Implicit Hyperbolic Function

ekkilop

Hi all,

In studying the eigenvalues of certain tri-diagonal matrices I have encountered a problem of the following form:

{(1+a/x)*2x*sinh[n*arcsinh(x/2)] - 2a*cosh[(n-1)*arcsinh(x/2)]} = 0

where a and n are constants. I'm looking to find n complex roots to this problem, but isolating x is troublesome. I attempted to use the implicit derivatives to obtain an expression for x in terms of a and n but it didn't lead me anywhere.

Is there a general approach to finding the roots of equations of this type? If not, can one find any general properties of the roots, e.g. if they belong to a certain half-plane etc.

The problem may be simplified somewhat if we choose a=-2 and try to find x as a function of n but even here the roots are hard to find.

Any advice would be much appreciated.
Thank you.

tiny-tim

hi ekkilop!

have you tried simplifying it by putting x = 2sinhy (and maybe a = 2b) ?

ekkilop

Hi tiny-tim,
Thanks for your reply.
I did try this and it cleans things up a bit. In particular it becomes clear that a=-2 is a convenient choice since we get

0 = (4+2a)sinh(n*y)sinh(y) + 2a[sinh(n*y) - cosh(n*y)cosh(y)]

after expanding cosh((n-1)y). However, it is not clear to me how to proceed from here. Is there perhaps some other substitution that would make life easier? Or maybe there's a standard form to rewrite sinh(n*y) and cosh(n*y). My attempts from here just seems to make things more complicated...

tiny-tim

hi ekkilop!
the only thing i can suggest is to divide throughout by cosh(n*y)cosh(y),

and get tanh(ny) = a function of tanh(y) and sech(y)

ekkilop

This is not a bad idea. All in all I can boil things down to

tanh(ny) = cosh(y)

which has the expected roots (found numerically). Solving for n is straight forward but inverting seems impossible, at least in terms of standard functions.
If n is a positive integer, what can be said about y?
Thank you for all your help!

tiny-tim

i don't think we can go any further than that

Mute

In this particular case that you have simplified to, perhaps the best you could do now is rewrite this equation as a polynomial. If you set ##x = e^y##, you can rearrange the equation into a degree 2n+2 polynomial, which you could then perhaps study to see if you can say anything useful about the roots for different n. (Remember that since x = exp(y), only positive valued x's are valid solutions to the polynomial, the rest are erroneous).

You should be able to write down a polynomial for the case ##a \neq -2## as well, but I'm not sure how helpful that will ultimately be, as you will have to vary both a and n.

ekkilop

Hi Mute!
Thanks for your reply. The problem is actually a result of a polynomial of degree n, which has been rewritten in it's present form. The coefficients of all n+1 terms are non-zero integers dependent on a, except for the leading term. I could probably do a more thorough analysis on the bounds of the solutions though, so thank you for the inspiration!