What is Hyperbolic: Definition and 346 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.
Homework Statement
On July 1, 2004, the Cassini spacecraft approached
Saturn with hyperbolic excess velocity 5.5 km/s to
swing by the planet at the closest approach distance
rp = 80,680 km. Compute the impulsive ΔV
required for a maneuver performed at the closest
approach to Saturn to...
Hello!
A book on calculus was introducing hyperbolic functions and pointed out that the identities such as cosh x and sinh x, etc. for hyperbolic functions were analogous to cos x and sin x for circular functions. I tried finding some internet sources explaining why this is so, but they tend to...
Kind of an odd question, but here goes: can anyone offer any examples of physical systems that include the use of the hyperbolic tangent function in their mathematical solution?
Hey guys,
I was doing some work on hyperbolic functions and teaching myself to solve some equations. One of the questions in the book really has me stumped:
Express using exponential definitions of cosh(x) and sinh(x) find the exact solution of:
tanh(x) + sinh(x) = 3
I had a go at...
Homework Statement
Hello,
Thanks for taking some time to help me out...and I have to apologize for posting a graphic of my logic and attempted answer instead of using LateX (It would take me a very long time just to get this problem viewable)
Please help me check my work and logic...
I am a bit confused over the shape of matterless spacetime:confused:: on one side Minkowski space is described as hyperbolic, and deSitter space is (hyper)spherical. Both are used, yet how can you have spacetime being both hyperbolic and spherical?
Homework Statement
Limit of x2cosh(1/x) as x approaches 0
Homework Equations
The Attempt at a Solution
I did some algebra,made into the form infinity over infinity and used hopital's rule. I got limit x2e^(1/x).
Then I again made into the form infinity over infinity(by making it 1/x2).
Then...
Given a triangle on a hyperbolic surface with all angles and edge length known the area is given by R^2 x(PI - a - b - c), where a, b and c are the angles and R is the radius of curvature of the surface. What if you don't know R?
Same question for a triangle on a spherical surface where R is...
A train is traveling at the speed of a bullet.
A man, stood on top of the train, fire's a gun in the direction from which the train has come from. (facing rearward)
He pulls the trigger the instant the train passes station 'A'.
The velocity of the bullet relative to the ground is 0
To an...
Hyperbolic and Inverse Trigonometric Functions
How extensive is the use of hyperbolic and inverse trigonometric functions in upper-level calculus and mathematics? I've taken 3 semesters of calculus, and not one of my teachers has gone over hyperbolic functions, and barely touched on inverse...
If this is in the wrong section, I apologize.
I've been doing some independent research on the applications of hyperbolic geometry, and I
ve hit a rather difficult snag. I've seen in a few places on line that Mercury's orbit can be more accurately calculated using hyperbolic geometry as...
Given the quantity
Cosh(x)*Cosh(y)
where x and y are two indipendent real variables is it possible to write it only in function of
k=Cosech(x)*Cosech(y)
?
It could seem a quite easy problem but I spent a few days between the proprieties of hyperbolic functions and I really...
Hello,
I would like to do some stuff with modeling geometry in hyperbolic space in software. When I look up information on hyperbolic space, however, I tend to find only information on working with models of hyperbolic space. For example I find lots of information on the poincare disc model...
Orthogonality Property of Hyperbolic functions ?
Hi all,
I have seen Orthogonal property for trigonomeric functions but I am unsure if there is something similar for sinh() , cosh() ? . I know that the integral of inner product of the two functions should be zero for them to be...
1. I am asked to calculate \int \frac{dx}{cosh(x)}
2. Homework Equations
3. The Attempt at a Solution
I know that \frac{1}{cosh(x)} is equivalent to "sech(x)" which by definition is \frac{2}{e^x+e^{-x}}.
I'm confused & I don't know which one I need to use for this question...
Homework Statement
Prove that:
(1+tanhx)/(1-tanhx)=e^(2x)
Homework Equations
The Attempt at a Solution
I tried substituting tanhx for (e^x-e^(-x))/(e^x+e^(-x)) and for (e^(2x)-1)/(e^(2x)+1))
But I really have no clue how to continue...
