Hello, To measure the atomic force with an AFM. One can use the frequency shift of a cantilever. This change of frequency is linked to the atomic force by what we called the Franz J. Giessibl formula in the community. z is the AFM tip-sample distance. frequency(z) is the change of frequency of the cantilever vs the distance. Forces(z) is the atomic force versus the distance. a is a parameter linked to the drive of the cantilever. If I put all the useless constants equals to 1, I have: frequency (z)=(1/a)*Integrale[ Force[z+a(1+u)]*u / Sqrt[1-u^2] , du from -1 to 1] I use u for the integration This integral is some kind of functional linking frequency(z) and Force(z). The Force(z) function looks like this: http://www.teachnano.com/education/i/F-z_curve.gif [Broken] I would like to proove that for any function Force[z] that is real and ( z is positif ) and that goes through a negative minimum called Fmin, there is a frequency(z) function that goes through a minimum which is freq_min. It works numerically. I think there is one-to-one relationship between F_min and freq_min but I'd like it to be more than a feeling. Any ideas?