quaternionic analysis

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Quaternionic analysis

A. Sudbery [[email protected]]

Math. Proc. Camb. Phil. Soc. (1979), 85, 199-225

1. Introduction. The richness of the theory of functions over the complex field

makes it natural to look for a similar theory for the only other non-trivial real asso-

ciative division algebra, namely the quaternions. Such a theory exists and is quite

far-reaching, yet it seems to be little known. It was not developed until nearly a cen-

tury after Hamiltons discovery of quaternions. Hamilton himself (1) and his principal

followers and expositors, Tait (2) and Joly (3), only developed the theory of functions

of a quaternion variable as far as it could be taken by the general methods of the theory

of functions of several real variables (the basic ideas of which appeared in their mod-

ern form for the first time in Hamiltons work on quaternions). They did not delimit a

special class of regular functions among quaternion-valued functions of a quaternion

variable, analogous to the regular functions of a complex variable.

This may have been because neither of the two fundamental definitions of a regularfunction of a complex variable has interesting consequences when adapted to quater-

nions; one is too restrictive, the other not restrictive enough. The functions of a quater-

nion variable which have quaternionic derivatives, in the obvious sense, are just the

constant and linear functions (and not all of them); the functions which can be repre-

sented by quaternionic power series are just those which can be represented by power

series in four real variables.

In 1935, R. Fueter (4) proposed a definition of regular for quaternionic functions

by means of an analogue of the Cauchy-Riemann equations. He showed that this defi-

nition led to close analogues of Cauchys theorem, Cauchys integral formula, and the

Laurent expansion (5). In the next twelve years Fueter and his collaborators developed

the theory of quaternionic analysis. A complete bibliography of this work is contained

in (6), and a simple account (in English) of the elementary parts of the theory has been

given by Deavours (7).

The theory developed by Fueter and his school is incomplete in some ways, and

many of their theorems are neither so general nor so rigorously proved as present-day

standards of exposition in complex analysis would require. The purpose of this paper

is to give a self-contained account of the main line of quaternionic analysis which

remedies these deficiencies, as well as adding a certain number of new results. By

using the exterior differential calculus we are able to give new and simple proofs of

most of the main theorems and to clarify the relationship between quaternionic analysis

and complex analysis.

In Section 2 of this paper we establish our notation for quaternions, and introduce

the quaternionic differential formsdq,dq dqand Dq, which play a fundamental role

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in quaternionic analysis. The 1-formdqand the 3-formDqhave simple geometricalinterpretations as the tangent to a curve and the normal to a hypersurface, respectively.

Section 3 is concerned with the definition of a regular function. The remarks in the

second paragraph of this introduction, about possible analogues of the definition of a

complex-analytic function, are amplified (this material seems to be widely known, but

is not easily accessible in the literature); then Fueters definition of a regular function,

by means of an analogue of the Cauchy-Riemann equations, is shown to be equivalent

to the existence of a certain kind of quaternionic derivative. Just as, for a function

f : C C, the Cauchy-Riemann equation f/x+ i f/y = 0 (the variablebeingz = x +iy) is equivalent to the existence of a complex number f(z)such thatdf=f(z) dz, so for a functionf : H H, the Cauchy-Riemann-Fueter equation

f

t +i

f

x+ j

f

y +k

f

z = 0 (1.1)

(the variable beingq= t + ix +jy + kz) is equivalent to the existence of a quaternionf(q)such thatd(dq dqf) =Dq f(q).

Section 4 is devoted to the quaternionic versions of Cauchys theorem and Cauchys

integral formula. If the functionfis continuously differentiable and satisfies (1.1),Gausss theorem can be used to show that

C

Dq f= 0, (1.2)

whereCis any smooth closed 3-manifold in H, and that ifq0 lies insideC,

f(q0) = 1

22

C

(q q0)1|q q0|2 Dq f(q). (1.3)

We will show that Goursats method can be used to weaken the conditions on the con-

tourCand the functionf, so thatCneed only be assumed to be rectifiable and thederivatives off need not be continuous. From the integral formula (1.3) it follows,as in complex analysis, that iff is regular in an open set U then it has a power se-ries expansion about each point ofU. Thus pointwise differentiability, together withthe four real conditions (1.1) on the 16 partial derivatives off, are sufficient to ensurereal-analyticity.

In Section 5 we show how regular functions can be constructed from functions

of more familiar type, namely harmonic functions of four real variables and analytic

functions of a complex variable, and how a regular function gives rise to others by

conformal transformation of the variable.

The homogeneous components in the power series representing a regular function

are themselves regular; thus it is important to study regular homogeneous polynomials,

the basic functions from which all regular functions are constructed. The corresponding

functions of a complex variable are just the powers of the variable, but the situation

with quaternions is more complicated. The set of homogeneous regular functions of

degreenforms a quaternionic vector space of dimension 12

(n+ 1)(n+ 2); this is truefor any integern if for negative n it is understood that the functions are defined andregular everywhere except at 0. The functions with negative degree of homogeneity

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correspond to negative powers of a complex variable and occur in the quaternionic

Laurent series which exists for any function which is regular in an open set except at

one point. Fueter found two natural bases for the set of homogeneous functions which

play dual roles in the calculus of residues. (He actually only proved that these bases

form spanning sets.) In Section 6 we study homogeneous regular functions by means of

harmonic analysis on the unit sphere in H, which forms a group isomorphic toSU(2);this bears the same relation to quaternionic analysis as the theory of Fourier series does

to complex analysis. In Section 7 we examine the power series representing a regular

function and obtain analogues of Laurents theorem and the residue theorem.

