Tagged Questions

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

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Distance between points lying on a hyperbola?

The question is rather simple but I can't find the answer I'm looking for anywhere. On an ordinary 1-dimensional hyperbola, given two points on the hyperbola, what is the length of the path between ...
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Hyperbolic sin derivation

https://www.youtube.com/watch?v=zd3RyRk6wYI On Khan Academy, Sal derives the hyperbolic function of sin in terms of $i\theta$. My question is, how did he get rid of the $i$ in the denominator? I know ...
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Complex conjugate of the logarithm of the hyperbolic tangent

Given the Schwarz reflection principle, I would aytomaticaly write down that the complex conjugate of the following function: $$ln[tanh(z)]$$, where z is a complex number, is: $$ln[tanh(\bar{z})]$$....
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Let $s(x)=\sum_{n=0}^\infty \frac1{(2n+1)!}x^{2n+1}$, $c(x)=\sum_{n=0}^\infty \frac1{(2n)!}x^{2n}$ prove that: $c(x)^2-s(x)^2=1$

Let $$s(x)=\sum_{n=0}^\infty \frac1{(2n+1)!}x^{2n+1}$$ $$c(x)=\sum_{n=0}^\infty \frac1{(2n)!}x^{2n}$$ prove that $$c(x)^2-s(x)^2=1$$ I know that the following series are representations of the cosh ...
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If $I_n$ is defined as $\int^1_0\sinh^nx$ show that $nI_n+(n-1)I_{n-2}=\cosh1\sinh^{n-1}1$

I've tried evaluating the first three terms, so I have the results: $I_1=\cosh1-1$ $I_2=\frac{1}{4}\sinh2-\frac{1}{2}$ $I_3=\frac{1}{12}\cosh3-\frac{3}{4}\cosh1+\frac{2}{3}$ These do satisfy the ...
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Integrate $\int\frac{dx}{a^2-x^2}$

For what values of $x$ is this valid? $$\int\frac{dx}{a^2-x^2}=\frac{1}{a}\tanh^{-1}\frac{x}{a}+C$$ I think the anwer should be $-a<x<a$ because of the domain of $tanh^{-1}$. Is this correct? ...
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$\tanh(x)$ is bijective, where to get continuity?

I'm trying to show $\tanh(x)$ is bijective using the intermediate value theorem. It works by noting $\tanh(x)$ as strictly increasing by differentiating $\tanh(x)$ and then surjective using limits to ...
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Applied Hyperbolic sinh(x) question - Getting Started

I'm REALLY stuck on this question as I don't really know how to begin, I understand that it has something to do with: $$\sinh⁡ x=\frac{e^x-e^{-x}}{2}$$ I'm definitely not asking for someone to do it ...
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Finding limit of hyperpolic expression.

I am having trouble of how to solve this kind of problem. I have to show the limit of the function: $f(x)=\frac{1 - \tanh x}{e^{-2x}}$ $\lim_{x\to\infty} f(x)$ I am to do this without using ...
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Reduction of expression algebraically

I have asked this question before and it helped me get a little further, but not at a solution. I have to algebraically reduce the expression: $\sinh(2 \cdot \sinh^{-1}(y))$ Now i had the idea of ...
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Prove cosh(x) and sinh(x) are continuous.

I failed this task at my univiersity and i do not understand why. No feedback was given. I have to prove that cosh(x) and sinh(x) are continious. I proved it for cosh(x) and said the same principles ...
I have been working on this one for a couple of hours and i just get stuck on every attempt i make. I have to reduce the formula algebraically: $\sinh(2 \cdot \sinh^{-1}(y))$ And I just can't seem ...