh(z) = -2$ $z = \cosh^{-1}(-2)$ $z = \ln(-2 \pm i\sqrt{4-1})$ $z = \ln(-2 \pm \sqrt{3}) $ -> I wolfram this and it choose only $-2 - \sqrt{3}$ not $-2 + \sqrt{3}$ I would like to know what is ...
3
votes
1answer
57 views

manipulations with Taylor expansions for log and sinh

How could we derive the equality $$ \frac14 \sum_{m=1}^\infty \frac1m \frac{1}{\sinh^2 \frac{m\alpha}2} = - \sum_{n=1}^\infty n\log (1-q^n)$$ where $q=e^{-\alpha}$ ?
0
votes
1answer
27 views

Solution of $A = e^{\alpha t}\cos(\omega t + \phi)$

I would like to find the real roots of the function $$i(t) = \frac{\hat{V}}{R}\left(\frac{\omega^2}{(\alpha^2 + \omega^2)} \cos\left(\omega t + \tan^{-1}\left(\frac{\alpha}{\omega}\right)\right) + ...