Questions tagged [hyperbolic-functions]

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

The hyperbolic functions are analogs of the usual trigonometric functions; we may define such functions as the hyperbolic sine

$$\sinh{x} = \frac{e^x - e^{-x}}{2}$$

and the hyperbolic cosine

$$\cosh{x} = \frac{e^x + e^{-x}}{2}$$

as well as the hyperbolic tangent

$$\tanh{x} = \frac{\sinh{x}}{\cosh{x}}=\frac{e^x - e^{-x}}{e^x + e^{-x}}$$

These functions are differentiable. More precisely, $\cosh'=\sinh$, $\sinh'=\cosh$, and $\tanh'=1-\tanh^2$.

Just as the point $(\cos{t}, \sin{t})$ describes a point on the unit circle $x^2 + y^2 = 1$, the point $(\cosh{t}, \sinh{t})$ defines a point on the unit hyperbola $x^2 - y^2 = 1$.

Reference: Hyperbolic function.

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What do sinh and cosh have to do with exp?

My friend told me that $\sinh$ and $\cosh$ result from an exponential function, but I can't figure out why
frank
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Show that $ 2 \sinh(z)=\exp(z)-\exp(-z)$

$ 2 \sinh(z)=\exp(z)-\exp(-z)$; $ 2 \cosh(z)=\exp(z)+\exp(-z)$ where $z \in \mathbb{C} $ $$\sin(z) := \sum_{k=0}^{\infty}\frac{(-1)^k}{2k+1!} z^{2k+1}$$ $$\cos(z) := \sum_{k=0}^{\infty}\frac{(-1)^k}{2k!} z^{2k}$$ I guess that I have to use a…
Herrpeter
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Find min distance from y = cosh x to y = x

The full question is this: Point P on curve y = cosh x is such that its perpendicular distance from the line y = x is a minimum. Show P's coordinates are (ln(1 + root 2), root 2). I am completely at a loss as to what to do. I cannot find any…
user546944
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Prove $\forall x \in \mathbb{R}$: $[\sinh(x)+\cosh(x)]^n = \cosh(nx)+\sinh(nx)$ ; $ n\in \mathbb{Q}$

How to prove? Prove $\forall x \in \mathbb{R}$: $[\sinh(x)+\cosh(x)]^n = \cosh(nx)+\sinh(nx)$ ; $ n\in \mathbb{Q}$
leotv
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Does this series $\cosh 1 + (\cosh 1-1)x^2+(\cosh1-1-\frac{1}{2!})x^4+(\cosh1-1-\frac{1}{2!}-\frac{1}{4!})x^6+\cdots$ converge?

Can somebody help me with this? $\cosh 1 + (\cosh 1-1)x^2+(\cosh1-1-\frac{1}{2!})x^4+(\cosh1-1-\frac{1}{2!}-\frac{1}{4!})x^6+\cdots$
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Can a sum of different exponentials be rewritten as a sum of trigonometric and hyperbolic functions?

In a textbook I recently read that $$ A_1e^{x}+A_2e^{-x}+A_3e^{ix}+A_4e^{-ix}$$ (where $A_n\in\mathbb{C}$, $A_i\neq A_j\;\forall\;i\neq j$) can be rewritten as $$ A'_1\sin x+A'_2\cos x+A'_3\sinh x+A'_4\cosh x$$ (where $A'_n\in\mathbb{R}$). Is that…
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Differentiating hyperbolic functions.

$\DeclareMathOperator{\sech}{sech}$Can anyhow me how to differentiate the following? I already tried using the product rule, but I can't quiet seem to succeed. $\sech^{2} x$. $2\bigl(\cosh(2x) - 2\bigr) \sech^{4}(x)$. Thanks a million!
user2250537
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Defining the unbounded integral $\int_0^\infty e^{-x} \tanh(S_1(t-x)) \mathrm{dx}$ in MATLAB

How to define the unbounded integral term $$\int_0^\infty e^{-x} \tanh(S_1(t-x)) \mathrm{dx}$$ where $S_1(t)$ is a function of $t$? in MATLAB. Can anyone provide MATLAB code for the same?
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Find $\lim_{x\to 0} \frac{\sinh x -\sin x}{\cosh x - \cos x}$

Find $$\lim_{x\to 0} \frac{\sinh x -\sin x}{\cosh x - \cos x}$$ I used Hopital and couldn't find the answer which is $0$.
mathletic
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How do I show this :$\int_{-\infty}^{+\infty} x^n 2\cosh( x)e^{-x^2}=0$ if it is true with $n$ odd positive integer?

I have looked to show that this integral: $$\int_{-\infty}^{+\infty} x^n 2\cosh( x)e^{-x^2}=0$$ for $n$ is an odd positive integer , but i don't succeed to show that using standard method for getting closed form , Wolfram alpha show that is $0$…
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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 logarithms but it doesn't seem to reduce to any simpler…
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$e^{-\beta \frac{1}{2}\hbar \omega} \frac{1}{1 - e^{\beta \hbar \omega}} = \frac{1}{2 \sinh \left( \frac{\beta \hbar \omega}{2}\right)}$

\begin{equation} e^{-\beta \tfrac{1}{2}\hbar \omega} \dfrac{1}{1 - e^{\beta \hbar \omega}} = \dfrac{1}{2 \sinh \left( \frac{\beta \hbar \omega}{2}\right)} \end{equation} I need to know how this equality works? Any help is really appreciated.
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Is $ \frac {1}{\sqrt{a^2-b^2}} $ = $\cosh^{-1}(\frac{a}{b})$?

Is $ \frac {1}{\sqrt{a^2-b^2}} $ = $\cosh^{-1}(\frac{a}{b})$ is this true or false?
Neels
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