Concept

Quaternionic analysis

Summary
In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called. As with complex and real analysis, it is possible to study the concepts of analyticity, holomorphy, harmonicity and conformality in the context of quaternions. Unlike the complex numbers and like the reals, the four notions do not coincide. Properties The projections of a quaternion onto its scalar part or onto its vector part, as well as the modulus and versor functions, are examples that are basic to understanding quaternion structure. An important example of a function of a quaternion variable is :f_1(q) = u q u^{-1} which rotates the vector part of q by twice the angle represented by u. The quaternion multiplicative inverse f_2(q) = q^{-1} is another fundamental function, but as with other number system
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