Download An Overview and Analysis of Quaternions Abstract:

An Overview and Analysis
of Quaternions
Stephen Gorgone
Department of Mathematics
Ithaca College
Ithaca, NY 14850 USA
[email protected]
Abstract:
The purpose of this paper is to better understand the quaternion number system. This
consists of understanding their discovery by Hamilton; their properties and numerous
representations compared to those of other number systems, especially their geometric
properties; their correspondence to the complex number system; and their calculus, of
course leading to an analogue of Cauchy’s Theorem (among others), the properties of
analytic functions, and conformal mappings.
Advised by: David Brown
Associate Professor,
Department of Mathematics
Ithaca College
Ithaca, NY 14850 USA
[email protected]
26 April 2013
Contents
1
Introduction
2
2
Complex Numbers and Rotations
2
3
Hamilton’s Discovery of the Quaternions
5
4
Axiomatic Properties of the Quaternions
9
5
Other Properties of Quaternions
11
6
Quaternion Derivatives
16
7
Cauchy’s Theorem
20
8
Construction of Regular Functions
23
9
Regular Functions and Conformal Mappings
25
10 Further Analogues
26
1
1
Introduction
When trying to understand numbers and number systems, we often begin with the most
basic number system, the natural numbers, and pass through the integers and the rationals until we arrive at the real numbers. After this, we study the complex numbers, the
number system that contains all of the reals. Following this progression, it makes sense
that the next step after studying the complex is to study the quaternions. Thus, in order to
fully understand the properties and tendencies of quaternions, we must first understand
the reals and complex so that we may see what changes and what stays the same. As
expected, a background knowledge of geometry, abstract algebra, linear algebra, real and
complex analysis, and calculus is required for the purposes of this paper. Further, some
topological concepts are introduced in later sections. Although the purely theoretical use
of quaternions is limited since most of its applications involve higher level physics, their
study aptly connects the knowledge obtained in most undergraduate mathematics curricula.
The historical background regarding Hamilton’s discovery of the quaternions is taken
mainly from [1]. The properties of the quaternions are from [2] while their linear algebra
representation are from [3]. Some additional properties and views of quaternion functions
are taken from [4] and the entire construction of calculus and Cauchy’s Theorem is from
[5].
2
Complex Numbers and Rotations
The ordinary complex numbers may be defined in √
a few different ways. The first is the
algebraic point of view of (a + bi), formerly a + b −1, where i2 = −1. The rule for
multiplication is
(a + bi)(c + di) = ac + adi + bci + bdi2 = (ac − bd) + (ad − bc)i.
Alternately, they may be defined as couples (a, b) where (1, 0) = 1 and (0, 1) = i, a rewriting of the basis vectors of R2 , thus naturally forming an isomorphism between C and R2 .
Here, multiplication is defined as
(a, b)(c, d) = (ac − bd, ad − bc).
As is apparent, the two definitions lead to the same result. They are merely written in
two different ways.
When viewing the plane C of complex numbers, rotation about O by angle θ is the
same as multiplication by
eiθ = cos θ + i sin θ.
since eiθ = −1 .
The set of all such numbers is the unit circle or 1 − dimensional-sphere
S1 = {z : |z| = 1}.
2
where |z| = z for z ≥ 0 and |z| = | − z| for z ≤ 0.
S1 is not only a geometric object, but also an algebraic structure, a group, under complex number multiplication since we know from complex analysis that it is closed (multiplying a distance from 0 by 1 also yields a distance from 0 by 1), is associative (as can
be quickly shown algebraically and geometrically), has an identity element (since multiplying by ei0 ∈ C is the same as multiplying by 1 ∈ R), and has inverse elements for
each element of the group (as can be shown algebraically and geometrically). Further,
eiθ1 · eiθ2 = ei(θ1 +θ2 ) and (eiθ )−1 = ei(−θ) depend on θ. This makes S1 a Lie group since
the exponential function is smooth, or infinitely differentiable, in C and the previous sentence is composed of functions based on the exponential function. S1 is also commutative
because multiplicative commutativity holds for complex numbers.
We may extend these ideas to the four-dimensional algebra of quaternions and the
three-dimensional sphere S3 of unit quaternions. Here, S3 is a noncommutative Lie group
called SU(2), closely related to the group of space rotations.
A rotation of the plane R2 about the origin O by angle θ is a linear transformation Rθ
that sends the basis vectors (1, 0) and (0, 1) to (cos θ, sin θ) and (− sin θ, cos θ), respectively
as shown:
Figure 1: Rotation of the plane through angle θ
It follows by linearity that Rθ sends the general vector
(x, y) = x(1, 0) + y(0, 1) to (x cos θ − y sin θ, x sin θ + y cos θ)
and that Rθ may be represented by the matrix
cos θ − sin θ
Rθ =
.
sin θ cos θ
Applying a rotation to (x, y) is the same as multiplying the column vector
by the matrix Rθ such that
x
cos θ − sin θ
x
x cos θ − y sin θ
Rθ
=
=
.
y
sin θ cos θ
y
x sin θ + y cos θ
3
x
y
on the left
Since matrices are applied from the left, applying Rϕ then Rθ is the same as applying
the product matrix Rθ Rϕ (even though this matrix also equals Rϕ Rθ since both equal
Rθ+ϕ ). Thus, we may combine successive rotations merely by multiplying matrices since
the multiplication of matrices for rotations is the same as adding the angles of rotation.
The matrices {Rθ : θ ∈ R} form a group called the special orthogonal group SO(2),
the rotation to the 2-dimensional space R2 . Each rotation Rθ of R2 may be represented by
the complex number
xθ = cos θ + i sin θ
to get the required rotation of an arbitrary point (x, y) = x + iy as shown:
zθ (x + iy) = (cos θ + i sin θ)(x + iy)
= x cos θ − y sin θ + i(x sin θ + y cos θ)
= (x cos θ − y sin θ, x sin θ + y cos θ).
Further, the product zθ zϕ represents the successive rotations of Rθ and Rϕ .
Matrix representation of complex numbers
θ − sin θ
To see why the matrices Rθ = cos
behave the same as the complex numbers zθ =
sin θ cos θ
cos θ + i sin θ, we write Rθ as the linear combination
1 0
0 −1
Rθ = cos θ
+ sin θ
0 1
1 0
of the basis matrices
1=
1 0
0 1
and i =
0 −1
.
1 0
It is obvious through matrix multiplication that the following hold:
12 = 1, 1i = i1 = 1, and i2 = −1
such that the matrices 1 and i behave the same as the complex numbers 1 and i, respectively. To generalize this further, the matrices
a −b
= a1 + bi, where a, b ∈ R,
b a
behave the same as the complex numbers a + bi since their properties are the same. Thus,
we may represent all complex numbers by 2 × 2 real matrices as well as the complex
numbers zθ that represent rotations. This permits us to explain certain properties of the
complex numbers using linear algebra, i.e.:
• The squared absolute value, |a+ bi|2 = a2 + b2 , of a + bi ∈ C is the determinant of
the corresponding matrix ab −b
a .
4
• The multiplicative property of the absolute value, |z1 z2 | = |z1 ||z2 |, follows the multiplicative property of determinants
det(A1 A2 ) = det(A1 ) det(A2 )
where A1 is the matrix representing z1 and A2 is the matrix representing z2 , a property that may also be shown algebraically.