Many of the algebraic and geometric properties of complex analytic functions are

not present in quaternionic analysis. Because quaternions do not commute, regular

functions of a quaternion variable cannot be multiplied or composed to give further reg-

ular functions. Because the quaternions are four-dimensional, there is no counterpart

to the geometrical description of complex analytic functions as conformal mappings.

The zeros of a quaternionic regular function are not necessarily isolated, and its range

is not necessarily open; neither of these sets need even be a submanifold ofH. There is

a corresponding complexity in the structure of the singularities of a quaternionic regu-

lar function; this was described by Fueter (9), but without giving precise statements or

proofs. This topic is not investigated here.

2. Preliminaries. We denote the four-dimensional real associative algebra of the

quaternions by H, its identity by 1, and we regard R as being embedded in H byidentifyingt Rwith1 H. Then we have a vector space direct sum H = R P,whereP is an oriented three-dimensional Euclidean vector space, and with the usualnotation for three-dimensional vectors the product of two elements ofPis given by

ab= a.b + a b. (2.1)

We choose an orthonormal positively oriented basis{i,j,k} for P, and write atypical quaternion as

q= t +ix +jy +kz (t,x,y,z R). (2.2)

On occasion we will denote the basic quaternions i ,j, k by ei and the coordinatesx,y,z by xi (i = 1, 2, 3) and use the summation convention for repeated indices.Then (2.2) becomes

q= t+eixi (2.3)and the multiplication is given by

eiej = ij+ ijkek. (2.4)

We will also sometimes identify the subfield spanned by1andiwith the complex fieldC, and write

q= v +jw (v, w C) (2.5)wherev = t +ixandw = y iz. The multiplication law is then

vj =j v (2.6)

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for allv C.We write

q= t ix jy kz, (2.7)|q| = qq=

t2 +x2 +y2 +z2 R, (2.8)

Re q= 12

(q+ q) =t R, (2.9)Pu q= 1

2(q q) =ix+jy +kz P, (2.10)

Un q= q

|q

| S, (2.11)

whereSis the unit sphere in H; and

q1, q2 = Re(q1q2) =t1t2+x1x2+y1y2+z1z2. (2.12)

Then we have

q1q2= q2q1, (2.13)

|q1q2| = |q1| |q2|, (2.14)Re(q1q2) = Re(q2q1) (2.15)

and

q1 = q

|q

|2

. (2.16)

Note that ifu1 andu2 are unit quaternions, i.e. |u1| =|u2| = 1, the mapqu1qu2 is orthogonal with respect to the inner product (2.12) and has determinant 1;conversely, any rotation ofH is of the formq u1qu2 for someu1, u2 H (see, forexample, (10), chap. 10).

The inner product (2.12) induces an R-linear map : H H, where

H = HomR(H,R)

is the dual vector space to H, given by

(), q =(q) (2.17)

forH,q

H. Since

{1, i , j , k

}is an orthonormal basis for H, we have

() =(1) +i (i) +j (j) +k (k).

The set ofR-linear maps from H to H forms a two-sided vector space over H of di-

mension4, which we will denote by F1. It is spanned (over H) by H, so the map

can be extended by linearity to a right H-linear mapr :F1 H and a left-linear map

l : F1 H.They are given by

r() =(1) +i (i) +j (j) +k (k) (2.18)

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and

l() =(1) +(i) i+(j)j+(k) k (2.19)

for any F1.The geometric terminology used in this paper is as follows:

An orientedk-parallelepipedin H is a map C : Ik H, whereIk Rk is theclosed unitk-cube, of the form

C(t1,...,tk) =q0+t1h1+...+tkhk.

q0

H is called theoriginal vertexof the parallelepiped, andh1,...,hk

H are called

its edge-vectors. A parallelepiped isnon-degenerate if its edge-vectors are linearly

independent (over R). A non-degenerate 4-parallelepiped ispositively orientedif

v(h1,...,h4)> 0,

negatively oriented ifv(h1,...,h4) < 0, wherev is the volume form defined below(equation (2.26)).

We will sometimes abuse notation by referring to the image C(Ik)as simplyC.Quaternionic differential forms. When it is necessary to avoid confusion with other

senses of differentiability which we will consider, we will say that a functionf : H H isreal-differentiable if it is differentiable in the usual sense. Its differential at a point

q H is then an R-linear mapdfq : H H. By identifying the tangent space at eachpoint ofH with H itself, we can regard the differential as a quaternion-valued 1-form

df= f

t dt+

f

xdx +

f

ydy+

f

zdz. (2.20)

Conversely, any quaternion-valued 1-form = a0dt +aidxi(a0, ai H) can beregarded as the R-linear map : H H given by

(t+xiei) =a0t+aixi (2.21)

Similarly, a quaternion-valued r-form can be regarded as a mapping from H to thespace of alternating R-multilinear maps from H ... H (rtimes) to H. We define theexterior product of such forms in the usual way: if is anr-form andis ans-form,

(h1,...,hr+s) = 1

r!s!

