# What is Hyperbolic: Definition and 344 Discussions

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, just as the derivatives of sin(t) and cos(t) are cos(t) and –sin(t), the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t).
Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.
The basic hyperbolic functions are:
hyperbolic sine "sinh" (),
hyperbolic cosine "cosh" (),from which are derived:
hyperbolic tangent "tanh" (),
hyperbolic cosecant "csch" or "cosech" ()
hyperbolic secant "sech" (),
hyperbolic cotangent "coth" (),corresponding to the derived trigonometric functions.
The inverse hyperbolic functions are:
area hyperbolic sine "arsinh" (also denoted "sinh−1", "asinh" or sometimes "arcsinh")
area hyperbolic cosine "arcosh" (also denoted "cosh−1", "acosh" or sometimes "arccosh")
and so on.
The hyperbolic functions take a real argument called a hyperbolic angle. The size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector.
In complex analysis, the hyperbolic functions arise as the imaginary parts of sine and cosine. The hyperbolic sine and the hyperbolic cosine are entire functions. As a result, the other hyperbolic functions are meromorphic in the whole complex plane.
By Lindemann–Weierstrass theorem, the hyperbolic functions have a transcendental value for every non-zero algebraic value of the argument.Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert. Riccati used Sc. and Cc. (sinus/cosinus circulare) to refer to circular functions and Sh. and Ch. (sinus/cosinus hyperbolico) to refer to hyperbolic functions. Lambert adopted the names, but altered the abbreviations to those used today. The abbreviations sh, ch, th, cth are also currently used, depending on personal preference.

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1. ### A Numerically solving a transport equation

I'm using a downwind'' approximation for the spatial derivative: \frac{\partial v}{\partial x}\approx -\frac{3}{2h}v_{j}+\frac{2}{h}v_{j-1}-\frac{1}{2h}v_{j-2} I'm using the usual approximation for the time derivative, I get the following for a stencil...
2. ### Write the given hyperbolic function as simply as possible

My take; ##2\cosh x = e^x +e^{-x}## I noted that i could multiply both sides by ##e^x## i.e ##e^x⋅2\cosh x = e^x(e^x +e^{-x})## ##e^x⋅2\cosh x = e^{2x}+1## thus, ##\dfrac{e^x}{1+e^{2x}}=\dfrac{\cosh x + \sinh x}{e^x⋅2\cosh x}## ##= \dfrac{\cosh x +...
3. ### Prove the hyperbolic function corresponding to the given trigonometric function

##8 \sin^4u = 3-4\cos 2u+\cos 4u## ##8 \sinh^4u = 3-4(1+2\sinh^2 u)+ \cosh ( 2u+2u)## ##8 \sin^4u = 3-4-8\sinh^2 u+ \cosh 2u \cosh 2u + \sinh 2u \sinh 2u## ##8 \sinh^4u = 3-4+1-8\sinh^2 u+ 4\sinh^2u +4\sinh^4 u + 4\sinh^2 u + 4\sinh^4 u## ##8 \sinh^4u = -8\sinh^2 u+ 8\sinh^2u +8\sinh^4 u##...
4. ### Find the roots of the given hyperbolic equation

This is a textbook question and i have no solution. My attempt: We know that ##\cosh x = \dfrac{e^x + e^{-x}}{2}## and ##\cosh u = \dfrac{{x^2 + 1}}{2x}## it therefore follows that; ##e^{2u} = x^2## ##⇒u = \dfrac {2\ln...
5. ### Comparing Hyperbolic and Cartesian Trig Properties

I came across this question; i noted that the hyperbolic trigonometry properties are somewhat similar to what i may call cartesian trigonometry properties... My approach on this; ##\tanh x = \sinh y## ...just follows from ##y=\sin^{-1}(\tan x)## ##\tan x = \sin y## Therefore...
6. ### MHB How to prove that the Euler-Bernoulli equation is hyperbolic

Hello, I would like to prove that the following partial differential equation is hyperbolic. u_{tt} (x,t)+ u_{xxxx} (x,t)= 0 with x \in \left[0 , 1\right] and x \in \left[0 , T \right ] . Can anyone help me? Thank you.
7. ### I Is a horseshoe orbit a hyperbolic orbit?

Epimetheus and Janus switch places periodically, because they follow a horseshoe orbit around Saturn, which is considered a "pseudo-orbit" around each other. I'm thinking that if you look at the conic sections - taking an elliptical orbit of two moons to greater and greater extremes until they...
8. ### I Deriving Lorentz Transformations: Hyperbolic Functions

While deriving Lorentz transformation equations, my professor assumes the following: As ##\beta \rightarrow 1,## $$-c^2t^2 + x^2 = k$$ approaches 0. That is, ##-c^2t^2 + x^2 = 0.## But the equation of the hyperbola is preserved in all inertial frames of reference. Why would ##-c^2t^2 + x^2##...
9. ### I Looking for what this type of PDE is generally called

We have this type of very famous nicely symmetric pde in our area. However, no one knows how to handle it properly since it is a nonlinear pde. Suggestions on how it is called in general would help us further googling. I already tried keywords like "bilinear", "dual", "double", but by far could...
10. ### Calculus Textbook for Integration using Hyperbolic substitution

Can someone please tell me the book that contain integration using hyperbolic substitution for beginner? I know that hyperbolic functions is taught in Calculus book but most of them is only some identities and inverses of hyperbolic functions.
11. ### I Integration Using Hyperbolic Substitution

Can someone please show me an example of integration using hyperbolic substitution? Thank you
12. ### I Does the Topology of AdS4 Affect Global Hyperbolicity?