Homework Statement
Evaluate:
\int\\{1}/{\sqrt{x^2-1}} dx between -3, -2
I know I'm supposed to use hyperbolic substitution in the question.
Homework Equations
edit: cosh^2(t) - sinh^2(t) = 1
The Attempt at a Solution
let x = -cosht, inside the integral let dx = sinh(t) dt
int (...
Homework Statement
Question(1) : Find the Cartesian equation of Re[ z - i / z + 1 ] = 0. If the locus is a circle, give its radius and the coordinates of its center.
The Attempt at a Solution
Workings : So I attempted to solve the problem and my workings are as below
... Since Re = Real...
Homework Statement
power series expansion of:
((cosh x)/(sinh x)) - (1/x)
Homework Equations
cosh x = (1/2)(ex + e-x)
sinh x = (1/2)(ex - e-x)
The Attempt at a Solution
what i have so far:
I simplified the first part of the eq to read :
e2x-1
e2x-1
now I am stuck...
Homework Statement
I don't understand how to take the derivative of inverse hyperbolic functions such as sinh^{-1}(x). I know that the derivative of sinh(x) is cosh(x) but don't know what to do with the inverse.
Homework Equations
The Attempt at a Solution
I'm completely at a...
I like to learn some basic hyperbolic geometry!
Starting with the hyperbolic plane, the upper half plane with the hyperbolic lines being all half-lines perpendicular to the x-axis, together with all semi-circles with center on the x-axis.
Why and how are there always infinitely many...
Homework Statement
What energy (in eV) should a beam of electrons have so that 0.1% of them are able to tunnel through a barrier of height 7.0eV and 1.0 nm wide? Start with the equation for T(E) and set it up with 1/T(E) on one side and let E/U=x for the unknown. Solve the equation for x and...
the questions below for scientific debate
1.how to understand SL(2,R)/SO(2,r)=H^2(2-dim hyperbolic space)
2.Let M be a compact manifold ,T compact lie group,what can we say about
M^T(the set of fixed points of M under the action of T),is M^T manifold?
how to argue?further,if T is torus,and M is...
\equivHomework Statement
Hi, I've been given a hyperbolic identity to prove:
2sinhAsinhB \equiv Cosh(A+B) - Cosh(A-B)
Homework Equations
Cos(A\pm B) \equiv CosACosB \mp SinASinB
The Attempt at a Solution
I have Cosh(A+B) and - Cosh(A-B) so i can kind of see that there will be...
I recently came across a problem where I was able to show that \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \tanh \left( \frac{n \pi}{2} \right)=\frac{\ln 2- \pi}{4} through numerical approximation...However, I don't have much practice evaluating such summations analytically, and I was wondering if...
Hi all,
I have to rewrite a serie into a fraction of hyperbolic sine but I am lost... My problem looks like this
\Sigma_{n=0}^{\infty} exp(K*F)*exp(-F*B(n+0.5))
which can be rearranged into
exp(K*F)/(sinh(FB/2)
my problem is I cannot relate the serie...
Homework Statement
Prove that the magnitude of the impact parameter B equals the length (-b) of the hyperbolic semiminor axis.
Homework Equations
|B|=|b|=|a|sqrt(e^2-1)
The Attempt at a Solution
I really don't know where to start. I was thinking of finding a relation between a...
Hi,
Did anyone know how to do the Fourier transform of the hyperbolic
secant? I know the answer; it's given in the text (I'm reading
Ablowitz, Fokas, Complex Variables), it's another hyperbolic secant,
but I want to know how to do it. My dilemma is:
a) what contour to use? I'm having...
I hope this is the right place to ask this question.
Imagine a right triangle with vertices A,B and C and corresponding opposite sides a, b and c such that there is a right angle at B and side b is the hypontenuse. Let the length of side b = 1. If I label side a as sin(A) and side c as...
This is not homework. I've been analysing some space-time diagrams and found I need to know the biggest angle in the illustrated triangle in terms of the lengths and the angle A which are known.
