# Tagged Questions

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

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### 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 $\lambda \cosh(x) + \sinh(x)$, ...
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### 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 ...
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### 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"?
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### 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 ...
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### 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 ...
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### 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 ...
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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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### $\frac{d^n}{dx^n}$ 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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### 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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Can you help me with solving this integral $$\int \cosh^4(7x) \, dx \text{ ?}$$ I tried use subs but i got $$\cosh^4(t)=m$$ $$t = \pm\operatorname{arccosh}^{-1}(m^{1/4})$$ $$dt = ... 1answer 51 views ### \cos(y\,\operatorname{acosh}(\exp(x))) is real for all real x,y \cos(y\,\operatorname{acosh}(\exp(x))) is real for all real x,y even though \operatorname{acosh} is complex for x<1. I found it empirically but still can't prove it yet. Can someone please ... 1answer 31 views ### Integral representation of the modified Bessel function involving \sinh(t) \sinh(\alpha t) I've come across this peculiar integral representation for K_\alpha(x): \frac{\alpha}{x}K_\alpha(x) = \int_0^\infty dt \sinh(t) \sinh(\alpha t) e^{-x \cosh(t)} How does it come about? Are there ... 0answers 36 views ### Functional equation for \sum_{n=1}^{\infty} \sinh(cn)^{-s}? Does anyone know of any kind of functional equation (or closed form) for \sum\limits_{n=1}^{\infty}\sinh(cn)^{-s}, where c is an arbitrary constant? I've been messing around with it off and on for ... 1answer 22 views ### hyperbolic function simplification In taking a derivative, I end up with this epression: e^x*(1-e^x)-(1+e^x)*(-e^x) / (1-e^x)^2 However, from a calculator I see that the first line simplifies in nicer, simpler expression: ... 1answer 32 views ### Relating the argument of a hyperbolic trig function to area Based on the definition of the hyperbolic trig functions (in terms of e^x and e^{-x}) it's easy to show that the point \big(\cosh(\alpha),\sinh(\alpha)\big) falls on the unit hyperbola ... 2answers 113 views ### How to solve lim as x approaches infinity for [\tanh(x)]^x I got as far as lim x approaches infinity for \ln y = x \ln(\tanh x). I'm not sure what to do there. I know \tanh x as x approaches infinity is one but 1^\infty isn't the correct answer. So ... 1answer 173 views ### Is  \sum_{n=1}\limits^{\infty}\frac{1}{\sinh(2^{n})}  equal to  \frac{2}{e^{2}-1}? Show that$$ \sum\limits_{n=1}\limits^{\infty}\frac{1}{\sinh(2^{n})}= \frac{2}{e^{2}-1}. $$1answer 30 views ### Hyperbolic sine of a logarithm Re-express 11\sinh(\ln 8) in the form n/m where n and m are integers. I am not sure where to start. Never went over something like this, its probably very easy though. 2answers 38 views ### Differentiate [x^{5\coth(6x)}]' can you help me to differentiate this function?$$[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]'$$... 1answer 559 views ### Curious about an empirically found continued fraction for tanh First of all, and since this is my first question in this forum, I would like to specify that I am not a professional mathematician (but a philosophy teacher); I apologize by advance if something is ... 1answer 39 views ### Hint to show \tanh(z)=\frac{\sinh(2x)+i\sin(2y)}{\cosh(2x)+\cos(2y)}? I really can't figure out how to do this at all. I've been trying to show this for nearly 4 hours now. I've tried working from \tanh(z)=\frac{\sinh(z)}{\cosh(z)} and expanding the top and bottom, ... 1answer 50 views ### Find the limit as x approaches infinity$$\lim_{x\to \infty} {\cosh^{-1}(x^{3}) + \coth^{-1}(\sqrt{x^{2}+1}) - 3\sinh^{-1}(x)}$$Honestly, I don't really know how to approach this. I know the logarithmic formulae for the inverse hyperbolic ... 1answer 34 views ### Hyperbolic trigonometric functions identity$$\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 ...
Let $\alpha$ and $v_0$ be both positive. Consider a following integral: {\mathcal J}^\alpha_{1/2}(v_0) := \int\limits_0^\infty v^{\alpha-1} e^{-v} \frac{1}{\sqrt{v_0-v}} d v ...