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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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. The algebra of quater ...
s as the
domain A domain is a geographic area controlled by a single person or organization. Domain may also refer to: Law and human geography * Demesne, in English common law and other Medieval European contexts, lands directly managed by their holder rather ...
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 g ...
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 co ...
, 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\i ...
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 for ...
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 Quaternion#Norm, norm one, also known as a unit quaternion. Each versor has the form :u = \exp(a\mathbf) = \cos a + \mathbf \sin a, \quad \mathbf^2 = -1, \quad a \in ,\pi where the r2 = −1 conditi ...
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^ which rotates the vector part of ''q'' by twice the angle represented by the versor ''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 ra ...
f_2(q) = q^ is another fundamental function, but as with other number systems, f_2(0) 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 general ...
s of quaternions have the form :f_3(q) = aq + b, \quad a, b, q \in \mathbb.
Linear fractional transformation In mathematics, a linear fractional transformation is, roughly speaking, an inverse function, invertible transformation of the form : z \mapsto \frac . The precise definition depends on the nature of , and . In other words, a linear fractional t ...
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'') (alternat ...
M_2(\mathbb) operating on the projective line over \mathbb. For instance, the mappings q \mapsto u q v, where u and v are fixed
versor In mathematics, a versor is a quaternion of Quaternion#Norm, norm one, also known as a unit quaternion. Each versor has the form :u = \exp(a\mathbf) = \cos a + \mathbf \sin a, \quad \mathbf^2 = -1, \quad a \in ,\pi where the r2 = −1 conditi ...
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 numbers, then the complex conjugate of a + bi is 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 des ...
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 numbers, then the complex conjugate of a + bi is a - ...
, the quaternion conjugation can be expressed arithmetically, as f_4(q) = - \tfrac (q + iqi + jqj + kqk) This equation can be proven, starting with the basis : :f_4(1) = -\tfrac(1 - 1 - 1 - 1) = 1, \quad f_4(i) = -\tfrac(i - i + i + i) = -i, \quad f_4(j) = -j, \quad f_4(k) = -k . Consequently, since f_4 is
linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...
, :f_4(q) = f_4(w + x i + y j + z k) = w f_4(1) + x f_4(i) + y f_4(j) + z f_4(k) = w - x i - y j - z k = q^*. 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 functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
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 de ...
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 \mathbb appears as a union of complex planes, the following proposition shows that extending complex functions requires special care: Let f_5(z) = u(x,y) + i v(x,y) be a function of a complex variable, z = x + i y. Suppose also that u is an
even function In mathematics, an even function is a real function such that f(-x)=f(x) for every x in its domain. Similarly, an odd function is a function such that f(-x)=-f(x) for every x in its domain. They are named for the parity of the powers of the ...
of y and that v is an
odd function In mathematics, an even function is a real function such that f(-x)=f(x) for every x in its domain. Similarly, an odd function is a function such that f(-x)=-f(x) for every x in its domain. They are named for the parity of the powers of the ...
of y. Then f_5(q) = u(x,y) + rv(x,y) is an extension of f_5 to a quaternion variable q = x + yr where r^2 = -1 and r \in \mathbb. Then, let r^* represent the conjugate of r, so that q = x - yr^*. The extension to \mathbb will be complete when it is shown that f_5(q) = f_5(x - yr^*). Indeed, by hypothesis :u(x,y) = u(x,-y), \quad v(x,y) = -v(x,-y) \quad one obtains :f_5(x - y r^*) = u(x,-y) + r^* v(x,-y) = u(x,y) + r v(x,y) = f_5(q).


