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

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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 ...
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27 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 $$ ...
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1answer
44 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 $u=\tanh x$, ...
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2answers
17 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 ...
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0answers
69 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 ...
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2answers
151 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 ...
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1answer
25 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 ...
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1answer
56 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) ...
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1answer
224 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 ...
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3answers
81 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 ...
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1answer
57 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 ...
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2answers
212 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 ...
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3answers
366 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 ...
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4answers
41 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} ...
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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 ...
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1answer
29 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) ...
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15 views

implicit derivation of hyperbolic functions

derive $$\sinh(x+y)=\tanh^{-1}(\frac{x}{y})$$ $$\cosh(x+y)\cdot(1+f')=\frac{\frac{y-xf'}{y^2}}{1-(\frac{x}{y})^2}$$ $$\cosh(x+y)\cdot(1+f')=\frac{\frac{y-xf'}{y^2}}{\frac{y^2-x^2}{y^2}}$$ ...
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3answers
108 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: ...
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2answers
39 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} &= ...
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1answer
57 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 ...
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1answer
20 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 - ...
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24 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
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4answers
57 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 ...
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3answers
61 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 ...
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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 = ...
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1answer
51 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 ...
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1answer
30 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 ...
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0answers
35 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 ...
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1answer
22 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: ...
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1answer
31 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 ...
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2answers
105 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 ...
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1answer
172 views

Is $ \sum_{n=1}\limits^{\infty}\frac{1}{\sinh(2^{n})} $ equal to $ \frac{2}{e^{2}-1}$?

Show that $$ \sum\limits_{n=1}\limits^{\infty}\frac{1}{\sinh(2^{n})}= \frac{2}{e^{2}-1}. $$
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1answer
29 views

Hyperbolic sine of a logarithm

Re-express $11\sinh(\ln 8)$ in the form $n/m$ where n and m are integers. I am not sure where to start. Never went over something like this, its probably very easy though.
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2answers
38 views

Differentiate $[x^{5\coth(6x)}]'$

can you help me to differentiate this function? $$[x^{5\coth(6x)}]'$$ My steps: $$[x^{5\coth(6x)}*\ln(x)]*[5(1-\coth^2(6x))]*[6]$$ I dont know what formula i should use $$[x^n]'$$ or $$[a^x]'$$ ...
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0answers
425 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 ...
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1answer
38 views

Hint to show $\tanh(z)=\frac{\sinh(2x)+i\sin(2y)}{\cosh(2x)+\cos(2y)}$?

I really can't figure out how to do this at all. I've been trying to show this for nearly 4 hours now. I've tried working from $\tanh(z)=\frac{\sinh(z)}{\cosh(z)}$ and expanding the top and bottom, ...
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1answer
50 views

Find the limit as x approaches infinity

$$\lim_{x\to \infty} {\cosh^{-1}(x^{3}) + \coth^{-1}(\sqrt{x^{2}+1}) - 3\sinh^{-1}(x)}$$ Honestly, I don't really know how to approach this. I know the logarithmic formulae for the inverse hyperbolic ...
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1answer
34 views

Hyperbolic trigonometric functions identity

$$\cosh(\sinh^{-1}x) = \sqrt{x^{2}+1}$$ I used the fact that $\cosh(x) = \frac{1}{2}(e^{x}+e^{-x})$ and that $\sinh^{-1}(x) = \ln(x + \sqrt{x^{2}+1})$ Eventually I simplified to ...
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1answer
43 views

A improper integral with integrable singularities

Let $\alpha$ and $v_0$ be both positive. Consider a following integral: \begin{equation} {\mathcal J}^\alpha_{1/2}(v_0) := \int\limits_0^\infty v^{\alpha-1} e^{-v} \frac{1}{\sqrt{v_0-v}} d v ...
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1answer
42 views

How to simplify expressions like $\sinh(4\,\text{arcsinh}(x))$?

I understand that expressions like $\sinh(\text{arcsinh}(x))$ simplify immediately and expressions like $\sinh(\text{arccosh}(x))$ tend to simplify after some algebra. However I cannot work out how ...
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1answer
103 views

Suppose $z=re^{iθ}$, prove : $|e^{iz}|=e^{-r\sinθ}$.

Suppose $z=re^{iθ}$, prove $|e^{iz}|=e^{-r\sin θ}$. I tried but the result was not as expected: $$e^{iz}=\cosh z+i(\sinh z)\\|e^{iz}|=(\cosh z)^2+[i(\sinh z)]^2=\cosh(2z)$$
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1answer
50 views

Exponential Proof

Let $c(x)=\dfrac{3^x+3^{-x}}{2}$ and $s(x)=\dfrac{3^x-3^{-x}}{2}$. Show that $(c(x))^2=\frac{1}{2}(c(2x)+1)$. How does one go about solving this? I have honesty tried substituting in ...
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0answers
57 views

Law of Cosines with imaginary arguments?

Does the law of cosines: $c^2 = a^2 + b^2 - 2 a b \cos \theta$ work with imaginary angles? to get something like: $c^2 = a^2 + b^2 - 2 a b \cosh \theta$ Alternatively, is there a hyperbolic ...
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0answers
49 views

Evaluate $\int\frac{1}{x}\coth(ax)\sin^2(\frac{xt}{2})dx$

I am trying to integrate this function: $\int_0^\infty\frac{1}{x}\coth(\frac{\hbar x}{2kT})\sin^2(\frac{xt}{2})dx$ which Wolframalpha (for me) returns nothing, just a blank screen. I thought that it ...
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3answers
30 views

Lateral limits of function involving hyperbolic trignometric functions

I am not being able to calculate the lateral limits at 0 of the following function $f(x) = \frac{\sinh(x)}{2\sqrt{\cosh(x) - 1}}$ I have tried both direct substitution (yields 0/0) and L'Hospital's ...
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0answers
8 views

Is the follow equation representative of a Hyperbolic One dimensional conservation law?

For $y : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ and let $j : \mathbb{R} \to \mathbb{R}$ $$\frac{\partial y}{\partial x_2} + \frac{\partial (j \circ y)}{\partial x_1} = 0$$
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0answers
35 views

Evaluating Hyperbolic Cotangent (coth) Integral

I am working on some simulation, and the paper that I am basing some of the work off of involves several complex integrals. In particular, the one I am trying to solve is $\int_0^\infty ...
5
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2answers
122 views

Explicit Form for Coefficients of Extended Hyperbolic Secant Function

Consider the function: $$\frac{3}{e^x+e^{{w_3}x}+e^{{w_3^2}x}}=\sum_{n=0}^\infty{E_{3,n}\frac{x^n}{n!}}$$ Note here that $w_3=e^{\frac{2i\pi}{3}}$ I am trying to get an explicit formula for the ...
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1answer
19 views

Integration of complicated hyperbolic functions

I have a complicated integral as below. I'd be appreciated if anyone could help me to find the answer. $$ I=\int_{0}^{U}du [(\partial_uq)^2+w^2q^2] $$ and the $q(u)$ is defined as $$ ...
5
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1answer
81 views

Closed form for finite sum of ${\rm csch}^2$

In a recent problem I was attempting to solve, I hit a road block when I reduced the problem to that of finding a closed form for the following sum $$ \mathcal{S}_n(x)\equiv\sum_{k=1}^n{\rm ...