• The inverse, z −1 =
a−bi
,
a2 +b2
of z = a + bi 6= 0 corresponds to the inverse matrix,
−1
1
a −b
a b
= 2
.
b a
a + b2 b a
The two-square identity
If we let z1 = a1 + ib1 and z2 = a2 + ib2 , then we can calculate the product of z1 and z2 and
their absolute values
(a21 + b21 )(a22 + b22 ) = (a1 a2 − b1 b2 )2 + (a1 b2 + a2 b1 )2 .
This shows that the sum of two squares times the sum of two squares equals another
sum of two squares for a1 , b1 , a2 , b2 ∈ Z. Nearly 2000 years ago, Diophantus noticed
this in Book III, Problem 19, of his Arithmetica. Nothing is mentioned about the sum
of three squares since there is no three-square identity. Because of this, there can be
no three-dimensional numbers, i.e. a number represented as a triplet, nor can there
be any n-dimensional numbers for n > 2. Fortunately, for quadruples, q = (a, b, c, d)
with a, b, c, d ∈ R, we may define addition and multiplication with all the basic laws of
arithmetic save multiplicative commutativity. This system of arithmetic for quadruples is
called quaternion algebra.
3
Hamilton’s Discovery of the Quaternions
When William Rowan Hamilton discovered the quaternions–the number system defined
either as quadruplets, (a, b, c, d), rather than just couplets; as a + bi + cj + dk with a, b, c, d ∈
R and i2 = j 2 = k2 = −1; as v + jw where v, w ∈ C, v = t + ix, and w = y − iz; or as the
−b−ic
matrix a+id
b−ic a−id –on 16 October 1843, he devised another set of rules for multiplication:
i2 = j 2 = k 2 = −1
ij = k
jk = i
ki = j
ji = −k
kj = −i
ik = −j
Quaternions may be added, subtracted, multiplied, and divided (excluding division by
zero), thus forming a division algebra. Unfortunately, as seen above, multiplicative commutativity does not hold since ij = −ji, jk = −kj, and ki = −ik.
5
Hamilton emphasized the definition of the ordinary complex numbers, the couplet,
(a, b), and hoped to find how to multiple number-triplets, (a, b, c), in the same manner as
couplets. He wrote the triplets as (a + bi + cj), represented the unit vectors 1, i, and j
as mutually perpendicular ”directed segments,” and hoped to represent a product (a +
bi + cj)(x + yi + zj) as vectors in the same space. He required that multiplication be
possible term by term and that the length of the product of the vectors equal the product
of the lengths (the ”law of the moduli”). In 1898, Hurwitz proved, with the help of matrix
multiplication, that these two requirements only work in spaces with dimensions of 1, 2, 4,
and 8 (the real numbers, complex numbers, quaternions, and octonions). Thus, although
Hamilton was not familiar with this result, he was unable to reach this goal since, as he
found out, triplets are not closed under multiplication.
In order to fulfill with law of the moduli, he extended the law for complex numbers
a + bi where ii = −1 and made it so that, for numbers a + cj, jj = −1. Assuming that
ij = ji, he calculated that
(a + bi + cj)(x + yi + zj) = (ax − by − cz) + (ay + bx)i + (az + cx)j + (bz + cy)ij.
The only thing left to do is to simply ij = ji into something of the form α + βi + γj.
Hamilton tried in many ways to so this:
1. As i2 = j 2 = −1, (ij)2 = 1. Since this is true, either ij = 1 or ij = −1 and the law
of the moduli holds if the sum the squares of the coefficients of the original and the
product are equal, so their difference must be 0.
(a) Let ij = 1. In this case,
(a + bi + cj)(x + yi + zj)
= (ax − by − cz) + (ay + bx)i + (az + cx)j + (bz + cy)(1)
= (ax + bz + cy − by − cz) + (ay + bx)i + (az + cx)j.
And [(ax+bz +cy −by −cz)2 +(ay +bx)2 +(az +cx)2 ]−[(a2 +b2 +c2 )(x2 +y 2 +z 2 )]
= ax2 + ay 2 + az 2 + 2aybx + bx2 − 2axby + by 2 + 2axbz − 2bybz + bz 2 +
2azcx + cx2 + 2axcy − 2bycy + 2bzcy + cy 2 − 2axcz + 2bycz − 2bzcz − 2cycz +
cz 2 − a2 x2 − b2 x2 − c2 x2 − a2 y 2 − b2 y 2 − c2 y 2 − a2 z 2 − b2 z 2 − c2 z 2
6= 0
(b) Let ij = −1. In this case,
(a + bi + cj)(x + yi + zj)
= (ax − by − cz) + (ay + bx)i + (az + cx)j + (bz + cy)(−1)
= (ax − by − bz − cy − cz) + (ay + bx)i + (az + cx)j.
and [(ax−by −bz −cy −cz)2 +(ay +bx)2 +(az +cx)2 ]−[(a2 +b2 +c2 )(x2 +y 2 +z 2 )]
= ax2 + ay 2 + az 2 + 2aybx + bx2 − 2axby + by 2 − 2axbz + 2bybz + bz 2 + 2azcx +
cx2 − 2axcy + 2bycy + 2bzcy + cy 2 − 2axcz + 2bycz + 2bzcz + 2cycz + cz 2 −
a2 x 2 − b 2 x 2 − c 2 x 2 − a2 y 2 − b 2 y 2 − c 2 y 2 − a2 z 2 − b 2 z 2 − c 2 z 2
6= 0
6
2. As the moduli were not equal in the previous attempt, Hamilton considered the
simplest case:
(a + bi + cj)2 = (a2 − b2 − c2 ) + (2ab)i + (2ac)j + (2bc)ij.
Calculating the sum of the squares of the coefficients of just 1, i, and j on the right
hand side, he found that
(a2 − b2 − c2 )2 + (2ab)2 + (2ac)2 = (a2 + b2 + c2 ).
Thus, the product rule is only fulfilled if ij = 0. Furthermore, if we construct a plane
that intersects the points 0, 1, and 1 + bi + cj, then the construction of the product
will remain in this plane. In order words, the vector (a + bi + cj)2 lies in the same
plane as (a + bi + cj) and the angle it makes with the vector 1 is twice that of its
original.
3. As the previous attempt where ij = 0 did not feel right, Hamilton took instead that
ij = −ji, named ij = k, and named ji = −k. He then pondered whether it was
possible that k = 0. Fortunately, he was correct in moving away from this since
letting ij = 0 means that |ij| = 0 6= (1)(1) = |i||j|.
4. Hamilton multiplied another simple version of the numbers such that both the two
multiplied segments would lie in one plane spanned by 0, 1, and bi + cj and they
would yield the result
(a + bi + cj)(x + bi + cj) = (ax − b2 − c2 ) + ((a + x)b)i + ((a + x)c)j + (bc − bc)k.
In this case, the coefficient of k still equals 0 while it is confirmed that ij = −ji.
After discovering that ij = −ji yields results, he calculated the general case of
(a + bi + cj)(x + yi + zj) = (ax − by − cz) + (ay + bx)i + (az + cx)j + (bz − cy)k.
Setting k = 0 such that
(a + bi + cj)(x + yi + zj) = (ax − by − cz) + (ay + bx)i + (az + cx)j,
he found the law of the moduli was not satisfied, thereby proving that k 6= 0.