Homographies

In the following, colons and square brackets are used to denote homogeneous vectors. The
rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
about axis ''r'' is a classical application of quaternions to
space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
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 : :1\beginu & 0\\0 & u \end = u : u\thicksim ^qu : 1, where u = \exp(\theta r) = \cos \theta + r \sin \theta is a
versor In mathematics, a versor is a quaternion of Quaternion#Norm, norm one, also known as a unit quaternion. Each versor has the form :u = \exp(a\mathbf) = \cos a + \mathbf \sin a, \quad \mathbf^2 = -1, \quad a \in ,\pi where the r2 = −1 conditi ...
. If ''p'' * = −''p'', then the translation q \mapsto q + p is expressed by : : 1begin1 & 0 \\ p & 1 \end = + p : 1 Rotation and translation ''xr'' along the axis of rotation is given by : : 1beginu & 0 \\ uxr & u \end = u + uxr : u\thicksim ^qu + xr : 1 Such a mapping is called a
screw displacement In kinematics, Chasles' theorem, or Mozzi–Chasles' theorem, says that the most general rigid body displacement can be produced by a screw displacement. A direct Euclidean isometry in three dimensions involves a translation and a rotation. The ...
. In classical
kinematics In physics, kinematics studies the geometrical aspects of motion of physical objects independent of forces that set them in motion. Constrained motion such as linked machine parts are also described as kinematics. Kinematics is concerned with s ...
, 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, a ...
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 :\begin1 & 0 \\ -s & 1 \end \beginu & 0 \\ 0 & u \end \begin1 & 0 \\ s & 1 \end = \beginu & 0 \\ z & u \end, where z = u s - s u = \sin \theta (rs - sr) = 2 t \sin \theta . Now in the (''s,t'')-plane the parameter θ traces out a circle u^ z = u^(2 t \sin \theta) = 2 \sin \theta ( t \cos \theta - s \sin \theta) in the half-plane \lbrace wt + xs : x > 0 \rbrace . Any ''p'' in this half-plane lies on a ray from the origin through the circle \lbrace u^ z : 0 < \theta < \pi \rbrace and can be written p = a u^ z , \ \ a > 0 . Then ''up'' = ''az'', with \beginu & 0 \\ az & u \end 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 change o ...
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 is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
from the path that a differential follows toward zero is too restrictive: it excludes even \ f(q) = q^2\ 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 a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative int ...
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 function \ f: \mathbb H \rightarrow \mathbb H\ is called ''differentiable on the set'' \ U \subset \mathbb H\ , if at every point \ x \in U\ , an increment of the function \ f\ corresponding to a quaternion increment \ h\ of its argument, can be represented as : f(x+h) - f(x) = \frac \circ h + o(h) where : \frac:\mathbb H\rightarrow\mathbb H 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 mapping V \to W between two vector spaces that p ...
of quaternion algebra \ \mathbb H\ , and \ o:\mathbb H\rightarrow \mathbb H\ represents some continuous map such that : \lim_ \frac = 0\ , and the notation \ \circ h\ denotes ... The linear map \frac is called the derivative of the map \ f ~. On the quaternions, the derivative may be expressed as : \frac = \sum_s \frac \otimes \frac Therefore, the differential of the map \ f\ may be expressed as follows, with brackets on either side. :\frac\circ \operatorname d x = \left(\sum_s \frac \otimes \frac\right)\circ \operatorname d x = \sum_s \frac \left( \operatorname d x \right) \frac The number of terms in the sum will depend on the function \ f ~. The expressions ~~ \frac ~~ \mathsf ~~ p = 0, 1 ~~ are called components of derivative. The derivative of a quaternionic function is defined by the expression : \frac\circ h = \lim_\left(\ \frac\ \right) where the variable \ t\ is a real scalar. The following equations then hold: : \frac = \frac + \frac : \frac = \frac\ g(x) + f(x)\ \frac : \frac \circ h = \left(\frac\circ h\right )\ g(x) + f(x) \left(\frac\circ h\right) : \frac = a\ \frac\ b : \frac\circ h = a \left(\frac\circ h\right) b For the function \ f(x) = a\ x\ b\ , where \ a\ and \ b\ are constant quaternions, 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^\ , 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


Notes


Citations


References

* * * * * * * * * . * * * * * * {{Analysis in topological vector spaces Articles containing proofs Functions and mappings
analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...