A new series expansion for slice regular functions

作者：Stoppato Caterina^*

来源：Advances in Mathematics, 2012, 231(3-4): 1401-1416.

DOI：10.1016/j.aim.2012.05.023

• Functions of a hypercomplex variable
• Slice regular function
• Quaternionic series expansion
• Quaternionic analytic function

A promising theory of quaternion-valued functions of one quaternionic variable, now called slice regular functions, has been introduced by Gentili and Struppa in 2006. The basic examples of slice regular functions are the power series of type Sigma(n is an element of N)q(n)a(n) on their balls of convergence B(0, R) = {q is an element of H : vertical bar g vertical bar %26lt; R}. Conversely, if f is a slice regular function on a domain Omega subset of H then it admits at each point q(0) is an element of Omega an expansion of type f(q) = Sigma(n is an element of N) (q-q(0))*(n)a(n) where (q-q(0))*(n) denotes the nth power of q-q(0) with respect to an appropriately defined multiplication *. However, the information provided by such an expansion is somewhat limited by a fact: if q(0) does not lie on the real axis then the set of convergence of the series in the previous equation needs not be a Euclidean neighborhood of q(0). We are now able to construct a new type of expansion that is not affected by this phenomenon: an expansion into series of polynomials valid in Open subsets of the domain. Along with this construction, we present applications to the computation of the multiplicities of zeros and of partial derivatives.

• 出版日期2012-11