I found the equivalent of the cosine rule of Euclidean space for the hyperbolic plane but I'm...
integrate (x^2) / (4+x^2)^(3/2)
Im not allowed to apply hyperbolic functions to this and have been trying to solve applying to a 90 deg. angle.
x = 2tan(theta)
x^2 = 4tan^2(theta)
dx = 2 sec^2(theta)
Hopefully you can se where I am going with this (trigonomic substitution)
Im...
The question is as follows (by the way I'm asking here, cause the calculus and beyond forum seems to be primarily concerned with Calculus,DE, LA and AA):
let f(z)=(2z+1)/(z+1) be an isometry of the hyperbolic plane H={z| Im(z)>0}.
let l be a hyperbolic line in H which is invariatn under f...
I have this question which is rather simple, basically reiterating a general theorem.
Show that S={z in H||z-i|=3/5} is a hyperbolic circle S={w in H| p(w,w0)=r}
for r>0 and find sinh(r/2) and w0.
Now to show that it's hyperbolic is the easy task, I just want to see if I got my...
Let b>a>0 real numbers, and z a point in H={z in CU{infinity}| Im(z)>0}
let I be in H, a euclidean interval which connects az with bz. find the point w in I such that it divides I into two intervals with the same hyperbolic length.
Now what I did is as follows:
let f be a mobius...
I have a picture of a problem and the answer. But a place in the answer I can't understand how they get from one function to the next, could you guys please explain it? I have marked it in red?
http://img381.imageshack.us/img381/1839/hyperik2.png
Homework Statement
Find the derivative of y=ln(cosh(2x^3))
The attempt at a solution
is this the same as saying (1/(cosh(2x^3)) ?
The correct answer is 6x^2 - ((12x^2)/(e^4x^3 + 1))... how do you derive this?
I am really stuck on this question I didn't learn about hyperbolic functions in...
Hello,
Given the three maps x_{n+1}=Ax_n with
A_1=\begin{pmatrix} 1&-1\\1&1 \end{pmatrix}, A_2=\begin{pmatrix} 1/2&1/2\\-1&1 \end{pmatrix}, A_3=\begin{pmatrix} 3&2\\5/2&2 \end{pmatrix},
describe the dynamics, and say whether or not the dynamics is hyperbolic.
Finding eigenvalues...
Homework Statement
Given the trigonometric identity cos(x+y)... use Osborn's rule to write down the corresponding identity for cosh(x+y)... Use the definitionis of the hyperbolic functions to prove this identity
Homework Equations
The Attempt at a Solution
I can use Osborns rule...
[SOLVED] Mathematica does not like hyperbolic functions
So, consider the equation cosh(x)=n*x
For a given n, the equation has 0, 1, or 2 possible values of x. If n is below the critical value, the equation has no solutions. If n is above the critical value, the equation has two solutions...
[SOLVED] Hyperbolic functions
As part of a long winded "show that" question I've ended up at the point where I have xcothx and I want to show that this is equal to cothx - \frac{1}{x} only I have no ideas how to get there. I can't see any reason why this should be so, but I'm pretty confident...
Homework Statement
a) Find y' if y = x^2arcsinh(2x)
b) Find y' if y = xarctanh(x) + ln[(1-x^2)^1/2]
c) Find y' if y = arccoth[sqrt(x^2+4)]
Homework Equations
d/dx arcsinh(x) = 1/sqrt(1+x^2)
d/dx arctan(x) = 1/(1-x^2)
d/dx arccoth(x) = 1/(1-x^2)
The Attempt at a Solution
I am still...
Homework Statement
(I haven't got the thing to write equations so this is a little difficult to understand sorry!)
Show that the integral of (4x^2 - 1)^-1/2 from 0.625 to 1.3 is equal to 1/2(ln 5 - ln 2)
Homework Equations
my formula book states:
the integral of (x^2 - a^2)^-1/2 is...
Homework Statement
Show that: (I don't have any way of showing it properly so have written it in words)
'The integral from 5 to 3 of (sq root (x^2 - 9) ) with respect to x' = 10 - (9/2) * ln 3
I hope that makes sense!
Homework Equations
Well I'm not sure.. I *think* that the fact the...