()(h(1),...,h(r))(h(r+1),...,h(r+s)), (2.22)

where the sum is over all permutations ofr + s objects, and() is the sign of.Then the set of allr-forms is a two-sided quaternionic vector space, and we have

a( ) = (a) ,( )a= (a),(a) = (a)

(2.23)

for all quaternionsa, r-forms and s-forms . The space of quaternionicr-formshas a basis of realr-forms, consisting of exterior products of the real 1-formsdt,dx,

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dy,dz ; for such forms left and right multiplication by quaternions coincide. Note thatbecause the exterior product is defined in terms of quaternion multiplication, which is

not commutative, it is not in general true that = for quaternionic 1-formsand.

The exterior derivative of a quaternionic differential form is defined by the usual

recursive formulae, and Stokess theorem holds in the usual form for quaternionic in-

tegrals.

The following special differential forms will be much used in the rest of the paper.

The differential of the identity function is

dq= dt+i dx+j dy +k dz; (2.24)

regarded as an R-linear transformation ofH, dq is the identity mapping. Its exteriorproduct with itself is

dq dq= 12ijkeidxj dxk =i dy dz+j dz dx+k dx dy, (2.25)

which, as an antisymmetric function on HH, gives the commutator of its arguments.For the (essentially unique) constant real 4-form we use the abbreviation

v= dt dx dy dz, (2.26)

so thatv(1, i , j , k) = 1. Finally, the 3-formDqis defined as an alternating R-trilinear

function by h1, Dq(h2, h3, h4) =v(h1, h2, h3, h4) (2.27)for all h1,...,h4 H. Thus Dq(i,j,k) = 1 and Dq(1, ei, ej) = ijkek. Thecoordinate expression forDqis

Dq= dx dy dz 12ijkei dt dxj dxk=dx dy dz i dt dy dz j dt dz dx k dt dx dy. (2.28)

Geometrically,Dq(a,b,c)is a quaternion which is perpendicular to a,b andc and hasmagnitude equal to the volume of the 3-dimensional parallelepiped whose edges area,bandc. It also has the following algebraic expression:

PROPOSITION 1.Dq(a,b,c) = 12

(cab bac).Proof. For any unit quaternionu, the mapq uqis an orthogonal transformation

ofH with determinant1; hence

Dq(ua,ub,uc) = u Dq(a,b,c).

Takingu= |a|a1, and using the R-trilinearity ofDq, we obtain

Dq(a,b,c) = |a|2a Dq(1, a1b, a1c). (2.29)

Now sinceDq(1, ei, ej) = ijkek = 12(ejei eiej), we have by linearity

Dq(1, h1, h2) = 12

(h2h1 h1h2) (2.30)

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for allh1, h2 H. HenceDq(a,b,c) = 1

2|a|2a(a1ca1b a1ba1c)

= 12

(cab bac). Two useful formulae were obtained in the course of this proof. The argument lead-

ing to (2.29) can be generalized, using the fact that the mapquqv is a rotation forany pair of unit quaternionsu,v, to

Dq(ah1b,ah2b,ah3b) =

|a

|2

|b

|2a Dq(h1, h2, h3)b; (2.31)

and the formula (2.30) can be written as

1Dq= 12dq dq, (2.32)

wheredenotes the usual inner product between differential forms and vector fieldsand1denotes the constant vector field whose value is 1.

Since the differential of a quaternion-valued function onHis an element ofF1, themapr can be applied to it. The result is

r(df) = f

t +i

f

x+j

f

y +k

f

z. (2.33)

We introduce the following notation for the differential operator occurring in (2.33),and for other related differential operators:

lf= 12

r(df) = 1

2

f

t +ei

f

xi

,

lf = 1

2

f

t ei f

xi

,

rf= 12

l(df) = 1

2

f

t +

f

xiei

,

rf = 1

2

f

t f

xiei

,

f = 2f

t2 +2f

x2 +2f

y2 +2f

z2 .

(2.34)

Note thatl, l,r and r all commute, and that

= 4rr = 4ll. (2.35)

3. Regular functions. The requirement that a function of a complex variable z =x+iy should be a complex polynomial, i.e. a sum of terms anzn, picks out a propersubset of the polynomial functionsf(x, y) + ig(x, y). The corresponding requirementof a function of a quaternion variable q = t+ ix + j y+ kz, namely that it shouldbe a sum of monomialsa0qa1...ar1qar, places no restriction on the function; for in

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h

t =i

h

x = g

y =i

g

z.

In terms of complex derivatives, these can be written as

g

v =

h

w =

h

v =

g

w = 0, (3.4)

g

v =

h

w (3.5)

andh

v = g

w . (3.6)

Equation (3.4) shows that g is a complex-analytic function ofv and w, and h is acomplex-analytic function ofv andw. Hence by Hartogss theorem ((11), p. 133)gandh have continuous partial derivatives of all orders and so from (3.5)

2g

v2 =

v

h

w

=

w

h

v

= 0.

Suppose for the moment that Uis convex. Then we can deduce thatg is linear in w,his linear inw and h is linear inv. Thus

g(v, w) =+v +w+vw,

h(v, w) =+v+w +vw,

where the Greek letters represent complex constants. Now (3.5) and (3.6) give the

following relations among these constants:

= , = , = = 0.Thus

f=g +jh = +j + (v+jw)(j)=a+qb,

where a= +j and b= j; sofis of the stated form ifUis convex. The generalconnected open set can be covered by convex sets, any two of which can be connected

by a chain of convex sets which overlap in pairs; comparing the forms of the function

fon the overlaps, we see thatf(q) =a +qb with the same constantsa,b throughoutU.