[(a2 + b2 + c2 )(x2 + y 2 + z 2 )] − [(ax − by − cz)2 + (ay + bx)2 + (az + cx)2 ]
= (bz − cy)2
6= 0
Given this new insight, Hamilton realized that he must introduce a fourth dimension
of space in order to calculate the triplets. This realization, although around the same
time as Mr. Cayley published a paper on Analytical Geometry of n dimensions in the
Cambridge Mathematical Journal from May 1843, was made independently of Cayley.
7
Hoping that he could rely on multiplicative associativity, he proceeded to procure
different products with k:
ik = (i)(ij) = (ii)(j) = −j
ki = (−ji)(i) = −(j)(ii) = −(j)(−1) = k
Further, since ij = −ji, he figured that jk = −kj and ik = −ki.
To solve for k 2 , he determined with associativity that
k 2 = (k)(k) = (ij)(ij) = i(ji)j = i(−ij)j = −(ii)(jj) = −(−1)(−1) = −1.
With k 2 = −1, Hamilton also found that it fulfilled the law of the moduli. Thus, he
assumed that
i2 = j 2 = k 2 = −1
i = jk = −kj
j = ki = −ik
k = ij = −ji.
To check that these assumptions fulfilled the law of the moduli, he multiplied two arbitrary quaternions
(a, b, c, d)(a0 , b0 , c0 , d0 ) = (a00 , b00 , c00 , d00 )
and solved from there to obtain a00 , b00 , c00 , and d00 :
(a00 + b00 i + c00 j + d00 k) = (a + bi + cj + dk)(a0 + b0 i + c0 j + d0 k)
= aa0 + ab0 i + ac0 j + ad0 k + bia0 + bib0 i + bic0 j + bid0 k
+cja0 + cjb0 i + cjc0 j + cjd0 k + dka0 + dkb0 i + dkc0 j + dkd0 k
= aa0 + ab0 i + ac0 j + ad0 k + ba0 i − bb0 + bc0 k − bd0
+ca0 j − cb0 − cc0 + cd0 i + da0 k + db0 j − dc0 i − dd0
= (aa0 − bb0 − cc0 − dd0 ) + (ab0 + ba0 + cd0 − dc0 )i
+(ac0 − bd0 + ca0 + db0 )j + (ad0 + bc0 − cb0 + da0 )k.
He then checked to see that the sum of the squares of each were equal, thus fulfilling the
law of the moduli:
(a2 + b2 + c2 + d2 )((a0 )2 + (b0 )2 + (c0 )2 + (d0 )2 )
= [(a2 + b2 + c2 + d2 )(a02 + b02 + c02 + d02 )]
−[(aa0 − bb0 − cc0 − dd0 )2 + (ab0 + ba0 + cd0 − dc0 )2
+(ac0 − bd0 + ca0 + db0 )2 + (ad0 + bc0 − cb0 + da0 )2 ]
= 0
The process by which he discovered and calculated these formulas is found in his
notebook as well as a letter to John T. Graves on 17 October 1843 when he announced his
discovery of the quaternions to Graves.
8
In addition to the law of the moduli for complex numbers,
(a2 + b2 )(c2 + d2 ) = (ac − bd)2 + (ad + bc)2 ,
Euler discovered a similar formula for the sum of 4 squares,
(a21 + ... + a24 )(b21 + ... + b24 ) = (c21 + ... + c24 )
where ci equals some linear combination of each a and b, which he stated in a letter to
Goldbach of 4 May 1748. The law of the moduli for octonions, the sum of 8 squares, was
also previously discovered in 1818 by C. P. Degen who believed that he could generalize
the theorem for all 2n squares.
The law of the moduli for triplets, or the sum of 3 squares, was shown impossible by
Legendre in his Théorie des nombres. Remarking that 3 = 1 + 1 + 1 and 21 = 16 + 4 + 1
may be represented as the sum of three squares, he proved that 3 × 21 = 63 cannot be
represented as such since it is of the form 8n + 7 and it has been proven that no number
of this form may be expressed as such. From this, it follows that forming the identity
(a2 + b2 + c2 )(x2 + y 2 + z 2 ) = (u2 + v 2 + w2 )
is impossible with 63 when assuming rational values for u, v, and w and when applying
Hurwitz’s theorem.
4
Axiomatic Properties of the Quaternions
Quaternions are defined H = {Q = 1 · q1 + i · q2 + j · q3 + k · q4 : q1 , q2 , q3 , q4 ∈ R} where 1
is the multiplicative identity and i, j, and k have the following properties:
i2 = j 2 = k 2 = −1
ij = k, ji = −k
jk = i, kj = −i
ki = j, ik = −j.
As with complex numbers, we can associate the ordered quadruple (q1 , q2 , q3 , q4 ) with the
matrix
q1 + iq4 −q2 − iq3
Q=
.
q2 − iq3 q1 − iq4
We call any matrix of this form a quaternion. Moreover, the complex numbers are the
special case of the quaternions with q3 = q4 = 0.
It should be noted that
q1 + q4 i −q2 − q3 i
= q1 1 + q2 i + q3 j + q 4 k
q2 − q3 i q 1 − q 4 i
where
1=
1 0
0 −1
0 −i
i 0
, i=
, j=
, k=
.
0 1
1 0
−i 0
0 −i
9
Thus, each matrix behaves as its algebraic unit where 1 behaves like 1 and i2 = j2 =
k2 = −1. To visually understand the products of any two distinct element i, j, k, use
the following figure where the product of two elements is the third element in the circle,
positive if the arrow points from the first element to the second element and negative if
the arrow points from the second element to the first element.
Figure 2: Products of imaginary unit quaternions.
As the extension of an algebra should strive to preserve the operators defined in the
original algebra, we strive to preserve the properties of the real and complex numbers
when working with quaternions.
Addition is component-wise as follows:
P +Q = (p1 +ip2 +jp3 +kp4 )+(q1 +iq2 +jq3 +kq4 ) = (p1 +q1 )+i(p2 +q2 )+j(p3 +q3 )+k(p4 +q4 )
This provides a consistent behavior for the subset of quaternions to respond both to real
numbers and to complex numbers and also is consistent with matrix addition. Further,
this rule preserves the associative and commutative properties of addition.
Multiplication is the same as for polynomials combined with the properties of i, j, and
k and is also equivalent to the matrix multiplication as follows:
P Q = (p1 + ip2 + jp3 + kp4 )(q1 + iq2 + jq3 + kq4 )
= (p1 q1 − p2 q2 − p3 q3 − p4 q4 ) + i(p1 q2 + p2 q1 + p3 q4 − p4 q3 )
+j(p1 q3 + p3 q1 + p4 q2 − p2 q4 ) + k(p1 q4 + p4 q1 + p2 q3 − p3 q2 ).
This and future properties may be shown algebraically as well as with linear algebra and
are left to the reader to verify. Again, this rule is consistent with the subset of real and
complex numbers. Although associativity for this operation is preserved, commutativity is not. This is more visually apparent with the previous representation of imaginary
quaternion units. Moreover, this failure is partially what enables the quaternions to represent other noncommutative things, i.e. rotations in three and four dimensions.
The conjugate of a given quaternion Q is defined as
Q = (q1 + iq2 + jq3 + kq4 ) = (q1 − iq2 − jq3 − kq4 )
and is interpreted geometrically in the same manner as with complex numbers with additional axes.
10
As with addition and multiplication, this is consistent with complex numbers in that
Qc = (q1 + iq2 + j0 + k0) = (q1 − iq2 ).
