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.
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
f1(q)=uqu-1
f2(q)=q-1
f2(0)
Affine transformations of quaternions have the form
f3(q)=aq+b, a,b,q\inH.
M2(H)
H
q\mapstouqv,
u
v
Quaternion variable theory differs in some respects from complex variable theory. For example: The complex conjugate mapping of the complex plane is a central tool but requires the introduction of a non-arithmetic, non-analytic operation. Indeed, conjugation changes the orientation of plane figures, something that arithmetic functions do not change.
In contrast to the complex conjugate, the quaternion conjugation can be expressed arithmetically, as
f4(q)=-\tfrac{1}{2}(q+iqi+jqj+kqk)
This equation can be proven, starting with the basis :
f4(1)=-\tfrac{1}{2}(1-1-1-1)=1, f4(i)=-\tfrac{1}{2}(i-i+i+i)=-i, f4(j)=-j, f4(k)=-k
f4
f4(q)=f4(w+xi+yj+zk)=wf4(1)+xf4(i)+yf4(j)+zf4(k)=w-xi-yj-zk=q*.
The success of complex analysis in providing a rich family of holomorphic functions for scientific work has engaged some workers in efforts to extend the planar theory, based on complex numbers, to a 4-space study with functions of a quaternion variable. These efforts were summarized in .
Though
H
Let
f5(z)=u(x,y)+iv(x,y)
z=x+iy
u
y
v
y
f5(q)=u(x,y)+rv(x,y)
f5
q=x+yr
r2=-1
r\inH
r*
r
q=x-yr*
H
f5(q)=f5(x-yr*)
u(x,y)=u(x,-y), v(x,y)=-v(x,-y)
f5(x-yr*)=u(x,-y)+r*v(x,-y)=u(x,y)+rv(x,y)=f5(q).
In the following, colons and square brackets are used to denote homogeneous vectors.
The rotation about axis r is a classical application of quaternions to space mapping.In terms of a homography, the rotation is expressed
[q:1]\begin{pmatrix}u&0\\0&u\end{pmatrix}=[qu:u]\thicksim[u-1qu:1],
u=\exp(\thetar)=\cos\theta+r\sin\theta
q\mapstoq+p
[q:1]\begin{pmatrix}1&0\ p&1\end{pmatrix}=[q+p:1].
[q:1]\begin{pmatrix}u&0\ uxr&u\end{pmatrix}=[qu+uxr:u]\thicksim[u-1qu+xr:1].
Consider the axis passing through s and parallel to r. Rotation about it is expressed by the homography composition
\begin{pmatrix}1&0\ -s&1\end{pmatrix}\begin{pmatrix}u&0\ 0&u\end{pmatrix}\begin{pmatrix}1&0\ s&1\end{pmatrix}=\begin{pmatrix}u&0\ z&u\end{pmatrix},
z=us-su=\sin\theta(rs-sr)=2t\sin\theta.
Now in the (s,t)-plane the parameter θ traces out a circle
u-1z=u-1(2t\sin\theta)=2\sin\theta(t\cos\theta-s\sin\theta)
\lbracewt+xs:x>0\rbrace.
Any p in this half-plane lies on a ray from the origin through the circle
\lbraceu-1z:0<\theta<\pi\rbrace
p=au-1z, a>0.
Then up = az, with
\begin{pmatrix}u&0\ az&u\end{pmatrix}
Since the time of Hamilton, it has been realized that requiring the independence of the derivative from the path that a differential follows toward zero is too restrictive: it excludes even
f(q)=q2
A continuous map
f:H → H
U\subsetH
x\inU
f
f(x+h)-f(x)= | df(x) |
dx |
\circh+o(h)
df(x) | |
dx |
:H → H
H
o:H → H
\lima →
|o(a)| | |
|a| |
=0
df(x) | |
dx |
f
On the quaternions, the derivative may be expressed as
df(x) | |
dx |
=\sums
ds0f(x) | |
dx |
⊗
ds1f(x) | |
dx |
f
df(x) | |
dx |
\circdx=\left(\sums
ds0f(x) | |
dx |
⊗
ds1f(x) | |
dx |
\right)\circdx=\sums
ds0f(x) | |
dx |
dx
ds1f(x) | |
dx |
The number of terms in the sum will depend on the function f. The expressions
dspdf(x) | |
dx |
,p=0,1
The derivative of a quaternionic function holds the following equalities
df(x) | |
dx |
\circh=\limt\to(t-1(f(x+th)-f(x)))
d(f(x)+g(x)) | |
dx |
=
df(x) | + | |
dx |
dg(x) | |
dx |
df(x)g(x) | |
dx |
=
df(x) | g(x)+f(x) | |
dx |
dg(x) | |
dx |
df(x)g(x) | |
dx |
\circh=\left(
df(x) | |
dx |
\circh\right) g(x)+f(x)\left(
dg(x) | |
dx |
\circh\right)
daf(x)b | |
dx |
=a
df(x) | |
dx |
b
daf(x)b | |
dx |
\circh=a\left(
df(x) | |
dx |
\circh\right)b
For the function f(x) = axb, the derivative is
=a ⊗ b |
\circdx=adxb |
and so the components are:
=a |
=b |
Similarly, for the function f(x) = x2, the derivative is
=x ⊗ 1+1 ⊗ x |
\circdx=xdx+dxx |
and the components are:
=x |
=1 | ||||||||
=1 |
=x |
Finally, for the function f(x) = x−1, the derivative is
=-x-1 ⊗ x-1 |
\circdx=-x-1dxx-1 |
and the components are:
=-x-1 |
=x-1 |