Find the roots of the given hyperbolic equation

chwala
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Homework Statement
Find the roots of the following equation in terms of ##u##.

##x^2-2x \cosh u +1 = 0##
Relevant Equations
hyperbolic trigonometric equations
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 x}{2}##

##u=\ln x##

##x=e^u ##

Your insight or any other approach welcome guys!
 
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chwala said:
Homework Statement:: Find the roots of the following equation in terms of ##u##.

##x^2-2x \cosh u +1 = 0##
Relevant Equations:: hyperbolic trigonometric equations

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 x}{2}##

##u=\ln x##

##x=e^u ##

Your insight or any other approach welcome guys!
What about ##x = e^{-u}##?
 
It's a quadratic; there should be two roots.

<br /> x^2 - 2x\cosh u + 1 = (x - \cosh u)^2 + 1 - \cosh^2 u = (x - \cosh u)^2 - \sinh^2 u. Hence <br /> x = \cosh u \pm \sinh u = e^{\pm u}.
 
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Simplest is perhaps <br /> x^2 - 2\cosh u + 1 = x^2 - (e^{u} + e^{-u})x + 1 = (x - e^u)(x - e^{-u}) since e^ue^{-u} = 1.
 
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pasmith said:
Simplest is perhaps <br /> x^2 - 2\cosh u + 1 = x^2 - (e^{u} + e^{-u})x + 1 = (x - e^u)(x - e^{-u}) since e^ue^{-u} = 1.
There's an x missing in the 2nd-term (compare with the previous post).

By the way,...
The equation looked familiar to me. It's related to special relativity.
It's the characteristic equation
0=\det\left( \begin{array}{cc} \cosh\theta -\lambda &amp; \sinh\theta \\ \sinh\theta &amp; \cosh\theta -\lambda \end{array}\right)
to find the eigenvalues of the Lorentz boost transformation
(where \theta is the rapidity, \tanh\theta =v/c is the dimensionless-velocity, and \cosh\theta =\gamma=\frac{1}{\sqrt{1-(v/c)^2}} is the time-dilation factor ).
The eigenvalues \lambda_{\pm}= e^{\pm \theta} are the Doppler factor (Bondi k-factor) and its reciprocal.
The eigenvectors are along the light-cone \left(\begin{array}{c}1\\\pm 1\end{array}\right).
 
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There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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