Further, many other properties of the conjugate from complex numbers remain. Both
(Q + Q) and (QQ) are real. The conjugate is also distributive over addition with P + Q =
P + Q, but with multiplication yields P Q = QP .
The quaternion conjugate is not the result of taking the complex conjugate of each
individual entry in Q, but rather is the result of taking the complex conjugate of each
individual entry in the transposed matrix QT . Moreover, since (P Q)T = QT P T , we get
PQ = Q P.
Moreover, defining the absolute value, or norm, of Q to be
q
|Q| = q12 + q22 + q32 + q42 where the positive root is assumed
yields the formula QQ = QQ = |Q|2 .
The linear algebra equivalent of the norm is the determinant of Q. Further, the absolute value has the multiplicative property |Q1 Q2 | = |Q1 ||Q2 | as verified by the multiplicative property of the determinant, det(Q1 Q2 ) = det Q1 det Q2 .
5
Other Properties of Quaternions
General Properties
Perhaps the most important non-axiomatic property of the quaternions is that they form a
division ring, simply defined as a ring where division is possible, i.e. a ring where every
nonzero element has a multiplicative inverse. As multiplicative commutativity does not
hold, we define P/Q as the solution of Q−1 of the following two equations where QL−1 is a
left quotient and Q−1
R is a right quotient, also known as inverses:
−1
QQ−1
L = P and QR Q = P.
As QQ = QQ = |Q|2 and |Q|2 /|Q|2 obviously equals the multiplicative identity 1, multiplying both sides of each equations by Q/|Q|2 yields
Q−1
L =
PQ
QP
−1
and
Q
=
R
|Q|2
|Q|2
For example, let P = k and Q = i. Since ij = (−j)i = k, P/Q yields Q−1
L = j and
−1
QR = −j.
In order to distill these two distinct quotients into one, let P = 1 to give the multiplicative inverse of a quaternion,
−1
−1
Q−1
=
L = QR = Q
11
Q
.
|Q|2
Similar to the previous example, let P = 1 and Q = i. Since i(−i) = (−i)i = 1, P/Q yields
−1
−1
Q−1
= −i. Since we know that i = −i and |i| = 1, we also get that
L = QR = −i, so Q
Q−1 = −i =
i
Q
=
.
2
|i|
|Q|2
We also know that all nonzero quaternions Q have an inverse Q−1 , the matrix inverse
of Q. From our original matrix definition of Q, if Q = q1 1 + q2 i + q3 j + q4 k 6= 0, then
−1
q1 + iq4 −q2 − iq3
−1
Q
=
q2 − iq3 q1 − iq4
1
q1 − iq4
−(−q2 − iq3 )
=
q1 + iq4
det q −(q2 − iq3 )
1
q1 − iq4 q2 + iq3
= 2
q1 + q22 + q32 + q42 −q2 + iq3 q1 + iq4
1
= 2
(q1 1 − q2 i − q3 j − q4 k).
2
q1 + q2 + q32 + q42
As this is purely algebraic, the verification of any possible example is left to the reader.
Quaternion multiplication is also distributive over addition, an algebraic proof of
which is left to the reader.
The subspace Hu of unit quaternions satisfies the condition that |Qu | = 1 for Qu ∈ Hu
as well as some other properties. One is that
Q−1
u = Qu .
Another is that
Qu = Ru cos φ + Pu sin φ = cos φ + Pu sin φ
where Ru = (1, 0, 0, 0) = 1 + i0 + j0 + k0 is a real unit quaternion, Pu = (0, p2 , p3 , p4 ) =
0 + ip2 + jp3 + kp4 is a vector unit quaternion parallel to the vector part of Qu , and φ is
an arbitrary real number. We also get that |Qu |2 = 1. Further, φ may be interpreted as the
quantified ratio of the real part to the magnitude of the vector part of a quaternion. This
may be viewed in the same way as complex numbers. Unit quaternions also satisfy the
following:
q12 + q22 + q32 + q42 = 1.
Thus, they form the 3-sphere S3 in the space R4 of all quadruplets (q1 , q2 , q3 , q4 ). It follows
from quaternion multiplication and the formula for Q−1 that the product of a unit quaternion is a unit quaternion, so S3 is a group under quaternion multiplication. As with the
1-sphere S1 of the unit complex numbers, S3 embodies a group of rotations, so we can
show how the unit quaternions may represent rotations in R3 .
Vector Properties and Quaternions as Rotations in 3-Space
The quaternions Q = (q1 + iq2 + jq3 + kq4 ) may be interpreted as having a real part, q1 , and
a vector part, (iq2 + jq3 + kq4 ), where the elements i, j, k may be interpreted geometrically
12
as unit vectors along the x, y, and z axes, respectively. The subspace Hr = {Qr = (q1 + i0 +
j0 + k0}) of real quaternions may be regarded as being equivalent to the real numbers,
Qr = {q ∈ R} since it shares the same properties. Likewise, the subspace Hv = {Qv =
(iq2 + jq3 + kq)} of vector quaternions may be regarded as being equivalent to ordinary
vectors, Qv = {q = (iqx + jqy + kqz ) : qx , qy , qz ∈ R}. These form a three-dimensional space
denoted by Ri + Rj + Rk, or R3 . This space is the orthogonal complement to the line R1,
simply R, of quaternions of the form q1 1, simply q1 , called the real quaternions. In other
words, we have the real axis with the real quaternions and three imaginary axes with the
vector quaternions. There are many combinations of multiplication that have interesting
properties. Many are outlined below.
The product of real quaternions is real; thus, this product is commutative and associative in that
Pr Qr = pq = qp = Qr Pr
and
(Pr Qr )Rr = (pq)r = p(qr) = Pr (Qr Rr ).
The product of a real and a vector quaternion is a vector. Here, commutativity holds:
Pr Qv = (0 + i(p1 q2 ) + j(p1 q3 ) + k(p1 q4 )) = (0 + i(q2 p1 ) + j(q3 p1 ) + k(q4 p1 ) = Qv Pr
We say that quaternions P and Q are parallel, notated P ||Q, if their vector parts, Pve =
(P − P )/2 and Qve = (Q − Q)/2, are parallel. So, if (R − R) = 0 where R = Pve Qve , then
P and Q are parallel. Using linear algebra, we also know that P Q ∈ R1 ⇐⇒ P × Q = 0.
This occurs if and only if P and Q are parallel.
Similarly, P and Q are perpendicular, notated P ⊥ Q, if Pve and Qve are perpendicular.
So, if R + R = 0 where R = Pve Qve , then P and Q are perpendicular. As with the previous
linear algebra solution to finding if P and Q are parallel, we know that P Q ∈ Ri + Rj +
Rk ⇐⇒ P · Q = 0. Again, this occurs if and only if P is orthogonal to Q.
The product of vector quaternions is
Pv Qv = −p · q + p × q
where ”·” and ”×” are the ”dot” and ”cross” products, respectively. Thus, the product of
two vector quaternions is a general quaternion except when Pv ||Qv and Pv ⊥ Qv where
Pv Qv equals −p · q and p × q, respectively.
Further, if Qu is unit vector quaternion, then
Q2u = Qu Qu = −Qu · Qu + Qu × Qu = −Qu · Qu + 0 = −|Qu |2 = −1.
This tells us that every unit vector in Ri + Rj + Rk is a square root of −1.
