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

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6
votes
3answers
300 views

Is there a formula for the area under $\tanh(x)$?

I understand trigonometry but I've never used hyperbolic functions before. Is there a formula for the area under $\tanh(x)$? I've looked on Wikipedia and Wolfram but they don't say if there's a ...
-1
votes
1answer
17 views

How to find $x$ such that $f(x)$ takes a prescribed value [on hold]

Find $x$ such that \begin{equation} x\tanh(x\sqrt{2\alpha})=\frac{2}{\sqrt{2\alpha}} \end{equation}
0
votes
1answer
41 views

Finding if there is a maximum or minimum on a curve?

My apologies for being very brief with this question, the reason for this is because I don't know where to start. The question is as follows: A curve has the equation $\lambda \cosh(x) + \sinh(x)$, ...
11
votes
3answers
300 views

Integral $\int_0^\infty\frac{\tanh^2(x)}{x^2}dx$

It appears that $$\int_0^\infty\frac{\tanh^2(x)}{x^2}dx\stackrel{\color{gray}?}=\frac{14\,\zeta(3)}{\pi^2}.\tag1$$ (so far I have about $1000$ decimal digits to confirm that). After changing variable ...
3
votes
2answers
304 views

How to I show that $\lim_{x\to0} \frac{1}{x^2}\left(\frac{\sinh x}{x}-1\right) = \frac{1}{6}$

I can do this limit with a symbolic calculator and get the result. $$\lim_{x\to0} \left[ \frac{1}{x^2}\left(\frac{\sinh x}{x} - 1\right) \right] = \frac{1}{6}$$ But how would I do it by hand, and ...
0
votes
2answers
25 views

Difficulty finding the sum of a hyperbolic function.

Can someone please point out where I am (If I am) going wrong during the solution process of the following question: I have been presented with the following : $$4sinh(2ln(2))-cosh(ln2)$$ and told ...
0
votes
2answers
43 views

What's the relationship between hyperbola, hyperbolic functions and the exponential function?

The hyperbolic functions can be expressed using the exponential function. However how are these related to "hyperbolas"?
2
votes
3answers
60 views

How to evaluate these two integrals about hyperbolic functions?

While I was calculating the two integrals below \begin{align*} \mathcal{I}&=\int_{0}^{\infty }\frac{\cos x}{1+\cosh x-\sinh x}\mathrm{d}x\\ \mathcal{J}&=\int_{0}^{\infty }\frac{\sin x}{1+\cosh ...
1
vote
1answer
34 views

Length of the arc of a hypercycle

I am still puzzeling to get a nice equation for the arclength of an hypercycle. (I asked a similar question (less developed) about a year ago that was never answered, now i am a bit further, i ...
1
vote
3answers
75 views

Evaluating a complex limit

I would love some advice on how to approach the following limit: $$\lim_{z\to \infty} \frac{\sinh(2z)}{\cosh^2(z)}$$ or let $z= \dfrac{1}{t}$ then $$\lim_{t\to 0} ...
0
votes
1answer
13 views

Limit of complex hyperbolic tan

For the hyperbolic tan function, we have the property $\displaystyle \lim_{a \to \infty} \tanh(a(z-b)) = \mathrm{signum}(z-b)$ where $z$, $a$ and $b$ are real. But what happens if $b$ is complex ...
0
votes
0answers
38 views

Sum Calculation: $\sum_{n=1}^\infty \left(1- \frac{\cosh^{-1} n}{\log 2x}\right)$

I was investigating the asymptotic properties of the $\cosh$ functions and how they all strongly relate to $e^x$ In my studying, I found out that $\cosh x\sim \frac{e^x}{2}$ By that definition, that ...
1
vote
0answers
34 views

Prove the identity $\tanh(N\textrm{acosh}\;a) = \vert \frac{g^{2N}-1}{g^{2N}+1}\vert$

During my recent study, I found an Identity which is of the form $$ \tanh(N\textrm{acosh}\;a) = \left\vert \frac{g^{2N}-1}{g^{2N}+1}\right\vert $$ where $a\geq1$ and $g>0$ satisfy ...
2
votes
1answer
36 views