Is ##\mathrm{AdS_4}## globally hyperbolic?$$g = -\left(1+ \dfrac{r^2}{l^2} \right)dt^2 + \dfrac{dr^2}{1+ \dfrac{r^2}{l^2}}+ r^2d\Omega^2$$Letting ##r = l \tan \chi## then defining ##\tilde{g} = g \cos^2 \chi##\begin{align*} g &= \sec^2 \chi (-dt^2 + l^2 d\chi^2) + l^2 \tan^2 \chi d\Omega^2 \\...
13. ### MHB Definite integral involving sine and hyperbolic sine

Calculate $\displaystyle \int_0^{\infty} \frac{\sin x}{\cos x + \cosh x}\, \mathrm dx.$
14. ### MHB Fourier Series involving Hyperbolic Functions

Hello everyone first time here. don't know if it's the correct group... Am having some issues wiz my maths homework that going to count as a final assessment. Really Really need help. The function (f), with a period of 2π is : f(x) = cosh(x-2π) if x [π;3π].. I had to do a graph as the first...
15. ### I Hyperbolic Time Chamber: Is Time Dilation Theoretically Possible?

Has anyone ever watched dragon ball Z before? In this TV show, there is something called the hyperbolic time chamber. 1 day on Earth is equivalent to 1 Year in the time chamber. In other words, if you stayed in the chamber for a full 24hours than exactly 1 year would have passed when you left...
16. ### Integration and hyperbolic function problem

This question arose while studying Cosmology (section 38.2 in Lecture Notes in GR) but it is purely mathematical, that is why I ask it here. I do not see why the equation $$H^2 = H_0^2 \left[\left( \frac{a_0}{a}\right)^3 (\Omega_M)_0 + (\Omega_{\Lambda})_0 \right] \tag{1}$$ Has the following...
17. ### Elliptic, Parabolic, and Hyperbolic PDEs

(1) ok. (2) We start with ##\sigma(ξ) = a_{11} ξ_1^2 +2a_{12}ξ_1ξ_2 +a_{22}ξ_2^2ξ## and we replace every ##ξ_iξ_j## with ##\partial_i\partial_ju##, giving ##a_{11}\partial_x^2+2a_{12}\partial_x\partial_yu+1_{22}\partial_2^2## (3) The given equation is the following. ##\sigma(ξ) = ξ^t A ξ ##...
18. ### I Exploring the Nature of Capacitor Voltage: Is it a Hyperbolic Curve?

Ok Hi everyone! I was working on what would happen if you apply a linear increasing voltage to a series capacitor resistor. The question is : If the capacitor voltage is plotted, is the cap voltage curve hyperbolic? I've done some plots on the cap voltage and it sure looks hyperbolic but I...

42. ### I Hyperbolic space simulation in VR

I have come across an interesting simulation of hyperbolic space. I can't totally wrap my hand around it, though. But I am interested in technical part of it. As far as I see no object in the space would get curved when you are moving. The visual deformation is the result of bent light reached...
43. ### Wave Interference: Minimum Intensity/Hyperbolic

Homework Statement Two identical speakers, 10.0 m apart from each other, are stimulated by the same oscillator, with a frequency f, of 21.5 Hz, at a place where the speed of sound is 344 m/s. a) Show that a receiver at A will receive the minimum intensity of sound (Amin) due to the...
44. ### Hyperbolic Geometry (Rectangles)

Homework Statement Recall that in hyperbolic geometry the interior angle sum for any triangle is less than 180◦. Using this fact prove that it is impossible to have a rectangle in hyperbolic geometry. Homework EquationsThe Attempt at a Solution - I wanted to use the idea that rectangles are...
45. ### B Problem solving with hyperbolic functions

Mod note: Because his caps-lock key is stuck, it's OK for this post to be in all caps. FIRSTLY, MY LAPTOP'S CAPS LOCK IS BEHAVING REALLY WEIRD AND I HAVE NO CONTROL ON IT WHATSOEVER. SO SORRY FOR POSTING IN ALL CAPS/ALL SMALL LETTERS I HAVE RECENTLY LEARNED HYPERBOLIC FUNCTIONS. HOWEVER, I AM...
46. ### A Hyperbolic Coordinate Transformation in n-Sphere

##x= r Cosh\theta## ##y= r Sinh\theta## In 2D, the radius of hyperbolic circle is given by: ##\sqrt{x^2-y^2}##, which is r. What about in 3D, 4D and higher dimensions. In 3D, is the radius ##\sqrt{x^2-y^2-z^2}##? Does one call them hyperbolic n-Sphere? How is the radius defined in these...
47. ### Approximation of a hyperbolic function

Homework Statement Hy guys I am having an issue with approximating this first question, which I have shown below. Now my problem is not so much solving it but I have been thinking that if given the same question without knowing that it approximates to so for example the question I am...
48. ### Integration with hyperbolic secant

Homework Statement Solve ##\displaystyle{d\sigma = \frac{d\rho}{\cosh\rho}.}## Homework Equations The Attempt at a Solution The answer is ##\displaystyle{\sigma = 2 \tan^{-1}\text{sinh}(\rho/2)}##. See equation (10.2) in page 102 of the lecture notes in...
49. ### Overdamped oscillator solution as hyperbolic function?

Homework Statement Here is the equation for the general solution of an overdamped harmonic oscillator: x(t) = e-βt(C1eωt+C2e-ωt) Homework Equations β decay constant C1, C2 constants ω frequency t time The Attempt at a Solution I know (eωt+e-ωt)/2 = coshωt and (eωt-e-ωt)/2 = sinhωt but how do...
50. ### Fourier/heat problem involving hyperbolic sine

Homework Statement A rectangular box measuring a x b x c has all its walls at temperature T1 except for the one at z=c which is held at temperature T2. When the box comes to equilibrium, the temperature function T(x,y,z) satisfies ∂T/∂t =D∇2T with the time derivative on the left equal to zero...