The characteristic polynomial in calculation of exponential and elementary functions in Clifford algebras

Formulas to calculate multivector exponentials in a basis-free representation and orthonormal basis are presented for an arbitrary Clifford geometric algebra , . The formulas are based on the analysis of roots of characteristic polynomial of a multivector. Elaborate examples how to use the formulas in practice are presented. The results are generalised to arbitrary functions of multivector and may be useful in the quantum circuits or in the problems of analysis of evolution of the entangled quantum states.