Quaternion analysis

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 functions of 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. It is known that for the complex numbers, these four notions coincide; however, for the quaternions, and also the real numbers, not all of the notions are the same.

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

which rotates the vector part of q by twice the angle represented by u.

The quaternion multiplicative inverse ${\displaystyle f_{2}(q)=q^{-1}}$ is another fundamental function, but it raises difficult questions such as “What should ${\displaystyle f_{2}(0)}$ be?” and “What ${\displaystyle q}$ solves the equation ${\displaystyle f_{2}(q)=0}$?”

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