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

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17
votes
6answers
184 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
100 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
34 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
52 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
27 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 ...
2
votes
0answers
40 views

How to prove that $\frac{200}{\pi}\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)\cosh{((2k+1)\frac{\pi}{2})}}=25$ [closed]

A friend of mine asked me to prove: $$ \frac{200}{\pi}\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)\cosh{((2k+1)\frac{\pi}{2})}}=25 $$ but I don't know where to start (honestly I am not much good at math). ...
2
votes
0answers
32 views

Can I get $e^x=f(trigonometric function)$ without $i$ [closed]

I searched in many websites to get identity of $e^x$ with the trigonometric function but I didn't find. Can anyone help me to give me it if there is
1
vote
1answer
31 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
42 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: ...
5
votes
0answers
63 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
40 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
42 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
133 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
16 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
30 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
49 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
27 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
31 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
44 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
50 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
24 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
45 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
43 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
29 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
14 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
14 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
22 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
60 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
73 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
32 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
26 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
33 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
91 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
27 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
56 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
26 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) + ...
2
votes
2answers
20 views

Roots of $i(t) = Ae^{\alpha t}cos(\omega t + \phi)$

I would like to find the roots of the function $i(t) = Ae^{\alpha t}\cos(\omega t + \phi)$ in the form $t = f(A, \alpha, \omega, \phi)$.
2
votes
2answers
62 views

Simplify $\sinh (\log (x))$

$$\sinh (\log (x))=\frac{x^2-1}{2 x}$$ However I do not see how this is done, here is an idea I had but I'm probably way off: $$\sinh \left(\ln \left(\frac{1}{2} ...
5
votes
3answers
131 views

Find $\int \sinh^{-1}x\hspace{1mm}dx$

Find $\int \sinh^{-1}x\hspace{1mm}dx$ $ $ I am asked to use the following Equation: $$\int \tan^{-1}x\hspace{1mm}dx= x\tan^{-1}x-\ln(\sec(\tan^{-1}x))+C$$ $ $ The confusing part is : What has ...
7
votes
4answers
251 views
0
votes
2answers
26 views

Tangent - point of contact

Question: Tangent to the curve $y = x^2 + 6$ at point P(1, 7) touches the circle $x^2 + y^2 + 16x + 12y + c = 0$ at a point Q. Then the coordinates of Q are: 1) (-6, -11) 2) (-9, -13) 3) (-10, -15) 4) ...
0
votes
2answers
63 views

A Method For Calculating Large Exponents Quickly

I've derived a formula for calculating large exponents quickly: $$a^b = 2 \cosh( - b \log( a ) )$$ My question is: Has anyone seen anything similar? I am curious if either it's novel OR if I have ...
0
votes
0answers
74 views

Hyperbolic sinc function

Cardinal sine function or sinc function is defined by: \begin{equation} \mathrm{sinc}x=\begin{cases}\frac{\sin x}{x}, & x \neq 0,\\ 1, & x = 0,\end{cases} \end{equation} Is there any ...
1
vote
2answers
33 views

here is a theorem that the isometries of the Hyperbolic plane are generated by $PSL(2, \mathbb{R})$ and $z \rightarrow - \overline{z}$.

There is a theorem that the isometries of the Hyperbolic plane are generated by $PSL(2, \mathbb{R})$ and $z \rightarrow - \overline{z}$. My question is, isn't $z \rightarrow kz$ an isometry for ...
0
votes
0answers
10 views

How can I find a hyperbolic function denoting zoom levels?

I'm working between two values. The first ($m$) represents the number of meters wide an estimate of location accuracy is, and the other ($z$) represents a vague level of zoom as described below. This ...
2
votes
0answers
24 views

Proof that function is real part of $\sec(z)$ [duplicate]

I'm working on the following problem: I've deduced that the key is to show that $u$ is the real part of $\sec(z)$. But, I'm getting stuck in the algebra and am hoping someone can point me in the ...
0
votes
1answer
42 views

Proof that $\cos^2(x)\cosh^2(y) + \sin^2(x)\sinh^2(y) = -1 + \sin^2(x) - \sinh^2(y)$

Could anyone offer a proof that $$ \cos^2(x)\cosh^2(y) + \sin^2(x)\sinh^2(y) = -1 + \sin^2(x) - \sinh^2(y)? $$
0
votes
3answers
72 views

How does $\frac{1}{2}\cosh(2x) -1 = \sinh^2(x)$?

Using hyperbolic trigonometric function identities is there a way to prove the following equation? $$\frac{1}{2} (\cosh(2x)-1) = \sinh^2(x)$$
3
votes
2answers
746 views

Calculate cosh(x) given sinh(x)

Given the value of sinh(x) for example sinh(x) = 3/2 How can I calculate the value of cosh(x) ?
4
votes
4answers
225 views

What is the use of hyperbolic trigonometric functions if they are easily expressible algebraically?

I get that there are uses for $\sin(x)$ and $\cos(x)$ because they are defined with imaginary exponents which aren't as easily worked with but the hyperbolic functions are simply $\frac12(e^x\pm ...