The cross product is neither commutative nor associative. Instead, we have the antisymmetric property:
P × Q = −Q × P.
Since the quaternion product is the difference between the cross product and the dot
product, and since the dot product is commutative and associative, this helps to inform
us as to why the quaternion product is also antisymmetric.
13
We also have the Jacobi identity for the cross product:
P × (Q × R) + R × (P × Q) + Q × (R × P ) = 0.
Understanding the product of a unit quaternion and a perpendicular vector quaternion is helpful in understanding how to rotate vectors in 3-space. Let Sv be a vector
quaternion, Qu be a unit quaternion, and Sv ⊥ Qu . Then
T = Qu Sv = (cos φ + Pu sin φ)Sv = Sv cos φ + Pu Sv sin φ
where Pu ||Sv , As Sv is a vector, the first term, Tv(1) , is also a vector such that Tv(1) ||Sv . As
Sv ⊥ Pu , the second terms, Tv(2) is also a vector such that Tv(2) ⊥ Sv and Tv(2) ⊥ Qu ||Pu .
As the product T is the sum of two vectors, it is also a vector. Since Tv(1) and Tv(2) both
lie in a plane perpendicular to Qu , Tv = Tv(1) + Tv(2) may be interpreted geometrically as a
rotation of Sv by an angle φ about an axis parallel to Qu in the plane perpendicular to Qu .
Now consider a different product
Uv = Tv Q−1
u = Tv Qu = Tv cos φ + Tv P u sin φ = Tv cos φ − Tv Pu sin φ = Tv cos φ + Pu Tv sin φ.
This is another rotation of angle φ about Qu . As the rotation φ is the same for these two
products,
−1
Uv = Tv Q−1
u = Qu Sv Qu
performs a rotation of Sv about Qu by an angle of 2φ. This may be shown more clearly
when we recall that quaternions have a real and imaginary part, written in the quaternion
analogue of Euler’s formula as Qu = cos φ + Uv sin φ where Uv is a purely imaginary unit
vector and thus in Ri + Rj + Rk. We then have the following:
Qu Sv Q−1
=
u
=
=
=
=
=
=
(cos φ + Pu sin φ)Sv (cos φ − Pu sin φ)
(cos φ + Pu sin φ)(Sv cos φ − Sv Pu sin φ)
Sv cos2 φ − Sv Pu cos φ sin φ + Pu Sv cos φ sin φ − Pu Sv Pu sin2 φ
Sv cos2 φ − Sv Pu cos φ sin φ − Sv Pu cos φ sin φ + Sv Pu2 sin2 φ
Sv cos2 φ − 2Sv Pu cos φ sin φ − Sv sin2 φ
Sv (cos2 φ − sin2 φ) − 2Sv Pu cos φ sin φ
Sv cos 2φ − Sv Pu sin 2φ
As Qu Sv Q−1
u rotates a perpendicular vector Sv about a unit quaternion Qu , let us now
consider this operation with an arbitrary vector Vv . If we decompose Vv = Qv + Sv where
Wv ||Qu and Sv ⊥ Su , then
−1
−1
−1
−1
Qu Vv Q−1
u = Qu (Wv + Sv )Qu = Qu Wv Qu + Qu Sv Qu = Qu Wv Qu + Uv
where Uv may be geometrically interpreted as Sv rotated about Qu by an angle of 2φ. To
understand the first time as well, note that Wv = zPu for some z ∈ R and some unit vector
Pu ||Qu since Wv ||Qu . Also note that Qu Pu = Pu Qu since the product of parallel quaternions
14
is a real quaternion equal to −p · q = −q · p, so multiplication is commutative. With all of
this in mind, we get that
−1
−1
−1
Qu Wv Q−1
u = Qu zPu Qu = zQu Pu Qu = zPu Qu Qu = zPu = Wv .
Therefore,
Qu Vv Q−1
u = Wv + Uv .
In a manner similar as before, this may be interpreted as a rotation of Vv about Qu by an
angle of 2φ as follows:
Figure 3: Arbitrary vector Vv is rotated by unit quaternion Qu about a unit vector Pu ||Qu
through angle 2φ
As we started with an arbitrary vector, this rotation is the same for all vectors including
the unit vectors of a coordinate system. Thus, it may transform the coordinates of any
reference frame into a new frame of different orientation.
This whole process is known as conjugation by Qu . In fact, conjugation by Q−1
u performs the same rotation, so rotation is uniquely determined by pairs (Pu , φ) and (−Pu , −φ)
with ±Qu as the only unit quaternions that induce this rotation. These are known as antipodal pairs of unit quaternions.
If Q1 and Q2 represent two unit quaternions, the multiplication of which with an arbitrary vector Vv yields an arbitrary rotation in 3-space, then applying them in succession
to Vv yields the following:
−1
−1
−1 −1
= Qi Vv Q−1
Q2 (Q1 Vv Q−1
1 )Q2 = (Q2 Q1 )Vv (Q1 Q2 ) = (Q2 Q1 )Vv (Q2 Q1 )
i
where Qi = Q2 Q1 may be interpreted as the successive composition of two rotations. As
multiplication is associative for quaternions, this property may be generalized for any
number of rotations. Since the inverse of a rotation is also a rotation, we know that rotations form a group.
In this reverse order of composition, Qi = ...Q2 Q1 , each successive rotation is relative
to the initial reference frame as shown in Figure 4.
If the composition of the rotations were in the forward order such that Qc = Q1 Q2 ...,
then each successive rotation is relative to the current reference frame as shown in Figure
5.
As an interesting aside and geometric application of quaternions, in addition to rotations being the composition of rotations, we may also define them in terms of reflections.
15
Figure 4: 90◦ rotations of a reference frame about the initial x, y, z axes, respectively
Figure 5: 90◦ rotations of a reference frame about its current x, y, z axes, respectively
For the sake of nominal simplicity, let Rot(φ) be a rotation about the origin and Ref (φ) be
a reflection across a line through the origin. As matrices, we may then represent them as
such
cos φ − sin φ
cos 2φ sin 2φ
Rot(φ) =
and Ref (φ) =
sin φ cos φ
sin 2φ − cos 2φ
in order show that Ref (φ)Ref (θ) = Rot(2(φ − θ)). With this fact, it may be shown that
the reflection of two lines through any point P that meets at angle φ/2 is a rotation:
Ref (φ)Ref (φ/2) = Rot(2(φ − φ/2)) = Rot(2(φ/2)) = Rot(φ).
Thus, quaternions may be represented in terms of reflections.
6
Quaternion Derivatives
Introduction
As with complex numbers, functions can be defined on the quaternions and, as such,
should have derivatives and integrals. Since the number of dimensions is greater than
that of the complex and reals, the theory behind derivatives must be adapted to fit. Moreover, given the interesting properties of the quaternions, notably its anti-symmetric property, we must properly and consistently define what a derivative actually is in order to
16
create a calculus which will lead to a quaternion analogue of Cauchy’s Theorem, perhaps
the most important theorem in complex analysis.
Eberly [4] defines different quaternionic functions under different representations in
order to show a more calculable derivative similar to what we would assume from the
calculus of real numbers. For example, the power of a unit quaternion is defined as
q t = (cos φ + u sin φ)t = eutφ = cos (tφ) + u sin(tφ)
while the natural logarithm of a unit quaternion is
log(q) = log(cos φ + u sin φ) = log(euφ ) = uφ.
