There is a topic called analytic functions of a matrix that could be applied here, if you represent the quaternion a0 + a1 i + b0 j + b1 k by the matrix
Q =: 2 2 $ a , b, (- + b), - + a where a means a0 j. a1 and b means b0 j. b1 When Q has distinct eigenvalues e0 and e1 with corresponding eigenvectors v0 and v1, then f(Q) = V mp (f E) mp %. V when f is an analytic function, V is the matrix v0 ,. v1 and E is the diagonal matrix 2 2 % e0 , 0, 0, e1 and f E is the diagonal matrix 2 2 $ (f e0),0,0,f e1 . Function f has to be analytic in neighborhoods of e0 and e1. mp is the matrix product +/ . * Analytic functions include monadic ^ and ^. and functions like those found on the dictionary page for o. . They make this topic about as hard as a typical engineering course in complex variables for which I recommend Fundamentals of Complex Analysis by E. B. Saff and A. D. Snider, Third Edition, Prentice Hall 2003. You can have x^y when quaternion y is a complex number but thinking of z^y as an analytic function of z has the difficulties presented in a complex variables course like Saff and Snider. Kip Murray Raul Miller wrote: > I have in the past thought that quaternions would be a nice addition to J. > > They are very useful, for example, for expressing rotations of geometric > models. > > However, J of course needs its implementation to be generally correct > and not just correct for some specific application. > > So I have been thinking of how to model J's primitives that would need > to interpret quaternion values. > > This leads to questions such as: what does x^y mean when x and y are > quaternions? > > Apparently, this is a hard problem: > > http://www.zipcon.net/~swhite/docs/math/quaternions/analysis.html > > I have been trying to find a definition of quaternion exponentials > that that does not do something silly (like assume that quaternion > multiplication is commutative). So far, I have not found anything I > feel I could implement a J model for transcendental functions from. > Also, x^y for quaternion x and quaternion y would often enough have an > infinite set of valid results, which complicates things. > > So... does anyone have an good ideas about how to handle > transcendental functions that work with quaternions? > > Thanks, > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