We will now give a definition of regular for a quaternionic function which is

satisfied by a large class of functions and which leads to a development similar to the

theory of regular functions of a complex variable.

Definition. A function f : H H is left-regularatq H if it is real-differentiableatqand there exists a quaternionfl (q)such that

d(dq dq f) =Dq fl (q). (3.7)It isright-regularif there exists a quaternionfr(q)such that

d(f dq dq) =fr(q) Dq.

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Clearly, the theory of left-regular functions will be entirely equivalent to the theory

of right-regular functions. For definiteness, we will only consider left-regular func-

tions, which we will call simply regular. We writefl (q) =f(q)and call it thederiva-

tiveoffatq.An application of Stokess theorem gives the following characterization of the

derivative of a regular function as the limit of a difference quotient:

PROPOSITION 2. Suppose thatfis regular atq0 and continuously differentiablein a neighbourhood ofq0. Then given >0, there exists >0 such that ifCis a non-degenerate oriented 3-parallelepiped withq0 C(I3)andq C(I3) |q q0| < ,then

C

Dq

1

C

dq dq f

f(q0) < .

The corresponding characterization of the derivative in terms of the values of the

function at a finite number of points is

f(q0) = limh1,h2,h30

[Dq(h1, h2, h3)1{(h1h2 h2h1)(f(q0+h3) f(q0))

+(h2h3 h3h2)(f(q0+h1) f(q0))+(h3h1 h1h3)(f(q0+h2) f(q0))}].

(3.8)This is valid if it is understood thath1,h2,h3 are multiples of three fixed linearly

independent quaternions,hi= tiHi, and the limit is taken ast1, t2, t3

0.

By writing (3.7) asdq dq df=Dq f(q)

and evaluating these trilinear functions with arguments(i,j,k)and(1, i , j), we obtaintwo equations which give an expression for the derivative as

f = 2lf = ft

+i f

x+j

f

y +k

f

z (3.9)

and also

PROPOSITION 3. (the Cauchy-Riemann-Fueter equations). A real-differentiable

functionfis regular atqif and only iflf= 0, i.e.

f

t +i

f

x+j

f

y +k

f

z = 0. (3.10)

If we writeq= v +jw,f(q) =g(v, w) + jh(v, w)as in Theorem 1, (3.10) becomesthe pair of complex equations

g

v =

h

w,

g

w = h

v, (3.11)

which can be seen as a complexification of the Cauchy-Riemann equations for a func-

tion of a complex variable.

From Proposition 3 and (2.35) it follows that iffis regular and twice differentiable,thenf= 0, i.e.fis harmonic. We will see in the next section that a regular functionis necessarily infinitely differentiable, so all regular functions are harmonic.

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4. Cauchys theorem and the integral formula. The integral theorems for regular

quaternionic functions have as wide a range of validity as those for regular complex

functions, which is considerably wider than that of the integral theorems for harmonic

functions. Cauchys theorem holds for any rectifiable contour of integration; the in-

tegral formula, which is similar to Poissons formula in that it gives the values of a

function in the interior of a region in terms of its values on the boundary, holds for

a general rectifiable boundary, and thus constitutes an explicit solution to the general

Dirichlet problem.

The algebraic basis of these theorems is the equation

d(g D q f ) =dg Dq f g Dq df= {(rg)f+ g(lf)}v,

(4.1)

which holds for any differentiable functionsfandg. Takingg = 1and using Proposi-tion 3, we have:

PROPOSITION 4.A differentiable functionfis regular atqif and only if

Dq dfq = 0.From this, together with Stokess theorem, it follows that iffis regular and contin-

uously differentiable in a domainD with differentiable boundary, thenD

Dq f= 0.

As in complex analysis, however, the conditions onfcan be weakened by using Gour-sats dissection argument. Applying this to a parallelepiped, we obtain

LEMMA 1.Iffis regular at every point of the 4-parallelepiped C,C

Dq f= 0. (4.2)

The dissection argument can also be used to prove the Cauchy-Fueter integral for-

mula for a parallelepiped:

LEMMA 2.Iffis regular at every point of the positively oriented 4-parallelepipedC, andq0is a point in the interior ofC,

f(q0) = 1

22

C

(q q0)1

|q q0|2

Dq f(q). (4.3)

Proof. In (4.1) takeg(q) = (qq0)1

|qq0|2 = r

1|qq0|2

.

Then gis differentiable except at q0, andrg= 0; hence iffis regular d(g D q f ) =0except atq0. A dissection argument now shows that in the above integralCcan bereplaced by any smaller 4-parallelepipedC with q0 int C C, and since f iscontinuous atq0 we can chooseC so small thatf(q)can be replaced by f(q0). Sincethe 3-formg Dqis closed and continuously differentiable in H {q0}, we can replaceC

g Dqby the integral over a 3-sphereSwith centre atq0, on which

Dq= (q q0)|q q0| dS,

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wheredSis the usual Euclidean volume element on the 3-sphere. Hence

C

(q q0)1|q q0|2 Dq f(q) =

S

dS

|q q0|3 f(q0) = 22f(q0).

