In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, quaternionic analysis is the study of
functions with
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s as the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
and/or range. Such functions can be called functions of a quaternion variable just as
functions of a real variable or a
complex variable
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
are called.
As with
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
and
real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include converg ...
, it is possible to study the concepts of
analyticity,
holomorphy,
harmonicity and
conformality
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.
More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
in the context of quaternions. Unlike the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s 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
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Willi ...
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
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rat ...
is another fundamental function, but as with other number systems,
and related problems are generally excluded due to the nature of
dividing by zero.
Affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
s of quaternions have the form
:
Linear fractional transformation
In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form
:z \mapsto \frac ,
which has an inverse. The precise definition depends on the nature of , and . In other words, a linear fractional transf ...
s of quaternions can be represented by elements of the
matrix ring
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''U ...
operating on the
projective line over . For instance, the mappings
where
and
are fixed
versor
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Willi ...
s serve to produce the
motions of elliptic space.
Quaternion variable theory differs in some respects from complex variable theory. For example: The
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
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
Orientation may refer to:
Positioning in physical space
* Map orientation, the relationship between directions on a map and compass directions
* Orientation (housing), the position of a building with respect to the sun, a concept in building de ...
of plane figures, something that arithmetic functions do not change.
In contrast to the
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
, the quaternion conjugation can be expressed arithmetically, as
This equation can be proven, starting with the
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
:
:
.
Consequently, since
is
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
,
:
The success of
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
in providing a rich family of
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
s 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
appears as a union of complex planes, the following proposition shows that extending complex functions requires special care:
Let
be a function of a complex variable,
. Suppose also that
is an
even function
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power seri ...
of
and that
is an
odd function
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power seri ...
of
. Then
is an extension of
to a quaternion variable
where
and
.
Then, let
represent the conjugate of
, so that
. The extension to
will be complete when it is shown that
. Indeed, by hypothesis
:
one obtains
:
Homographies
In the following, colons and square brackets are used to denote
homogeneous vectors.
The
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
about axis ''r'' is a classical application of quaternions to
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
mapping.
In terms of a
homography
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
, the rotation is expressed
:
where
is a
versor
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Willi ...
. If ''p'' * = −''p'', then the translation
is expressed by
:
Rotation and translation ''xr'' along the axis of rotation is given by
:
Such a mapping is called a
screw displacement
A screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation and the line along which translation of a body occurs. Chasles' theorem shows that each Euclidean displacement in three-dimensional space has a screw ...
. In classical
kinematics
Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...
,
Chasles' theorem states that any rigid body motion can be displayed as a screw displacement. Just as the representation of a
Euclidean plane isometry In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types: translations, rotations, reflections, ...
as a rotation is a matter of complex number arithmetic, so Chasles' theorem, and the
screw axis
A screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation and the line along which translation of a body occurs. Chasles' theorem shows that each Euclidean displacement in three-dimensional space has a screw ...
required, is a matter of quaternion arithmetic with homographies: Let ''s'' be a right versor, or square root of minus one, perpendicular to ''r'', with ''t'' = ''rs''.
Consider the axis passing through ''s'' and parallel to ''r''. Rotation about it is expressed by the homography composition
:
where
Now in the (''s,t'')-plane the parameter θ traces out a circle
in the half-plane
Any ''p'' in this half-plane lies on a ray from the origin through the circle
and can be written
Then ''up'' = ''az'', with
as the homography expressing
conjugation
Conjugation or conjugate may refer to:
Linguistics
* Grammatical conjugation, the modification of a verb from its basic form
* Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
* Complex conjugation, the chang ...
of a rotation by a translation p.
The derivative for quaternions
Since the time of Hamilton, it has been realized that requiring the independence of the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
from the path that a differential follows toward zero is too restrictive: it excludes even
from differentiation. Therefore, a direction-dependent derivative is necessary for functions of a quaternion variable.
Considering the increment of
polynomial function
In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
of quaternionic argument shows that the increment is a linear map of increment of the argument. From this, a definition can be made:
A continuous map
is called differentiable on the set
, if, at every point
, the increment of the map
can be represented as
:
where
:
is
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
of quaternion algebra
and
is a continuous map such that
:
The linear map
is called the derivative of the map
.
On the quaternions, the derivative may be expressed as
:
Therefore, the differential of the map
may be expressed as follows with brackets on either side.
:
The number of terms in the sum will depend on the function ''f''. The expressions
are called
components of derivative.
The derivative of a quaternionic function holds the following equalities
:
:
:
:
:
:
For the function ''f''(''x'') = ''axb'', the derivative is
and so the components are:
Similarly, for the function ''f''(''x'') = ''x
2'', the derivative is
and the components are:
Finally, for the function ''f''(''x'') = ''x''
−1, the derivative is
and the components are:
See also
*
Cayley transform In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform ...
*
Quaternionic manifold In differential geometry, a quaternionic manifold is a quaternionic analog of a complex manifold. The definition is more complicated and technical than the one for complex manifolds due in part to the noncommutativity of the quaternions and in p ...
Notes
Citations
References
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{{Analysis in topological vector spaces
Articles containing proofs
Functions and mappings
Quaternions