Hyperbolic Definition and 347 Threads

  1. O

    Computing arc length in Poincare disk model of hyperbolic space

    I am reading Thurston's book on the Geometry and Topology of 3-manifolds, and he describes the metric in the Poincare disk model of hyperbolic space as follows: ... the following formula for the hyperbolic metric ds^2 as a function of the Euclidean metric x^2: ds^2 = \frac{4}{(1-r^2)^2} dx^2...
  2. T

    Proving an identity involving hyperbolic functions

    Homework Statement Prove sin(x-iy) = sin(x) cosh(y) - i cos(x) sinh(y) Homework Equations The Attempt at a Solution I tried to prove it by developing sinh into it's exponential form, but I get stuck. sinh(x-iy) = [ ei(x-iy) - e-i(x-iy) ] /2i = [ eixey - e-ix e-y ] /2i...
  3. T

    Critical points of a Hyperbolic function

    Homework Statement I am trying to find the critical points of the following hyperbolic function: f(x) = a / (b + x) Homework Equations Critical points--> where f '(x) = 0 One of the points on the graph is a/2b The Attempt at a Solution I am not sure how to proceed with this...
  4. J

    Proving Inverse Hyperbolic Function

    Can someone please do a step by step proof for this or send me a link to a step by step webpage. Thank you very much Cosh^(-1 ) A/x=( x √(X^( 2 )+A ))/2a my attempt=> Cosh^(-1 ) = y A/x = Cosh^(-1 ) A/x = Sechy A/x= (2/(e^2 + e^-2))
  5. E

    Arc Length and Surface question about hyperbolic function

    If the circumference of the region bounded by the curve y=cosh(x) and the lines y=0 x=a and x=-a is 2a+4, where a>0 find the area of the surface obtained by rotating the part of the curve y=cosh(x) between x=a x=-a and around the x axis. This is my homework question.I tried to solve it.I...
  6. N

    Solving Complex Hyperbolic functions

    Homework Statement I am a little confused on the steps to take to solve these kinds of functions. Solve: cosh z = 2i The Attempt at a Solution We were given identities for sinh z = 0 and cosh z = 0 and also other identities like cosh(z) = cos (iz) So cos (iz) = 2i cos...
  7. DryRun

    Integral of hyperbolic function

    Homework Statement \int^4_3 \frac{1}{\sqrt{3x^2-6x+1}}\,.dx The attempt at a solution I complete the square for the quadratic: \sqrt{3x^2-6x+1} \\=\sqrt{3(x^2-2x+\frac{1}{3})} \\=\sqrt 3 \times \sqrt{(x-1)^2-\frac{2}{3}} \int^4_3 \frac{1}{\sqrt{3x^2-6x+1}}\,.dx \\=\frac{1}{\sqrt 3}\int^4_3...
  8. DryRun

    Integral involving hyperbolic functions

    Homework Statement Find \int \frac{x}{\sqrt{2x^2-2x+1}}\,dx The attempt at a solution First, i complete the square for the quadratic expression: 2x^2-2x+1=2((x-\frac{1}{2})^2+\frac{1}{4}) \int \frac{x}{\sqrt{2x^2-2x+1}}\,dx=\int \frac{x}{\sqrt 2 \sqrt{(x-\frac{1}{2})^2+\frac{1}{4}}}\,dx...
  9. DryRun

    Homework SolutionProve Hyperbolic Function: Solving for x in Terms of y

    Homework Statement If \sinh^{-1}x=2\cosh^{-1}y, prove that x^2=4y^2(y^2-1) The attempt at a solution I re-wrote \sinh^{-1}x and 2\cosh^{-1}y in terms of x and y. \sinh^{-1}x=\ln(x+\sqrt{x^2+1}) \\2\cosh^{-1}y=2\ln(y+\sqrt{y^2-1})=\ln(y+\sqrt{y^2-1})^2...
  10. L

    Understanding AdS Global Hyperbolicity: Insights from Hawking and Ellis

    Is there an easy way to see this? can it be see via the penrose diagram? does it have anything to do with how when a stone is thrown in AdS it will alwayscome backk in finite time..
  11. C

    MHB Help with Hyperbolic Functions

    Bany's question from Yahoo Questions: CB
  12. K

    Riemann function for a second order hyperbolic PDE

    Homework Statement Find the Riemann function for uxy + xyux = 0, in x + y > 0 u = x, uy = 0, on x+y = 0 Homework Equations The Attempt at a Solution I think the Riemann function, R(x,y;s,n), must satisfy: 0 = Rxy - (xyR)x Rx = 0 on y =n Ry = xyR on x = s R = 1 at (x,y) = (s,n) But I...
  13. N

