Thank you for your comment, I will try to better specify the problem that I am dealing with:
I am seeking a solution for the biharmonic equation
that governs problems in plane elasticity.
I am seeking for solutions in the form
where k is a constant that may be complex.
Substituting into (1) f(x,y) by g(x,y) yields the following fourth order equation
The domain for F is [-h/2,h/2]
Assuming for boundary conditions
Which correspond admitting a traction free condition on the edge of a membrane [0,L]x[-h/2,h/2]
An eigenvalue problem associated with the differential equation is obtained and the corresponding complex eigenvalues are the nonzero roots of the transcendental equations