Possible analytical solution of transcendental equation by an assumption.

[Azorspace]

Is it possible to solve the next transcendental equation analytically (obviously for k):

sinh(k)=(b/2)(1+(e^(-2kl)))

making the assumption that (bl >>1). I think that is not possible, but in an article that i found,
they solve it by making that assumption, and they reach to the solution:

E=Eo+(d/2)

where:

E=-2cosh(k), Eo=-{(4+b²)^(1/2)}, and d=(2b²/|Eo|)(e^(-bl))

physically this problem corresponds to that of finding the energy of bound states in an infinite one dimensional regular lattice, with two embedded impurities.

Thanks everybody in advance.

 

[coelho]

Well... the solution they get isn't exact. But its precision is as great as the "goodness" of their assumption. In other words, as they assumed that bl>>1, as greater bl were, as precise their solution will be.