We will use the following special notation for the function occurring in the integral

formula:

G(q) =q1

|q|2 .

This function is real-analytic except at the origin; hence in (4.3) the integrand is a

continuous function of(q, q0)inC int Cand, for each fixed q C, a real-analyticfunction ofq0inint C. It follows ((12), p. 7) that the integral is a real-analytic functionofq0 in int C. Thus we have

THEOREM 1.A function which is regular in an open setUis real-analytic inU.This makes it valid to apply Stokess theorem and so obtain Cauchys theorem for

the boundary of any differentiable 4-chain. It can be further extended to rectifiable

contours, defined as follows:

Definition. LetC :I3 H be a continuous map of the unit 3-cube into H, and letP : 0 = s0< s1< ... < sp= 1,Q: 0 =t0< t1< ... < tq = 1and

R: 0 =u0< u1< ... < ur= 1

be three partitions of the unit intervalI. Define

(C; P,Q,R) =

p1l=0

q1m=0

r1n=0

Dq(C(sl+1, tm, un) C(sl, tm, un),

C(sl, tm+1, un) C(sl, tm, un),C(sl, tm, un+1) C(sl, tm, un)).

Cis a rectifiable 3-cell if there is a real number Msuch that (C; P,Q,R)< Mfor allpartitions P, Q,R. If this is the case the least upper bound of the numbers (C; P,Q,R)is called thecontentofCand denoted by(C).

Letf andg be quaternion-valued functions defined onC(I3). We say thatf Dq gis integrableoverCif the sum

p1l=0

q1m=0

r1n=0

f(C(sl, tm,un))Dq(C(sl+1, tm, un) C(sl, tm, un),

C(sl, tm+1, un) C(sl, tm, un),C(sl, tm, un+1) C(sl, tm, un))g(C(s, tm,un)),

wheresl sl sl+1, tm tm tm+1 andun un un+1, has a limit in thesense of Riemann-Stieltjes integration as |P|, |Q|, |R| 0, where

|P| = max0lp1

|sl+1 sl|

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measures the coarseness of the partition P. If this limit exists, we denote it byC

f Dq g.We extend these definitions to define rectifiable 3-chains and integrals over rectifi-

able 3-chains in the usual way.

Just as for rectifiable curves, we can show that f Dq gis integrable over the 3-chainCiff andg are continuous andCis rectifiable, and

C

f Dq g

(maxC |f|)(maxC |g|)(C).Furthermore, we have the following weak form of Stokess theorem:

STOKESS THEOREM FOR A RECTIFIABLE CONTOUR.LetCbe a rectifi-able 3-chain in H withC= 0, and supposefandg are continuous functions definedin a neighbourhoodUof the image ofC, and thatf Dq g = d where is a 2-formonU. Then

C

f Dq g = 0.

The proof proceeds by approximatingCby a chain of 3-parallelepipeds with vertices atthe pointsCa(sl, tm, un)whereCa is a 3-cell inCandsl,tm,unare partition pointsin I. Stokess theorem holds for this chain of 3-parallelepipeds and we can use thesame argument as for rectifiable curves (see, for example, ( 13), p. 103).

We can now give the most general forms of Cauchys theorem and the integral

formula.

THEOREM 2. (Cauchys theorem for a rectifiable contour).Supposef is regular in an open setU, and letCbe a rectifiable 3-chain which ishomologous to0 in the singular homology ofU. Then

C

Dq f= 0.

Proof. First we prove the theorem in the case whenU is contractible. In this case,sinced(Dq f) = 0 andfis continuously differentiable (by Theorem 1), Poincareslemma applies and we haveDq f =d for some 2-form on D. But C= 0, so byStokess theorem

C

Dq f = 0.In the general case, supposeC=C whereC is a 4-chain inU. We can dissect

C as

C

=n C

n,

where each Cnis a 4-cell lying inside an open ball contained in Uand Cnis rectifiable.

Hence by the first part of the theoremC

n

Dqf= 0, and therefore

C

Dq f =n

C

n

Dqf = 0.

For the general form of the integral formula, we need an analogue of the notion of

the winding number of a curve round a point in the plane. Letqbe any quaternion,and letCbe a 3-cycle in H {q}. ThenCis homologous to n C0, whereC0 is a

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positively oriented 4-parallelepiped in H{q}, andnis an integer (independent of thechoice ofC0), which we will call thewrapping numberofCaboutq.

THEOREM 3 (the integral formula for a rectifiable contour).

Suppose f is regular in an open setU. Letq0 be a point in U, and letC be arectifiable 3-chain which is homologous, in the singular homology ofU {q0}, to adifferentiable 3-chain whose image isBfor some ballB U. Then

1

22

C

(q q0)1|q q0|2 Dq f(q) =nf(q0)

wheren is the wrapping number ofCaboutq0

.

Many of the standard theorems of complex analysis depend only on Cauchys inte-

gral formula, and so they also hold for quaternionic regular functions. Obvious exam-

ples are the maximum-modulus theorem (see, for example, (14), p. 165 (first proof))

and Liouvilles theorem ((14), p. 85 (second proof)). Moreras theorem also holds for

quaternionic functions, but in this case the usual proof cannot easily be adapted. It can

be proved (8) by using a dissection argument to show that iffis continuous in an opensetUand satisfies

C

Dq f= 0

for every 4-parallelepipedCcontained inU, thenfsatisfies the integral formula; andthen arguing as for the analyticity of a regular function.

