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.

1108 questions
-1
votes
1 answer

Find the integral in a closed form : $\int\limits_0^\infty\text{arctanh}(x^{2})e^{-ax^{2}}dx$

Evaluate : $\displaystyle\int_{0}^{\infty}x\text{arctanh}(x^{2})e^{-ax^{2}}dx$ where $a>1$ I don't know if we can find closed form or not but I have some results for…
Thê Kîng
  • 149
  • 2
-1
votes
1 answer

A simple Variation on the Imaginary Unit i

I think a more appropriate tag would have been 'quasicomplex numbers' rather than 'hypercomplex numbers'. I'm normally perfectly comfortable with the correspondence between hyperbolic functions & circular functions - the way one is just the other…
AmbretteOrrisey
  • 961
  • 3
  • 12
-1
votes
1 answer
-1
votes
2 answers

Integral of $\arccos(x + 1)$

I'm trying to work out how to find the indefinite integral of $\operatorname{arccosh}(x + 1)$ I have been using integration by parts to get it down to $$x\operatorname{arccosh}(x + 1) - \int \frac{x}{\sqrt{\left({(x+1)^2} - 1 \right)}} \,…
user589159
-1
votes
1 answer

Hyperbolic inverse Function

how to find the inverse function of the arcsinh x also the domainof arcsinh
-1
votes
2 answers

how do we prove integral sechx?

$$\int \text{sech} x\, dx = 2\arctan(\tanh x/2) $$ how do we prove this in step by step process?? Is $\arctan(\sinh x)$ equal to $2\arctan(\tanh x/2)$ ??
IreneManiac
  • 27
  • 1
  • 2
-1
votes
3 answers

Hyperbolic differentiation of $\sinh^{-1}(x/a)$

So as the title states I'd like to find the derivative. I've used different methods but upon looking at the formula I noticed a difference between the author's approach and mine.…
-1
votes
1 answer

Showing a function is one-to-one

So I know for one-to-one I need to show Ψ(u1,v1)=Ψ(u2,v2) but I am unsure how to go to about it for this function...
KM9
  • 135
  • 7
-1
votes
2 answers

Solve $4\sinh (x)+3\cosh (x)=0$ for $x$

Solve the following for $x$ giving your answer to $3$ significant figures: $$4\sinh (x)+3\cosh (x)=0$$ I need help understanding hyperbolic functions.
curtis
  • 1
-1
votes
2 answers

two variables quadratic inequalities solution

Suppose there are $n$ quadratic inequalities, the form is $A_i x^2 + B_i y^2 + C_i xy + D_i x + E_i y + F_i \leq 0$, $(\forall i \in [1,n])$, where $x,y$ are two variables and $(A_i, B_i, C_i, D_i, E_i, F_i)$ are constant numbers, and all the range…
-1
votes
1 answer

Alternative methods to trigonometric equations

I've been given the the question The length $L$ of a heavy cable hanging under gravity is given you the equation $L=2\sinh x + 3\cosh x$ $L$ is given as $5$ I have working this out into a quadratic formula and found the answer. I have been asked…
-1
votes
1 answer

How to find $x$ such that $f(x)$ takes a prescribed value

Find $x$ such that \begin{equation} x\tanh(x\sqrt{2\alpha})=\frac{2}{\sqrt{2\alpha}} \end{equation}
Ruth90
  • 277
  • 2
  • 11
-1
votes
3 answers

Absolute value of hyperbolic function

Is this statement true, if yes, can anyone show me why? $$ \cosh(z)\cosh(z^*) = |\cosh(z)|^2 $$
camzor00
  • 11
  • 4
-1
votes
1 answer

Hyperbolic functions and sequence and series

Using these identities: sinh(mx+x)=cosh(mx)sinh(x)+sinh(mx)cosh(x) cosh(mx+x)=cosh(mx)cosh(x)+sinh(mx)sinh(x) Express the following sums in terms of just cosh((n+1)x), sinh((n+1)x), cosh(x) and…
Alexio
  • 17
  • 6
-1
votes
1 answer

hyperbolic isometry

I have a project I have to do. In order to do it I need to investigate this book. In page 94 they defined hyperbolic isometry on a metric s.t it possesses no fixed point in the tree. After that they claim that such isomtery possesses 2 fixed point…
wantToLearn
  • 1,275
  • 11
  • 17
1 2 3
73
74