Simplifying hyperbolic compositions like $\sinh (N \operatorname{acosh} a)$

In many occasions, we may meet hyperbolic functions, as well as their combined ones. I want to simplify expressions like $$ \tanh\left( N\left(\textrm{acosh}~ a\right)\right) $$ and $$ \sinh\left( ...
1
vote
1answer
27 views

Hyperbolic equation

I have the following hyperbolic identity, which I solved, analytically: $6~\text{sech}^2 x$ $= 4 + \tanh x$ The two solutions which I get are: $x=\frac{1}{2} \ln3$ and $x=-\frac{1}{2} \ln5$ These ...
0
votes
0answers
25 views

What the inverse function of $_2F_1(a,b;c;z)$

What the inverse function of the function $f(z)$ given by $$ f(z) = \, _2F_1(a,b;c;z), \quad \mid z \mid <1, $$ where is the Gauss hypergeometric function given by $$ ...
2
votes
1answer
24 views

Relating the argument of a hyperbolic trig function to area

Based on the definition of the hyperbolic trig functions (in terms of $e^x$ and $e^{-x}$) it's easy to show that the point $\big(\cosh(\alpha),\sinh(\alpha)\big)$ falls on the unit hyperbola ...
1
vote
1answer
28 views

Integral representation of the modified Bessel function involving $\sinh(t) \sinh(\alpha t)$

I've come across this peculiar integral representation for $K_\alpha(x)$: $\frac{\alpha}{x}K_\alpha(x) = \int_0^\infty dt \sinh(t) \sinh(\alpha t) e^{-x \cosh(t)}$ How does it come about? Are there ...
3
votes
0answers
104 views

Analytic forms of $\frac{\sinh(2\pi/7)}{\sinh^{2}(3\pi/7)} - \frac{\sinh(\pi/7)}{\sinh^{2}(2\pi/7)} + \frac{\sinh(3\pi/7)}{\sinh^{2}(\pi/7)}$

On page 183 of Berndt's Ramanujan's Notebooks Vol. 4, eq. 32.34 reads: $$ \frac{\sin(2\pi/7)} {\sin^{2}(3\pi/7)} - \frac{\sin(\pi/7)}{\sin^{2}(2\pi/7)} + \frac{\sin(3\pi/7)} {\sin^{2}(\pi/7)} = ...
21
votes
6answers
361 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 ...
0
votes
1answer
33 views

Integral of a hyperbolic function

$$\int \tanh(x) - \tanh^3(x)\,dx$$ I get the answer as $\tanh x + c$? I took out a factor of $\tanh x$, used the identity $1-\tanh^2 x=\text{sech}^2x$, used the substitution of $u=\tanh x$, ...
0
votes
1answer
54 views

Identity between $x=y+z$ and $\tan\left(\frac{\theta}{2}\right)=\tan\left(\frac{\nu}{2}\right)\tan\left(\frac{\pi/2-\epsilon}{2}\right) $

I would like to prove that (1) $$\begin{equation} \tan\left(\frac{\theta}{2}\right)=\tan\left(\frac{\nu}{2}\right)\tan\left(\frac{\pi/2-\epsilon}{2}\right) \end{equation}$$ can transformed to (2) ...
1
vote
3answers
54 views

Prove the identity $\tanh\left(\frac{x}{2}\right)=\frac{\cosh(x)-1}{\sinh(x)}$

Prove that $$\tanh\left(\frac{x}{2}\right)=\frac{\cosh(x)-1}{\sinh(x)}$$ I have started with: ...
1
vote
3answers
96 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 ...
4
votes
1answer
169 views

Prove the identity $\cosh(2x)=\cosh^2(x)+\sinh^2(x)$ using the Cauchy product. [closed]

Prove the identity $$\cosh(2x)=\cosh^2(x)+\sinh^2(x)$$ using the Cauchy product and the Taylor series expansions of $\cosh(x)$ and $\sinh(x)$. The relations involving the exponential function are ...
0
votes
2answers
12 views

Solve the value of a and b for a catenary (hyperbolic function question)