5. Construction of regular functions. Regular functions can be constructed from

harmonic functions in two ways. First, iff is harmonic then (2.35) shows thatlf isregular. Second, any real-valued harmonic function is, at least locally, the real part of

a regular function:

THEOREM 4. Letu be a real-valued function defined on a star-shaped open setU H. Ifu is harmonic and has continuous second derivatives, there is a regular

functionfdefined onUsuch thatRe f=u.Proof. Without loss of generality we may assume that Ucontains the origin and is

star-shaped with respect to it. In this case we will show that the function

f(q) =u(q) + 2Pu

10

s2lu(sq)q ds (5.1)

is regular inU.

Since

Re

10

s2lu(sq)q ds= 1

2

10

s2

tu

t(sq) +xi

u

xi(sq)

ds

= 1

2

10

s2 d

ds[u(sq)] ds

= 12

u(q) 10

su(sq) ds,

we can write

f(q) = 2

10

s2lu(sq)q ds+ 2

10

su(sq) ds. (5.2)

14


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Sinceu andlu have continuous partial derivatives in U, we can differentiate underthe integral sign to obtain, forq U,

lf(q) = 2

10

s2l[lu(sq)]q ds+

10

s2{lu(sq)+eilu(sq)ei} ds+2 10

s2lu(sq) ds.

But l[lu(sq)] = 14su(sq) = 0 sinceuis harmonic inU, and

lu(sq) +eilu(sq)ei= 2lu(sq)=

2lu(sq) sinceu is real.

Hencelf = 0 inUand sofis regular. If the regionUis star-shaped with respect not to the origin but to some other point

a, formulae (5.1) and (5.2) must be adapted by changing origin, thus:

f(q) =u(q) + 2Pu

10

s2lu((1 s)a+sq)(q a) ds (5.3)

= 2

10

s2lu((1 s)a+sq)(q a) ds+ 2 10

su((1 s)a+sq) ds. (5.4)

An example which can be expected to be important is the case of the function

u(q) = |q|2.This is the elementary potential function in four dimensions, aslog |z| is in the com-plex plane, and so the regular function whose real part is |q|2 is an analogue of thelogarithm of a complex variable.

We takeU to be the whole ofH except for the origin and the negative real axis.ThenUis star-shaped with respect to1, and |q|1 is harmonic inU. Put

u(q) = 1

|q|2 , lu(q) = q1

|q|2 , a= 1;

then (5.3) gives

f(q) = (qPu q)1 1

|Pu q

|2

arg q ifPu q= 0

= 1|q|2 ifqis real and positive,

(5.5)

where

arg q= log(Un q) = Pu q

| Pu q| tan1

| Pu q|Re q

, (5.6)

which isi times the usual argument in the complex plane generated by q. (In practicethe formulae (5.3) and (5.4) are not very convenient to use, and it is easier to obtain

(5.5) by solving the equations

.F= 2t(t2 +r2)2

15


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andF

t + F= 2r

(t2 +r2)2,

wheret = Re q, r = Pu qand r =|r|these express the fact that F : HP is thepure quaternion part of a regular function whose real part is |q|2and assuming thatFhas the formF(r)r.)

We will denote the function (5.5) by2L(q). The derivative ofL(q) can mosteasily be calculated by writing it in the form

L(q) = r2 +teixi2r2(r2 +t2)+

eixi2r3 tan

1 rt

; (5.7)

the result is

lL(q) =G(q) = q1

|q|2 . (5.8)

ThusL(q)is a primitive for the function occurring in the Cauchy-Fueter integral for-mula, just as the complex logarithm is a primitive for z1, the function occurring inCauchys integral formula.

Theorem 4 shows that there are as many regular functions of a quaternion variable

as there are harmonic functions of four real variables. However, these functions do not

include the simple algebraic functions, such as powers of the variable, which occur as

analytic functions of a complex variable. Fueter (4) also found a method for construct-

ing a regular function of a quaternion variable from an analytic function of a complexvariable.

For each q H, let q : C H be the embedding of the complex numbers inthe quaternions such thatqis the image of a complex number (q) lying in the upperhalf-plane; i.e.

q(x+iy) = x+ Pu q

| Pu q|y, (5.9)

(q) = Re q+i| Pu q|. (5.10)Then we have

THEOREM 5. Supposef : C C is analytic in the open setU C, and definef : H H by

f(q) =q

f

(q). (5.11)

Thenfis regular in the open set1(U) H, and its derivative isl(f) = f

, (5.12)

wheref is the derivative of the complex functionf.For a proof, see (7). Note that if we writef(x + iy) = u(x, y) + iv(x, y),t= Re q

andr = Pu q, then

f(q) =u(t, r) +Pu q

r v(t, r), (5.13)

f(q) = 2u2(t, r)

r +

2 Pu q

r

v2(t, r)

r v(t, r)

r2

, (5.14)

16


17/28

where the suffix 2 denotes differentiation with respect to the second argument.

