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.
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...
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 +...
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...
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...
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.
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...
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##...
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...
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.
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...
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...
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...
(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 ##...
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...
The hyperbolic function ##\cosh t \text{ and } \sinh t## respectively represent the x and y coordinate of the parametric equation of the parabola ##x^2-y^2=1##. The exponential expressions of these hyperbolic functions are
$$
\begin{cases}
\sinh x = {e^x-e^{-x}} /2; \: x \in \mathbb R, \: f(x)...
Let $$Y(t)=tanh(ln(1+Z(t)^2))$$ where Z is the Hardy Z function; I'm trying to calculate the pedal coordinates of the curve defined by $$L = \{ (t (u), s (u)) : {Re} (Y (t (u) + i s (u)))_{} = 0 \}$$ and $$H = \{ (t (u), s (u)) : {Im} (Y (t (u) + i s (u)))_{} = 0 \}$$ , and for that I need to...
In "The Geometry of Minkowski Space in Terms of Hyperbolic Angles" by Chung, L'yi, & Chung in the Journal of the Korean Physical Society, Vol. 55, No. 6, December 2009, pp. 2323-2327 , the authors define an angle ϑ between the respective inertial planes of two observers in Minkowski space with...
How to prove that every quadric surface can be translated and/or rotated so that its equation matches one of the six types of quadric surfaces namely 1) Ellipsoid
2)Hyperboloid of one sheet
3) Hyperboloid of two sheet 4)Elliptic Paraboloid
5) Elliptic Cone 6) Hyperbolic Paraboloid
The...
Can anyone derive the distance formula of a hyperbola for me, please? I have not found the derivation on the internet. I can't get any clue from the picture of hyperbola.
The integral of cothx is ln|sinhx|+C.
Does this mean the integral of coth2x is ln|sinh2x|+C?
If not, does anyone have a link to a page on how it is achieved - I'm trying to compile a list of all common hyperbolic function derivatives and integrals. However, I can't find anything to confirm if...
Hello all,
I am trying to solve a limit:
\[\lim_{x\rightarrow 0}\frac{sinh (x)}{x}\]
I found many suggestions online, from complex numbers to Taylor approximations.
Finally I found a reasonable solution, but one move there doesn't make sense to me.
I am attaching a picture:
I have marked...
Homework Statement
Homework Equations
So the question is asking to solve an integral and to use the answer of that integral to find an additional integral. With part a, I don't have much problem, but then I don't know how to apply the answer from it to part b. I know I should subsitute all...
$$V_{HE}=\sqrt{\frac{\mu}{a}}$$
What is the rationale for this formula when we can determine the change in velocity from Earth's orbit to transfer orbit using the vis-viva equation? Likewise, what is the use of defining the radius for the sphere of influence for interplanetary transfer...
Hi, I'm reading a book about numerical models for PDE and it says that a method is said to be stable if this condition holds:
$$|| \mathbf u^{n+1} || \le c_t || \mathbf u^{n} ||$$
where ##c_t## is a constant greater than zero, and ##u## is the numerical solution to the problem. (In particular...
I am trying to understand why maxwell equations are correct in any reference frames? While i started to understand of his laws of physics a bit i could not imagine why he uses hyperbolic functions such as coshw instead of spherical ones in position and time relation between moving frames...
Given ##ds^2 = y^{-2}(dx^2 + dy^2)##, I am trying to prove that a demicircle centred on the x-axis, written parametrically as ##x=r\cos\theta + x_0 ## and ##y= r \sin \theta ## are geodesics. Where ##r## is constant and ##\theta \in (0,\pi)##
I have already found the general form of the...
Hello,
Recently, a solar power tower plant was founded next to where I work.
Since it's the tallest object in the area, it's quite hard to miss it. But apart from that, every morning the reflected light is arranged in a hyperbolic- like way, as you can see in the picture.
Does anyone have a...
Greg Bernhardt submitted a new PF Insights post
Rindler Motion in Special Relativity: Hyperbolic Trajectories
Continue reading the Original PF Insights Post.
I thought this was nice. In what follows, ##c=1##.
For a particle undergoing constant proper acceleration ##\alpha## in the positive ##x##-direction, an inertial observer can use:
##x (\tau) = x_0 + \dfrac{\gamma(\tau) - \gamma_0}{\alpha}##,
where the Lorentz factor is:
##\gamma (\tau) =...
Homework Statement
Find a complete set of conditions on the constants a, b, c, n such that, for Cartesian coordinates (x, y, z), V = axn + byn + czn is a solution of Laplace’s equation ##∇^2V = 0##. A mass filter for charged particles consists of 4 electrodes extended along the z direction...
Today, I was at an award ceremony, where in one out of the two scientific lectures, the professor was teaching the basics of Hyperbolic Geometery. However, due to time constraints, he had to teach very fast, and there was no laser pointer, nor a chalkboard, so he couldn't explain very well...
This is a pannable simulation that will run in your browser. It shows where Pioneer 10 & 11, Voyager 1 & 2, New Horizons and ʻOumuamua are, how far they are from the Sun, and their speeds.
simulation: http://orbitsimulator.com/gravitySimulatorCloud/simulations/1511746688216_hyperbolic.html...
I am filling my circular hottub, and charting the water level height. Its sides have a small, constant slope from vertical - i.e. it is a truncated, inverted cone.
Imagining an ideal hottub of unlimited height*, the water level will always be increasing - but at a decreasing rate - it will...
Homework Statement
I'm rather confused as to where the orbited body is placed, and where the orbiting trajectory lies in this figure.
Is it right for me to say that if I placed say, the Earth on the left hand focus, a comet with a hyperbolic trajectory will travel a path defined by the...
Hello.
I have this function ## v(x) = -\sum_{i=1} x^i \sqrt{2}^{i-2} \int_{-\infty}^{\infty} m^{i-1} \cosh(m)^{-4} dm## which I can not seem to figure out how to simplify.I tried looking at some partial integration but repeated integration of ## \cosh ## gives polylogarithms which seemed to...
Homework Statement
Homework EquationsThe Attempt at a Solution
The attempt is in the picture. Is this the right method? Is there any faster method without cumbersome calculations?
Homework Statement
For the expression
$$r = \frac{i\kappa\sinh(\alpha L)}{\alpha\cosh(\alpha L)-i\delta\sinh(\alpha L)} \tag{1}$$
Where ##\alpha=\sqrt{\kappa^{2}-\delta^{2}}##, I want to show that:
$$\left|r\right|^{2} = \left|\frac{i\kappa\sinh(\alpha L)}{\alpha\cosh(\alpha...
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...
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...
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...
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...
##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...
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...
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...
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...
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...