Quaternionic analysis

In mathematics, a function of a quaternion variable is a function with domain and range in the quaternions H.

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. A very useful function of a quaternion variable is

f(q)=uqu^{-1}

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

The quaternion inversion $f(q)=q^{-1}$ is another fundamental, but it introduces questions f(0) = ? and “Solve f(q) = 0.” Using affine transformations

f(q)=aq+b,\quad a,b\in H

along with the reciprocation function, one obtains the function theory of inversive quaternion geometry.

Quaternion variable theory differs in some respects from complex variable theory as in this instance: The complex conjugate mapping of the complex plane is a powerful tool but requires the introduction of a non-arithmetic operation. Indeed, conjugation changes the orientation of plane figures, something that arithmetic functions do not change. In contrast, the quaternion conjugation can be expressed arithmetically:

Proposition: The function $f(q)=-{\frac {1}{2}}(q+iqi+jqj+kqk)$ is equivalent to quaternion conjugation.

Proof: f(1) = − (1/2)(1 − 1 − 1 − 1) = 1. f(i) = −(1/2)(i − i + i + i) = − i . f(j) = − j , and f(k) = −k. Consequently, since f is a linear function, f(q) = f(w + xi + yj + zk) = w f(1) + x f(i) + y f(j) + z f(k) = w − x i − y j − zk = q*.

The immense success of complex function theory in providing the scientific world with an array of models has encouraged some to seek extensions to quaternion variables. Some of these efforts were reviewed by Devours (see ref). For example, …

Proposition: Let $f(z)=u(x,y)+iv(x,y)$ be a complex variable z = x + i y function. Suppose also that u is an even function of y and that v is an odd function of y. Then $f(q)=u(x,y)+rv(x,y)$ is an extension of f to a quaternion variable $q=x+yr,\quad r^{2}=-1,\quad r\in H$ .

References

Deavours