Functions of the form fhave been taken as the basis of an alternative theory offunctions of a quaternion variable by Cullen (15). The following examples are inter-

esting: when

f(z) = z1, f(q) = 4G(q); (5.15)when

f(z) = log z, f(q) = 4L(q) (5.16)Given a regular function f, other regular functions can be constructed from it by

composing it with conformal transformations. The special cases of inversion and rota-

tion are particularly useful:

PROPOSITION 5. (i)Given a functionf : H H, letI f : H {0} H be thefunction

If(q) = q1

|q|2 f(q1). (5.17)

Iffis regular atq1,I fis regular atg.(ii)Given a function f : H H and constant quaternions a, b, letM(a, b)fbe the

function

[M(a, b)f](q) =bf(a1qb). (5.18)

Iffis regular ata1qb,M(a, b)fis regular atq.Proof. (i) By Proposition 4 it is sufficient to show that

Dq d(If)q = 0.NowI f = G(f ), whereG(q) = q1/|q|2 and : H {0} H is the inversionq q1. Hence

Dq d(If)q =Dq dGqf(q1) +Dq G(q)d(f )q=Dq G(q) qdfq1

sinceGis regular atq= 0. ButqDq(h1, h2, h3) =Dq(q1h1q1, q1h2q1, q1h3q1)

=

q1

|q|4Dq(h1, h2, h3)q

1

by (2.31). Thus

Dq G(q) = |q|2qqDqand so

Dq d(If)q = |q|2qq(Dq dfq1)= 0

iffis regular atq1.(ii) Let : H H be the mapq aqb. Then by (2.31)

Dq= |a|2|b|2a D q b

17


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and soDq d[M(a, b)f]q =Dq bqdf(q)

= |a|2|b|2a1(qDq)b1 bqdf(q)= |a|2|b|2a1q(Dq df(q))= 0

if f is regular at (q). It follows from Proposition 4 that M(a, b)f is regular atq.

The general conformal transformation of the one-point compactification ofH is of

the form (q) = (aq+b)(cq+d)1 (5.19)

witha1b= c1d. Such a transformation ((16), p. 312) is the product of a sequenceof transformations of the types considered in Proposition 5, together with translations

q q+ a (which clearly preserve regularity). The corresponding transformation ofregular functions is as follows:

THEOREM 6.Given a functionf : H H and a conformal transformationasin (5.19), letM()fbe the function

[M()f](q) = 1

|b ac1d|2(cq+d)1

|cq+d|2 f((q)).

Iffis regular at(q),M()fis regular atq.6. Homogeneous regular functions. In this section we will study the relations be-

tween regular polynomials, harmonic polynomials and harmonic analysis on the group

S of unit quaternions, which is to quaternionic analysis what Fourier analysis is tocomplex analysis.

The basic Fourier functions ein andein, regarded as functions on the unit circlein the complex plane, each have two extensions to harmonic functions on C {0};thus we have the four functionszn, zn,zn and zn. The requirement of analyticitypicks out half of these, namely zn and zn. In the same way the basic harmonicfunctions onS, namely the matrix elements of unitary irreducible representations ofS, each have two extensions to harmonic functions on H {0}, one with a negativedegree of homogeneity and one with a positive degree. We will see that the space of

functions belonging to a particular unitary representation, corresponding to the space

of combinations ofein andein for a particular value ofn, can be decomposed intotwo complementary subspaces; one (like ein) gives a regular function on H {0}when multiplied by a positive power of |q|, the other (likeein) has to be multipliedby a negative power of |q|.

Let Unbe the set of functions f : H{0} Hwhich are regular and homogeneousof degreenover R, i.e.

f(q) =nf(q) for R.Removing the origin from the domain offmakes it possible to consider both positiveand negativen (the alternative procedure of adding a point at infinity to Hhas disad-vantages, since regular polynomials do not necessarily admit a continuous extension to

18


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H{} =S4). LetWnbe the set of functions f : H{0} H which are harmonicand homogeneous of degreen overR. ThenUn and Wn are right vector spaces overH (with pointwise addition and scalar multiplication) and since every regular function

is harmonic, we haveUn Wn.Functions inUn and Wn can be studied by means of their restriction to the unit

sphereS= {q: |q| = 1}. LetUn= {f|S: f Un}, Wn= {f|S: f Wn};

thenUnand Un are isomorphic (as quaternionic vector spaces) by virtue of the corre-

spondencef Un f Un, where f(q) =rnf(u), (6.1)

using the notationr = |q| R,u = q/|q| S.SimilarlyWnand Wnare isomorphic.In order to express the Cauchy-Riemann-Fueter equations in a form adapted to the

polar decompositionq = ru, we introduce the following vector fieldsX0,...,X3 onH {0}:

X0f = d

d[f(qe)]=0, (6.2)

Xif = d

df[qexp(ei)]=0=

d

df[q(cos +eisin )]=0 (i= 1, 2, 3). (6.3)

These fields form a basis for the real vector space of left-invariant vector fields on

the multiplicative group ofH, and they are related to the Cartesian vector fields/t,/xiby

X0= t

t+ xi

xi, (6.4)

Xi= xi t

+ t

xi ijkxj

xk, (6.5)

t =

1

r2(tX0 xiXi), (6.6)

xi=

1

r2(ijkxjXk+tXi+xiX0). (6.7)

Their Lie brackets are

[X0, Xi] = 0, (6.8)[Xi, Xj ] = 2ijkXk. (6.9)

Using (6.6) and (6.7) the differential operators land can be calculated in termsofX0and Xi. The result is

l = 12 q1(X0+eiXi), (6.10)

= 1

r2{XiXi+X0(X0+ 2)} . (6.11)

The following facts about the space of harmonic functions Wnare well known (andfollow from (6.11); see, for example, (17), p. 71):

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PROPOSITION 6. (i) Wn= Wn2. (ii)dim Wn = (n+ 1)2. (iii)The elementsofWnare polynomials inq.