I am having trouble with the following question: A more general equation for a catenary is $y = a \cosh(x/b)$. Find $a$ and $b$ to match the following characteristics of a hanging cable. The ends ...
1
vote
0answers
61 views

Inverse function of sum of coth and tanh terms

In a publication I found an equation of the form $c_p = B + C \left( \frac{D/T}{\sinh(D/T)} \right)^2 + E \left( \frac{F/T}{\cosh(F/T)} \right)^2$ $c_p$ is the heat capacity, $T$ is the ...
9
votes
2answers
141 views

Calculate the sum: $\sum_{x=2}^\infty (x^2 \operatorname{arcoth}(x) \operatorname{arccot} (x) -1)$

$${\color\green{\sum_{x=2}^\infty (x^2 \operatorname{arcoth} (x) \operatorname{arccot} (x) -1)}}$$ This is an impressive sum that has bothered me for a while. Here are the major points behind the ...
8
votes
3answers
78 views

Prove that $\cosh^{-1}(1+x)=\sqrt{2x}(1-\frac{1}{12}x+\frac{3}{160}x^2-\frac{5}{896}x^3+…)$

How can we prove the series expansion of $$\cosh^{-1}(1+x)=\sqrt{2x}\left(1-\frac{1}{12}x+\frac{3}{160}x^2-\frac{5}{896}x^3+...\right)$$ I know the formula for ...
1
vote
1answer
20 views

Hyperbolic Cosine: System of Equations, Isolate Variables

Background information that may help you answer: Alright, so I'm working on a formula that posits that there are a unique pair of coordinates $(x_1, y_1)$ and $(x_2, y_2) $ on the hyperbolic cosine ...
3
votes
1answer
53 views

Reduction of $\tanh(a \tanh^{-1}(x))$

Given $x\in \Re$, $a \in \Re$ where $-1 \le x \le 1$ and $0 \le a \le 4$, is it possible to reduce the following expression: $\tanh(a \tanh^{-1}(x))$ E.g. to some kind of polynomium? I know that if ...
12
votes
2answers
206 views

Need help with $\int_0^1\frac{\log(1+x)-\log(1-x)}{\left(1+\log^2x\right)x}\,dx$

Please help me to evaluate this integral $$\int_0^1\frac{\log(1+x)-\log(1-x)}{\left(1+\log^2x\right)x}\,dx$$ I tried a change of variable $x=\tanh z$, that transforms it into the form ...
2
votes
4answers
72 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 ...
2
votes
4answers
39 views

Simplifying Dervatives of Hyperbolic functions

Last minute Calc I reviews have me stumbling on this question $$D_x\left[\frac {\sinh x}{\cosh x-\sinh x}\right] $$ I've solved the derivative as $$ y' = \frac{\cosh x}{\cosh x-\sinh x} ...
0
votes
0answers
20 views

$\frac{d^n}{dx^n}$ in term of $\frac{d}{d t}$ for $x= \frac{1}{\cosh^{2} (t)}$

Let: $$x= \frac{1}{\cosh^{2} (t)},$$ I want to express $\frac{d^n}{dx^n}$ in term of $\frac{d}{d t}$. we have $x = \cosh^{-2} (t)$, so \begin{align*} \frac{d}{dt} &= \frac{d}{dx} \frac{d x ...
0
votes
1answer
28 views

where is the singularity of this function?

Consider the following function $\ f:\mathbb{R} \to \mathbb{R}$ given by: $$f(x) = - \tanh\left(3 x - \tanh^{-1}( u_{0} )\right)$$ In the above, $u_{0}$ is a constant where $u_{0} \in (-\infty, -1) ...
0
votes
0answers
15 views

implicit derivation of hyperbolic functions

derive $$\sinh(x+y)=\tanh^{-1}(\frac{x}{y})$$ $$\cosh(x+y)\cdot(1+f')=\frac{\frac{y-xf'}{y^2}}{1-(\frac{x}{y})^2}$$ $$\cosh(x+y)\cdot(1+f')=\frac{\frac{y-xf'}{y^2}}{\frac{y^2-x^2}{y^2}}$$ ...
4
votes
4answers
467 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 ...
0
votes
2answers
37 views