    Proving the Complex Hyperbolic Property Using Trigonometric Construction

    Homework Statement Show: sinh(z + i2(pi)) = sinh(z) using sinh(z) = (ez - e-z)/2 Homework Equations The Attempt at a Solution So far I have (ex + i(2∏+y) - e-(x+i(2∏+y))/2. Need help proceeding from here. My thoughts were to define a z' = x + i(2∏+y) but I don't think that I can then say...
  14. F

    MHB How can I integrate Sinh4(x) using the hyperbolic sine method?

    integrate Sinh4(x) I have been struggling with this problem for a week. I know the answer because of wolfram but I cannot see how it gets it. Honestly, I can't even decide what to make my substitution as. I haven't really had problems with any other questions from our homework but this one and...
  15. B

    Why Isn't the Answer to the Hyperbolic Function e^10x?

    Homework Statement why is the answer not e^10x ? If you ignore the e's it should be 5x - 5x + 5x - - 5x, or 5x - 5x + 5x + 5x, which is 10x
  16. S

    Hyperbolic cosine identity help

    Homework Statement Show that cosh^2(x) = (cosh(2x) - 1)/2 Homework Equations cosh(x) = (e^x + e^-x)/2 The Attempt at a Solution I have attempted this multiple times and get the same results every time. Squaring cosh(x) I get 1/4(e^2x + e^-2x +2), which is i believe 1/4(cosh(2x) +2)...
  17. R

    Hyperbolic Boundary Valued Problem

    Hi, I am trying to understand solving boundary valued partial differential equations and it's relation to hyperbolic functions. In one of my problems, there is a PDE and the solution contains the hyperbolic function "cosh". I was just curious if anyone has any information for me to read up on...
  18. DryRun

    Differentiate hyperbolic function

    Homework Statement If lny = sinh^(-1)(x), prove that (1+ x^2)y'' + xy' - y = 0 The attempt at a solution I have tried various (unsuccessful) ways of doing this, but the basic procedure that I've done is: D.w.r.t.x for lny = sinh^(-1)(x) This gives: (1/y)y' = 1/(1 + x^2) To...
  19. DryRun

    How can the R.H.S. of a hyperbolic function be manipulated to match the L.H.S.?

    Homework Statement http://s1.ipicture.ru/uploads/20111203/B1Ax1OcU.jpg Frankly, I've been sitting staring at that problem for long enough, and it just can't be solved through the direct use of the standard hyperbolic identities. I need a hint.
  20. T

    Numerical solution to hyperbolic PDE - grid leapfrog - what to do at boundary

    Hi! I'm implementing a scheme to solve the following equation \frac{\partial \psi}{\partial t}=-c_{s} \cdot \frac{\partial \phi}{\partial x} \frac{\partial \phi}{\partial t}=-c_{s} \cdot \frac{\partial \psi}{\partial x} c_{s} is just the isothermal velocity of sound. The equations are for a...
  21. D

    Hyperbolic sine in Taylor Series

    I am reading through a worked example of the Taylor series expansion of Sinh(z) about z=j*Pi The example states: sinh(j*Pi)=cos(Pi)*Sinh(0) +jcosh(x)sin(y) I am unsure of this relation. I understand why the x terms are zero but don't know the relation to expand sinh. Can anyone shed...
  22. C

    Partial Sums resembling sums of secant hyperbolic

    Homework Statement Compute the following partial sum \sum_{k=0}^n\frac{1}{2^{2^k}+2^{-2^k}} Homework Equations The Attempt at a Solution So far, I've tried transforming the terms into secant hyperbolic functions...
  23. U

    Integral of inverse trig or inverse hyperbolic

    Homework Statement ∫5/(4x√(9-16x2)dx Homework Equations I am pretty sure this is in the form of ∫du/(u√(a2-u2) The Attempt at a Solution setting u=4x a=3 and du=4dx so 1/4du=dx I get: -5/12 sech-1(4x/3) + C Is this right or am I using the wrong definition? Just trying to...
  24. S

    Relationship between hyperbolic arctan and logarithm

    Homework Statement The relationship between arctanh and log is: arctanh(x)=\frac{1}{2}log(\frac{1+x}{1-x}) but if i take x=1.5, I have: arctanh(1.5)=0.8047 + 1.5708i and \frac{1}{2}log(\frac{1+1.5}{1-1.5})=0.8047 + 1.5708i as expected, but using the laws of logarithm, why does this...
  25. N

    Efficient Differentiation of Hyperbolic Integrals

    Hello again :) I get the feeling I'm missing some kind of 'trick', as this is proving a very difficult question :( I'll write out my frustration below; Homework Statement Find f'(x) if f(x) = \int^{cosh(x^{2})}_{0} tanh(t^2)dt Homework Equations --- The Attempt at a Solution My idea was...
  26. T

