Coordinate-free exponentials of general multivector in Cl(p,q) algebras
Closed form expressions in real Clifford geometric algebras Cl(0,3),
Cl(3,0), Cl(1,2), and Cl(2,1) are presented in a coordinate-free form
for exponential function when the exponent is a general multivector. The
main difficulty in solving the problem is connected with an entanglement
(or mixing) of vector and bivector components ai and
ajk in a form
(ai-ajk)2, i≠ j≠ k .
After disentanglement, the obtained formulas simplify to the well-known
Moivre-type trigonometric/hyperbolic function for vector or bivector
exponentials. The presented formulas may find wide application in
solving GA differential equations, in signal processing, automatic
control and robotics.