We consider the Clifford algebra C_n(F) where the field F are the real or the complex numbers. It is well known that an m-blade can be represented by the mth compound matrix of an n by m matrix relative to the basis of the underlying Grassmann algebra G. Since the Clifford product is simply related to the Grassmann product; the question of a corresponding representation of the Clifford product arises in a natural way.

We will show that the Clifford product of an odd (even) number of vectors corresponds to a linear combination of forms of odd (even) grade where the coefficients of these linear combinations are Pfaffians of certain matrices which can be understood as the skew symmetric counterpart of the corresponding Gramians. Based on this representation we calculate the mth Clifford power of a vector x which enables the extension of an analytical function to their corresponding Clifford function.

prells@penmaen.demon.co.uk