    Propagating a Hyperbolic Trajectory

    Hi, first post here at PF :) I have a problem here regarding orbit propagation. Basically my situation is as follows: I have coded a system that can track the orbital parameters from an object in a simulated orbit (Basic rigidbody physics). The system takes the state vectors of the orbiting...
  27. C

    Verify the hyperbolic identites

    Homework Statement verify these identities: 1) tanh^2 x + sech^2 x =1 2) sinh(x+y) = sinh cosh y + cosh x sinh y Homework Equations cosh2x - sinh2x = 1 sech2x + tanh2x = 1 coth2x - csch2x = 1 sinh (x ± y) = sinh x cosh y ± cosh x sinh y cosh (x ± y) = cosh x cosh y ± sinh x...
  28. N

    What is the geometry of the pseudo sphere and what dimension does it exist in?

    Most books and websites define the hyperbolic distance element and the corresponding shortest paths in the upper half plane with no explanation. I found a derivation for them in a book called Visual Complex Analysis by Needham and it relied on mapping the "pseudosphere" onto the upper half...
  29. M

    Inverse hyperbolic sin derivation

    Homework Statement derive the formula inverse sinhx = ln(x+sqrt(x^2+1)) for all real x Homework Equations sinhx=(e^x-e^-x)/2 ? The Attempt at a Solution i have been staring at this for awhile and i don't know how to start what should be the first step towards deriving that...
  30. M

    Finding Inverse Hyperbolic secant in terms of logarithms ?

    The Problem is when I Compute the Inverse I have to solutions sech^{-1}(x) = ln(\frac{1\pm \sqrt{1-x^{2}}}{x}) : 0<x\leq 1 And this not function which of them I will choose Another Question is how can I prove without the graph that csch (x) is one - to -one thanks
  31. L

    Delta-v for Hohmann transfer from hyperbolic trajectory to circular orbit

    I get different result than stated in the book. What am I doing wrong? Homework Statement A spacecraft returning from a lunar mission approaches Earth on a hyperbolic trajectory. At its closest approach A it is at an altitude of 5000 km, traveling at 10 km/s. At A retrorockets are fired to...
  32. M

    Integral using hyperbolic substitution

    Homework Statement \int\left(1+x^{2}\right)^{\frac{3}{2}}dx Homework Equations The hyperbolic functions. The Attempt at a Solution We've been going over hyperbolic substitutions in class so I assume I'm meant to use one of those, but I'm just not sure how to choose which one. Any help...
  33. I

    Electrostatics - Finding potential V(r,z) given hyperbolic boundry conditions.

    Homework Statement I'm trying to derive Equation (1) from the paper: http://idv.sinica.edu.tw/jwang/EP101/Paul-Trap/Winter%2091%20ajp%20demo%20trapping%20dust.pdf We are working with a cylindrically symmetric geometry along the z-axis. r^2 = x^2 + y^2 We have electrodes described by...
  34. F

    Hyperbolic Substituion. I am wrong?

    Homework Statement \int \;\sinh(2x) \cosh(2x) dx The Attempt at a Solution I let u = sinh(2x), du = 2cosh(2x)dx Integrating I shuold get\frac{1}{4} sinh^2 (2x) + C But http://www.wolframalpha.com/input/?i=Integrate[cosh%282x%29%28sinh%282x%29%29%2Cx] says I need to let u = 2x first. Why? I...
  35. A

    Differentiation Problem about a hyperbolic function

    Homework Statement How could one apply differentiation formulas on this one: \partial^{2}\left(2sinh(nx)\div\sqrt{sinh(2L)-2L}\right)\div\partial x^{2} Homework Equations The Attempt at a Solution is this differentiation formula enough to differentiate:
  36. mccoy1

    Transformation of expoential to hyperbolic

    Homework Statement The book has it exp(-MgbH/KT) =(sinh(2S+1)x/2)/(sinh(x/2)) for M=2S+1, and x = gbH/(kt). Homework Equations The Attempt at a Solution I'd have it as cosh(Mx)-sinh(Mx). How did they get the above result? Help please. Thanks.
  37. D

    Trignometric and hyperbolic equalities: Why the golden ratio?