While these simplifications are useful in terms of understanding how these functions act,
his attempt to define the derivative is incomplete. Therefore, we resort to Sudbery’s construction of Quaternionic Analysis [5] for a geometric and algebraic understanding of
the derivative and its applications, a more detailed way of providing a complex analysis equivalent of the quaternions than the previous analogues thus far. Since the proofs
of most of the theorems are at a higher level than this paper and not needed for understanding them and connecting them to complex analysis, refer to Sudbery’s paper for the
complete proofs.
The Quaternionic Derivative
The coordinate expression for the 3-form exterior derivative of a quaternion is
Dq = dx ∧ dy ∧ dz − i dt ∧ dy ∧ dz − j dt ∧ dz ∧ dx − k dt ∧ dx ∧ dy.
Geometrically, Dq(a, b, c) is a quaternion which is perpendicular to a, b, c ∈ H and has
a magnitude equal to the volume of the 3-dimensional parallelepiped whose edges are a,
b, and c. This may be algebraically expressed as
1
Dq(a, b, c) = 2 (cab − bac).
The dot product induces an R-linear map Γ : H∗ → H where H∗ = HomR (H, R), the
homomorphisms from H to R, is the dual of H given by
Γ(α) · q = α(q)
for α ∈ H∗ , q ∈ H. Further, since {1, i, j, k} is an orthonormal basis for H, we get
Γ(α) = α(1) + iα(i) + jα(j) + kα(k).
The set of R-linear maps of H into itself forms a two-sided vector space over H of dimension 4 is denoted F1 . As it is spanned over H by H∗ , Γ may be extended by linearity to a
right H-linear map Γr : F1 → H and a left H-linear map Γl : F1 → H given by
Γr (α) = α(1) + iα(i) + jα(j) + kα(k)
17
and
Γl (α) = α(1) + α(i)i + α(j)j + α(k)k
for any α ∈ F1 .
Since the differential of a quaternion-valued function on H is an element of F1 , the
maps Γr and Γl may be applied to it. The result is
Γr (df ) =
∂f
∂f
∂f
∂f
+i
+j
+k
∂t
∂x
∂y
∂z
Γl (df ) =
∂f
∂f
∂f
∂f
+
i+
j+
k.
∂t
∂x
∂y
∂z
and
We also get the following differential operators:
1 ∂f
∂f
1
∂ l f = 2 Γr (df ) =
+ en
,
2 ∂t
∂xn
1 ∂f
∂f
1
∂l f = 2 Γr (df ) =
− en
,
2 ∂t
∂xn
1 ∂f
∂f
1
∂ r f = 2 Γl (df ) =
+
en ,
2 ∂t
∂xn
∂f
1 ∂f
1
−
∂r f = 2 Γl (df ) =
en , and
2 ∂t
∂xn
∂ 2f
∂ 2f
∂ 2f
∂ 2f
+
+
+
∆f =
∂t2
∂x2
∂y 2
∂z 2
where en (for n = 1, 2, 3) stands for the basic quaternions i, j, and k and xn (for n = 1, 2, 3)
stands for the coordinates x, y, and z. Note that the partials and their moduli commute
and that
∆ = 4∂r ∂ r = 4∂l ∂ l .
This four-dimensional Laplace equation parallels the Complex Analysis two-dimensional
version. Further, the function is harmonic when ∆ = 0, i.e. when any of the partials or
conjugates of the partials equals 0.
With these preliminary functions, we may now begin to define specific quaternionic
functions in order to better understand all of these functions as a whole.
Definition: A quaternionic monomial is a function f : H → H of the form
f (q) = ao qa1 q...ar−1 qar
for some non-negative integer r (the degree of the monomial) and constant quaternions
a0 , ..., ar .
Definition: A quaternionic polynomial is a finite sum of quaternionic monomials.
Definition: A homogeneous polynomial function of degree r on H is a function f :
H → H of the form
f (q) = F (q, ..., q)
18
where F : H × ... × H (r times) → H is R-multilinear.
Definition: A polynomial function on H is a finite sum of homogeneous polynomial
functions of varying degrees. It is worth noting that every polynomial function on H may
be expressed as a quaternionic polynomial.
Definition: A function f : H → H is quaternion-differentiable on the left at q if the
limit
df
= lim [h−1 {f (q + h) − f (q)}]
dq h→0
exists. We may alternately define a function that is quaternion-differentiable on the right,
but the theory behind this parallels left-differentiability, so it makes sense to simply choose
one side since it follows from the antisymmetric property of the quaternions that only
some functions where multiplicative commutativity is not an issue will have the same
left and right derivatives. From this definition, we get the following
Theorem: If the function f is defined and quaternion-differentiable on the left throughout a connected open set U , then, on U , f has the form
f (q) = a + qb
for some a, b ∈ H.
This finding is important since it implies that the only quaternionic polynomials that
are quaternion-differentiable given our definition are linear quaternionic polynomials,
i.e. quaternionic polynomials with degree 1. Moreover, despite f being quaterniondifferentiable, it will not generally satisfy Cauchy’s theorem in the form
Z
dqf = 0
where the integral is around a closed curve. Moreover, the only functions that satisfy this
form of the theorem for all closed curves are constant functions as stated in the following
theorem:
Theorem: If the function f : U → H is real-differentiable in the connected open set U
and satisfies d(dqf ) = 0 in U , then f is constant on U .
Now that we know about differentiable functions, we may define what a regular function is, the equivalent of an analytic function in complex analysis:
Definition: A function f : H → H is left-regular at q ∈ H if it is real-differentiable at q
and there exists a quaternion fl0 (q) such that
d(dq ∧ dq f ) = −2Dqfl0 (q).
Definition: A function f : H → H is right-regular at q ∈ H if it is real-differentiable at
q and there exists a quaternion fr0 (q) such that
d(f dq ∧ dq) = −2fr0 (q)Dq.
Since the theory of left-regular functions and right-regular functions will be equivalent, let us consider only the left-regular functions, henceforth deemed regular. Further,
let f 0 (q) = fl0 (q) be the derivative of f at q.
19
7
Cauchy’s Theorem
With these definitions in place, we may begin to formulate the quaternionic analysis analogue of Cauchy’s Theorem:
Theorem (the Cauchy-Riemann-Fueter equations): A real-differentiable function f is
regular at q if and only if
Γr (dfq ) = 0.
This proof may be done similarly to the Cauchy-Riemann equations of complex analysis. We also get that, if f is regular and twice differentiable, then
∆f = 0,
so f is harmonic. Further, the derivative of a regular function may be characterized as the
limit of a difference quotient, a characterization similar to the definition of a derivative of
a complex-analytic function.
From complex analysis, we know that Cauchy’s Theorem was proved by Goursat by
filling in the open set with boxes and manipulating the sum of integrals over the boundaries of these boxes to equal 0. Since quaternions fill a greater dimension than complex
numbers, we must instead fill the open set with parallelepipeds where the sum of the
integrals over the boundaries of these parallelepipeds will be determined to be 0. Let us
then have the following:
Definition: An oriented k-parallelepiped in H is a map C : I k → H where I k ⊂ Rk is
the closed unit k-cube of the form
C(tq , ..., tk ) = q0 + t1 h1 + ... + tk hk .
q0 ∈ H is the original vertex of the parallelepiped and h1 , ..., hk ∈ H are its edge-vectors.