Calculation with Hyperbolic Cosine

Could you please check my work? $\cosh \left(\ln \sqrt{5}\right) =\ ?$ \begin{align*}\cosh(x) &= \frac{e^x + e^{-x}}{2} \\ \\ \frac{e^{\ln \sqrt{5}} + e^{-\ln \sqrt{5}}}{2} &= ...
2
votes
0answers
35 views

Find the positive root of the equation $\cosh x+\cos x-3=0$, other than numerically

I know you are able to find the root of the equation by using Newton-Raphson method. But is there any other way? $$\cosh x+\cos x-3=0$$ I thought maybe you could say that $-1\leq \cos x \leq 1$. So ...
1
vote
4answers
38 views

Integrate $\int \cosh^4(7x) dx$

Can you help me with solving this integral $$\int \cosh^4(7x) \, dx \text{ ?}$$ I tried use subs but i got $$\cosh^4(t)=m$$ $$t = \pm\operatorname{arccosh}^{-1}(m^{1/4})$$ $$dt = ...
0
votes
1answer
19 views

ODE with hyperbolics from J.D Murray's Mathematical Biology

After working through J.D Murray's Mathematical Biology, I have come across this differential equation during a derivation of the SIR model. $\frac{dR}{dt}$ = a$[N - S_0 + (\frac{S_0}{p}-1)R - ...
0
votes
0answers
21 views

How to prove that $(\frac{1+\tanh x}{1-\tanh x})^3=\cosh 6x+\sinh 6x$

How to prove that $$(\frac{1+\tanh x}{1-\tanh x})^3=\cosh 6x+\sinh 6x$$ I have tried using the Dmoivres theorme
0
votes
4answers
55 views

How to prove $\tanh ^{-1} (\sin \theta)=\cosh^{-1} (\sec \theta)$

As the question says How to prove $$\tanh ^{-1} (\sin \theta)=\cosh^{-1} (\sec \theta)$$ I have tried to solve it The end result that got for RHS $$=\log \frac{1+\tan\frac{\theta}{2}}{1-\tan ...
3
votes
3answers
56 views

How to prove $\sinh^{-1} (\tan x)=\log \tan (\frac{\pi}{4}+\frac{x}{2})$

Like the question says How to prove $$\sinh^{-1} (\tan x)=\log \tan (\frac{\pi}{4}+\frac{x}{2})$$ I have tried using many identity but in vain For reference $$\tanh ^{-1} x=\frac{1}{2} \log ...
0
votes
1answer
48 views

$\cos(y\,\operatorname{acosh}(\exp(x)))$ is real for all real $x,y$

$\cos(y\,\operatorname{acosh}(\exp(x)))$ is real for all real $x,y$ even though $\operatorname{acosh}$ is complex for $x<1$. I found it empirically but still can't prove it yet. Can someone please ...
1
vote
0answers
33 views

Functional equation for $\sum_{n=1}^{\infty} \sinh(cn)^{-s}$?

Does anyone know of any kind of functional equation (or closed form) for $\sum\limits_{n=1}^{\infty}\sinh(cn)^{-s}$, where $c$ is an arbitrary constant? I've been messing around with it off and on for ...
0
votes
1answer
19 views

hyperbolic function simplification

In taking a derivative, I end up with this epression: $e^x*(1-e^x)-(1+e^x)*(-e^x) / (1-e^x)^2$ However, from a calculator I see that the first line simplifies in nicer, simpler expression: ...
0
votes
2answers
86 views

How to solve lim as x approaches infinity for $[\tanh(x)]^x$

I got as far as lim x approaches infinity for $\ln y = x \ln(\tanh x)$. I'm not sure what to do there. I know $\tanh x$ as $x$ approaches infinity is one but $1^\infty$ isn't the correct answer. So ...
21
votes
0answers
354 views

Curious about an empirically found continued fraction for tanh

First of all, and since this is my first question in this forum, I would like to specify that I am not a professional mathematician (but a philosophy teacher); I apologize by advance if something is ...