    1. \sin \theta = \cos \theta \theta=\frac{\pi}{4} 2. \sin \theta = \tan \theta \theta = 0 3. \cos \theta = \tan \theta \theta =\arcsin (\varphi -1) 4. \sin \theta = \csc \theta \theta = \frac{\pi}{2} 5. \sin \theta =\sec \theta \theta does not exist. 6. \sin \theta =\cot...
  38. D

    Law of Sines (Elliptic, Hyperbolic, Euclidean)

    Well, I created this thread (under Geometry/Topology) about the Law of Sines, specifically for the three kinds of geometries. http://en.wikipedia.org/wiki/Law_of_sines http://mathworld.wolfram.com/LawofSines.html The Law of Sines states that, for a triangle ABC with angles A, B, C, and...
  39. A

    Solve the first order hyperbolic equation

    Solve the first order hyperbolic equation 3 du/dx + 2x du/dt =2u With initial condition: u(x,0) = 2x+4 My attempt at a solution I usually adopt the method of characteristics: dx/a = dt/b = du/c So from the above: a=3, b=2x and c=2u am I on the right track here?
  40. T

    Hyperbolic Equation Instability

    Hello, I'm trying to calculate the following equation which is the derivative in 'x' of a distribution function: d(dxF)/dt = d(Efield.(dvxF))/dx The problem comes because the right hand of the equation is solved by using central difference, but there is a zone where there is a...
  41. M

    What is a line in hyperbolic geometry?

    I'm reading a book on an introduction to non-Euclidean geometry, and it starts off with the usual Euclidean geometry. I didn't really need a line to be defined in that case, since it's obvious, but now that the parallel postulate has been replaced and we are working with non-Euclidean geometry...
  42. D

    Hyperbolic equations: domain of dependence

    In many cfd textbooks the domain of dependence is stated as the entire region emclosed by the characteristics. Is this correct? Isnt it only the values on the characteristics? Thanks!
  43. H

    Hyperbolic Function with Asymmetric Asymptotes

    Hello, I wish to find a function similar to, y^2-x^2=1 but instead of the slope of the asymptotes being +/- 1, I need one of the asymptotes to be of slope 0. That is, I wish to find a hyperbolic function with one horizontal asymptote and the other of slope 1. Is this possible?
  44. P

    Manipulating hyperbolic functions

    Homework Statement Express the function cosh(6x) in terms of powers of cosh(x) Homework Equations The Attempt at a Solution Okay the problem booklet also asks me to do the opposite. Express cosh(x)^6 as mutiples of cosh(x). I can do that fine, I just simply write it out as [1/2(e^x + e^-x)]^6...
  45. T

    Pitch for a 3d hyperbolic spiral

    Hi, have worked out the formula for a hyperbolic spiral space curve to be r(t)=1/t X cos(alpha)+1/t X sine(alpha) + t and obtained the tangent vector T, normal vector N, and curvature (kappa) with the z axis being the central axis of the spiral. Am having trouble finding the formula for the...
  46. R

    Inverse Hyperbolic Funcs. or Trigonometric Substitution?

    So, in integrals that lead to inverse hyperbolics and can be solved with trigonometric substitution i just get lost. I know how to use both of them but i don't know which to use. For the sake of simplicity i'll just go with this one ∫ dx/sqrt(4+x^2) We know this equals arcsinh(x/2)...
  47. O

    Rearrangeing Inverse Hyperbolic functions

    Hi, My brain is not working today. So can someone please tell me what I am doing wrong. (^2 = squared) coshy^2 - sinhy^2 = 1, how do I rearrange this for coshy^2 I keep getting: coshy^2 = 1 + Sinhy^2 The book that I'm looking at has it this way: coshy^2 = Sinhy^2 + 1 Thanks Obs
  48. K

    Hard hyperbolic tan integral :/

    Homework Statement tanh{[(2 *g*t)/(c)]+[(2*g*exp(-rt))/(r*c)]-[(2*)/(r*c)]} in terms of t Homework Equations when I plug into maple I get -(1/4)*c*ln(tanh(2*g*(t+1/r(e^(-rt))-1/r)/c)-1)/g-(1/4)*c*ln(tanh(2*g*(t+1/r(e^(-rt))-1/r)/c)+1)/g The Attempt at a Solution This cannot be...
  49. P

    What Is a Hyperbolic Structure in Conic Sections?

    I'm trying to understand how structures based on conic sections work. For example, when people speak of a parabolic mirror or a parabolic orbit for a satalite, I know what they mean, but when they speak of a hyperbolic mirror or a hyperbolic orbit, what does that actually mean? Is a hyperbolic...
  50. P

    Second-order, linear, homogeneous, hyperbolic PDE. Solvable?

    First, my deepest apologies if I am asking a trivial question, or asking it in the wrong forum. I am trying to solve a PDE, which I have already reduced to canonical form and simplified to the full extent of my abilities. The PDE is: u_xy + a(x,y) u_x + b u_y = 0, with a(x,y)=2/(x+y) and...
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