A k-parallelepiped is non-degenerate if its edge-vectors are linearly independent over
R. A non-degenerate 4-parallelepiped is positively oriented if v(h1 , h2 , h3 , h4 ) > 0 and
negatively oriented if v(h1 , h2 , h3 , h4 ) < 0.
From these definitions, we get the following equation useful in proving Cauchy’s Theorem:
Theorem: If f is regular at q0 and continuously differentiable in a neighborhood of
q0 , then, given > 0, there exists δ > 0 such that, if C is a non-degenerate oriented 3parallelepiped with q0 ∈ C(I 3 ) and q ∈ C(I 3 ) =⇒ |q − q0 | < δ, then
Z
−1 Z
0
Dq
dq ∧ dqf + 2f (q0 ) < C
δC
An analogue to complex analysis may be in the restatement as follows: If f is regular
at z0 , then, given > 0, there exists δ > 0 such that, if L is a directed line segment (i.e. an
oriented parallelepiped of codimension 1 in C) with zo ∈ L and z ∈ L =⇒ |z − z0 | < δ,
then
Z
−1 Z
0
dz
f
+
2f
(z
)
0 < .
L
δL
20
From these definitions and theorems, Sudbery goes on to find Cauchy’s Theorem and
integral formula for a 4-parallelepiped, arriving at a limited case of the more generalized
complex analysis equivalent, before he goes on to find its most general form. From this
process, however, we arrive at the following theorem:
Theorem: A function which is regular in an open set U is real-analytic in U .
This theorem enables us to find the most general form for Cauchy’s theorem for a
differentiable contour of integration:
Theorem (Cauchy’s theorem for a differentiable contour): If f is regular in an open
set U , and if C is a differentiable 3-chain in U which is homologous to 0 in the differentiable singular homology of U , i.e. C = ∂C 0 for some differentiable 4-chain C 0 , then
Z
Dq f = 0.
C
In this process, we find the quaternionic analysis equivalent of a winding number, a
very useful tool in complex analysis, and one required here to find the integral formula
for a differentiable contour:
Definition: Let q be any quaternion and let C be a closed 3-chain in H \ {q}. Then C
is homologous to a 4-chain C 0 : ∂I 4 → S where S is the unit sphere with center q. The
wrapping number of C about q is the degree of the map C 0 .
This leads us to the following:
Theorem (the integral formula for a differentiable contour): If f is regular in an
open set U , q0 ∈ U , and C is a differentiable 3-chain in U \ {q0 } which is homologous in
the differentiable singular homology of U \ {q0 } to a 3-chain whose image is ∂B for some
ball B ⊂ U , then
Z
(q − q0 )−1
1
Dq f (q) = n f (q0 )
2π 2 C |q − q0 |2
where n is the wrapping number of C about q0 .
This formula is extremely similar to the complex analysis integral formula, with one
large different of π 2 instead of π, a change due to the fact that integration occurs over a
volume rather than an area.
For a final generalization well above the level of this paper, Sudbery gives Cauchy’s
theorem and integral formula for a rectifiable contour:
Definition: Let C : I 3 → H be a continuous map of the unit 3-cube into H and let P :
0 = s0 < s1 < ... < sp = 1, Q : 0 = t0 < t1 < ... < tq = 1 and R : 0 = u0 < u1 < ... < ur = 1
be three partitions of the unit interval I. Define
σ(C; P, Q, R) =
p−1 q−1 r−1
X
XX
C(sl+1 , tm , un ) − C(sl , tm , un ),
Dq(
l=0 m=0 n=0
C(sl , tm+1 , un ) − C(sl , tm , un ),
C(sl , tm , un1 ) − C(sl , tm , un )).
C is a rectifiable 3-cell if there exists a real number M such that σ(C; P, Q, R) < M for
all partitions P , Q, R. If this is the case, then the least upper bound of the numbers
(C; P, Q, R) is called the content of C, denoted σ(C).
21
Let f and g be quaternion-valued functions defined on C(I 3 ). We say that f Dqg is
integrable over C if the sum
σ(C; P, Q, R) =
p−1 q−1 r−1
X
XX
C(sl+1 , tm , un ) − C(sl , tm , un ),
f (C(sl , tm , un ))Dq(
l=0 m=0 n=0
C(sl , tm+1 , un ) − C(sl , tm , un ),
C(sl , tm , un1 ) − C(sl , tm , un ))g(C(sl , tm , un )),
where sl ≤ sl ≤ sl+1 , sm ≤ tm ≤ tm+1 , and un ≤ un ≤ un+1 , has a limit in the sense of
Riemann-Stieltjes integration as |P |, |Q|, |R| → 0, where |P | = max0≤l≤p−1 |sl+1 −
R sl | measures the coarseness of the partition P . If this limit exists, then we denote it by C f Dqg.
We may extend these definitions to define rectifiable 3-chains and integrals over rectifiable 3-chains. In the same way as for rectifiable curves, we may show that f Dqg is
integrable over the 3-chain C if f and g are continuous, C is rectifiable, and
Z
f Dq g ≤ max |f | max |g| σ(C).
C
C
C
This is similar to the Complex Analysis finding pertaining to line integrals. Given
Z b
Z b
≤
g(t)dt
|g(t)|dt
a
a
and given a curve γ(t) : [a, b] → C with a length, we get that, where f (z) is continuous on
γ and γ is a simple curve,
Z
f (z)dz ≤ max |f (z)| (length of γ)
z∈γ
γ
We also get the weak form of Stoke’s theorem from this:
Theorem (Stoke’s theorem for a rectifiable contour): If C is a rectifiable 3-chain in H
with ∂C = 0, and if f and g are continuous functions defined in a neighborhood U of the
image of C and suppose that f Dqg = dω where ω is a 2-form on U , then
Z
f Dq g = 0.
C
With all of this, we may give the most general forms of Cauchy’s theorem and the
integral formula.
Theorem (Cauchy’s theorem for a rectifiable contour): If f is regular in an open set
U , and if C is a rectifiable 3-chain which is homologous to 0 in the singular homology of
U , then
Z
Dq f = 0.
C
Theorem (the integral formula for a rectifiable contour): If f is regular in an open
set U , and if q0 ∈ U and C is a rectifiable 3-chain in U \ {q0 } which is homologous in the
22
singular homology of U \ {q0 } to a differentiable 3-chain whose image is ∂B for some ball
B ⊂ U , then
Z
1
(q − q0 )−1
Dq f (q) = n f (q0 )
2π 2 C |q − q0 |2
where n is the wrapping number of C about q0 .
Although the mathematics behind these generalizations span more advanced topics
than previously covered (i.e. topology, more advanced algebra, etc.), the formulas are
very similar to those in complex analysis. Furthermore, as in complex analysis, weaker
forms of the theorem were discovered before more generalized versions were able to be
stated. We may now use this analogue of Cauchy’s Theorem, perhaps the most important
finding in all of complex analysis, to explore more properties of the quaternions.
Theorem (Liouville’s theorem): If f is any bounded entire function in H (i.e., if f is
any bounded function which is regular over the entirety of H), then f is constant.
This theorem follows from the fact that regular functions are harmonic and follows immediately from the Cauchy-Fueter integral formula. We also get the following theorem,
the converse of Cauchy’s theorem:
R
Theorem (Morera’s theorem): If f is continuous in an open set U , and if ∂C Dqf = 0
for every 4-parallelepiped C contained in U , then f is regular in U .
