Magnetization patterns in ferromagnetic nano-elements as functions of complex variable

Assumption of certain hierarchy of soft ferromagnet energy terms, realized in small enough flat nano-elements, allows to obtain explicit expressions for their magnetization distributions. By minimising the energy terms sequentially, from most to the least important, magnetization distributions are expressed as solutions of Riemann-Hilbert boundary value problem for a function of complex variable. A number of free parameters, corresponding to positions of vortices and anti-vortices, still remain in the expression. These parameters can be found by computing and minimizing the total magnetic energy of the particle with no approximations. Thus, the presented approach is a factory of realistic Ritz functions for analytical micromagnetic calculations. These functions are so versatile, that they may even find applications on their own (e.g. for fitting magnetic microscopy images). Examples are given for multi-vortex magnetization distributions in circular cylinder, and for 2-dimensional domain walls in thin magnetic strips.