8
Construction of Regular Functions
From harmonic functions, we may construct regular functions in two ways. If f is harmonic, then we get that ∂l f is regular. Alternately, any real-valued harmonic function is
the real part of a regular function since those real values are the same. Regardless, after
the following definition, we arrive at another theorem:
Definition: A star-shaped set in H is a set such that, if there exists a q0 ∈ H, then for
all q ∈ H the line segment from q0 to q is in H.
Theorem: Let u be a real-valued function defined on a star-shaped open set U ⊆ H. If
u is harmonic and has continuous second derivatives, then there exists a regular function
f defined on U such that Re f = u.
In this proof, we get that the function
Z
f (q) = u(q) + 2 Pu
1
s2 ∂l u(sq)q ds
0
where Pu denotes the vector component of a quaternion, is regular in U for some U containing the origin, and is star-shaped with respect to the origin. In an adaptation where
U is instead star-shaped with respect to some other other point, a, we get the following
adaptation:
Z 1
f (q) = u(q) + 2 Pu
s2 ∂l u((1 − s)a + sq)(q − a) ds
0
.
An example of this importance is in the function u(q) = |q|−2 , the elementary potential
function of four dimensions, analogous to log |z| in the two-dimensional complex plane.
23
Thus, the regular function whose real part is |q|−2 is the analogue of log z. Take U =
H \ (R− ∪ {0}). Then U is star-shaped with respect to 1 and |Q|−2 is harmonic in U . For
1
q −1
u(q) = 2 , ∂l u(q) = − 2 , a = 1;
|q|
|q|
our formula for f (q) yields the following:
(
−(q Pu q)−1 −
f (q) =
where
arg q = log
q
|q|
1
|Pu q|2
arg q if Pu q 6= 0
1
if q ∈ R+
|q|2
Pu q
=
tan−1
|Pu q|
|Pu q|
Re q
,
an argument i times the argument generated by q in the complex plane.
A useful formula related to this and previous equations with t = Re q, r = Pu q, and
r = |r| is as follows:
e n xn
r2 + ten xn
−1 r
+
tan
,
L(q) = − 2 2
2r (r + t2 )
2r3
t
the derivative of which is also important,
∂l L(q) = G(q) =
q −1
.
|q|2
L(q) is thus a primitive for the function occurring in the Cauchy-Feuter integral formula in the same way that log z is a primitive for z −1 , the function occurring in Cauchy’s
integral formula.
The previous theorem shows that there are as many regular functions of a quaternion
variable as there are harmonic functions of four real variables, functions not limited to
simple algebraic functions such as powers.
Feuter also developed a method for constructing regular quaternion functions from
analytic complex functions. For each q ∈ H, let ηq : C → H be the embedding of the
complex numbers in the quaternions such that q is the image of a complex number ζ(q)
lying in the upper half-plane. That is,
ηq (x + iy) = x +
Pu q
y,
|Pu q|
ζ(q) = Re q + i|Pu q|.
From this, we get the following:
Theorem: Suppose f : C → C is analytic in the open set U ⊆ C. Define f˜ : H → H by
f˜(q) = ηq ◦ f ◦ ζ(q).
Then ∆f˜ is regular in the open set ζ −1 (U ) ⊆ H and its derivative is
∂l (∆f˜) = ∆f˜0 ,
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where f 0 is the derivative of the complex function f .
From the proof of this theorem, we develop some new useful formulas. Let t = Re q,
r = Pu q, r = |r, u(x, y)| = Re f (xi y), v(x, y) = Im f (xi y) such that
r
f˜(q) = u(t, r) + v(t, r).
r
Using the fact that u and v are harmonic functions, we get
2u2 (t, r) 2r v2 (t, r) v(t, r)
˜
∆f (q) =
+
−
.
r
r
r
r2
Recalling that
G(q) = ∂l (
1
e n xn
1
r2 + ten xn
−1 r
+
,
)
=
−∂
(
)
and
L(q)
=
−
tan
r
|q|2
|q|2
2r2 (r2 + t2 )
2r3
t
we get the following interesting examples:
If f (z) = z −1 , then ∆f˜(q) = −4G(q).
If f (z) = log z, then ∆f˜(q) = −4L(q).
9
Regular Functions and Conformal Mappings
The following section parallels the study of Mobius transformations in Complex Analysis
in regard to how complex-analytic functions may be mapped conformally:
Theorem: Let H ∗ = H ∪ {∞} be the one-point compactification of H. If the mapping
f : H ∗ → H∗ is conformal and orientation-preserving, then f is of the form
f (q) = (aq + b)(cq + d)−1
for some a, b, c, d ∈ H. Conversely, any such mapping is conformal and orientation preserving.
From this theorem, we also get that
dfq = (ac−1 d − b)(cq + d)−1 cdq(cq + d)−1 ,
a function which is a combination of dilatation and rotation. For this proof, we use this
fact to show that the set of all orientation-preserving conformal mappings are equal to the
set of all quaternionic Mobius transformations.
Although multiplicative commutativity is not preserved, an alternate version may be
proven for
f (q) = (qc + d)−1 (qa + b).
The following shows how a regular function, by a conformal transformation of the
variable, gives rise to other regular functions:
Theorem:
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(i) Given a function f : H → H, let If : H \ {0} → H be the function
If (q) =
q −1
f (q −1 ).
|q|2
If f is regular at q −1 , then If is regular at q.
(ii) Given a function f : H → H and quaternions a, b, let M (a, b, )f be the function
[M (a, b)f ](q) = bf (a−1 qb).
If f is regular at aqb, then M (a, b)f is regular at q.
(iii) Given a function f : H → H and a conformal mapping ν : q 7→ (aq+b)(cq+d)−1 ,
let M (ν)f be the function
[M (ν)f ](q) =
(cq + d)−1
1
f (ν(q)).
|b − ac−1 d|2 |cq + d|2
If f is regular at ν(q), then M (ν)f is regular at q.
Essentially, this theorem uses multiple mappings to gradually generate one regular
function from another.
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Further Analogues
Since the quaternions are so similar to the complex numbers, the analogues between the
two are endless. It is true that antisymmetric property of the quaternions and the intricacies of dealing with more dimensions prevents all of the theorems from holding in their
exact form, but analysts have extended the type of work we have shown thus far in order to reproduce the study of complex analysis as applied to the quaternions. Sudbery
goes on to show such analogues as Fourier analysis leading into a result that enables us
to generate harmonic function from any regular function, a study of specific subspaces
of H where we conclude that every regular polynomial has an antiderivative, and the
power series representation of regular functions arriving at the Laurent series for those
functions with isolated singularities. While many applications exist in computer science
and physics, quaternionic analysis as a branch of pure mathematics is ever-expanding
as evidenced by the myriad of research ever since Sudbery’s influential paper just a few
decades ago.
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References
[1] B. L. van der Waerden, Hamilton’s Discovery of Quaternions, Mathematics Magazine,
Vol. 49, No. 5 (Nov., 1976), pp. 227–234.
[2] V. Chi, Tutorial on Quaternions and Rotations in 3-Space: How it Works, preprint (1998).
[3] J Stillwell, Naive Lie Theory, Springer, New York, 2008.
[4] D. Eberly, Quaternion Algebra and Calculus, preprint (2010).
[5] A. Sudbery, Quaternionic Analysis, preprint (1977).
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