We can now obtain the basic facts about the spaces Unof regular functions:THEOREM 7. (i) Wn = Un Un2. (ii)Un= Un3. (iii)dim Un = 12(n+

1)(n+ 2).Proof. (i) Equation (6.10) shows that the elements ofUn, which satisfyX0f =nf

and lf = 0, are eigenfunctions of = eiXi with eigenvalue n. Since the vectorfieldsXi are tangential to the sphere S, can be considered as an operator on Wn,and Unconsists of the eigenfunctions of with eigenvalue n. Using (6.9), it can beshown that

2 2 +XiXi= 0.Hence

f Wn (rnf) = 0 XiXif= n(n+ 2)f ( n 2)( +n)f= 0.

It follows that Wn is the direct sum of the eigenspaces ofwith eigenvalues nandn+ 2(these are vector subspaces ofWnsince the eigenvalues are real), i.e.

Wn= Un Un2.(ii) It follows from Proposition 5(i) that the mappingIis an isomorphism between

Unand Un3.(iii) Letdn= dim Un. By (i) and Proposition 6(ii),

dn+dn2= (n+ 1)2

and by (ii),

dn2= dn1.

Thus

dn+dn1= (n+ 1)2.

The solution of this recurrence relation, withd0= 1, is

dn= 12

(n+ 1)(n+ 2).

There is a relation between Proposition 5 (ii) and the fact that homogeneous regular

functions are eigenfunctions of. Proposition 5 (ii) refers to a representation Mof thegroup H

H defined on the space of real-differentiable functions f : H{0} Hby

[M(a, b)f](q) =b f(a1qb).

Restricting to the subgroup {(a, b) : |a| = |b| = 1}, which is isomorphic toSU(2) SU(2),

we obtain a representation ofS U(2) SU(2). Since the mapq aqb is a rotationwhen |a| =|b| = 1, the set W of harmonic functions is an invariant subspace under

20


21/28

this representation. NowW = H C Wc, whereWc is the set of complex-valuedharmonic functions, and the representation ofSU(2) SU(2)can be written as

M(a, b)(q f) = (bq) R(a, b)fwhere R denotes the quasi-regular representation corresponding to the action q aqb1 ofSU(2) SU(2)on H {0}:

[R(a, b)f]q= f(a1qb).

ThusM|W

is the tensor product of the representationsD

0

D1 and

R|Wc of

SU(2) SU(2),whereDn denotes the (n+ 1)-dimensional complex representation ofS U(2). Theisotypic components ofR|Wc are the homogeneous subspacesWcn, on whichR actsirreducibly asDn Dn; thusWnis an invariant subspace under the representation M,andM|Wn is the tensor product (D0 D1) (Dn Dn). Wn therefore has twoinvariant subspaces, on whichM acts as the irreducible representations Dn Dn+1and

Dn Dn1.These subspaces are the eigenspaces of. To see this, restrict attention to the secondfactor inSU(2)

SU(2); we have the representation

M(b)(q f) =M(1, b)(q f) = [D1(b)q] [R(1, b)f]whereD1(b)q= bq. The infinitesimal generators of the representationR(1, b)are thedifferential operatorsXi; the infinitesimal generators ofD1(b) are ei (by which wemean left multiplication byei). Hence the infinitesimal operators of the tensor productM areei+ Xi. The isotypic components ofWare the eigenspaces of the Casimiroperator

(ei+Xi)(ei+Xi) =eiei+XiXi+ 2.

Buteiei= 3, andXiXi= n(n + 2) onWn; hence(ei+Xi)(ei+Xi) = 2 n2 2n 3.

and so the isotypic components ofWn for the representationM are the eigenspacesof.Un, the space of homogeneous regular functions of degreen, has eigenvalue nfor, and soM|Unis the representationDn+1 ofSU(2).

Similar considerations lead to the following fact:

PROPOSITION 7.Iff is regular,qfis harmonic.The representation MofSU(2) SU(2)can also be used to find a basis of regular

polynomials. It belongs to a class of induced representations which is studied in ( 18),

where a procedure is given for splitting the representation into irreducible components

and finding a basis for each component. Rather than give a rigorous heuristic derivation

by following this procedure, which is not very enlightening in this case, we will state

the result and then verify that it is a basis.

21


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Since the functions to be considered involve a number of factorials, we introduce

the notation

z[n] = zn

n! ifn 0

= 0 ifn


23/28

Sincezj = j zfor anyz C, it can be seen from the definition (6.14) thatPnklj = j P

nnk,nl

and therefore

Qnklj = Qnnk+1,nl.

ThusUn is spanned by theQnkl (0 k l n), which therefore form a basis for

Un. Another basis forUnwill be given in the next section.We conclude this section by studying the quaternionic derivativel. Sincel is a

linear map fromUn intoUn1 anddim Un > dim Un1,l must have a large kerneland so we cannot conclude from lf = 0 thatfis constant. However, although theresult is far from unique, it is possible to integrate regular polynomials:

THEOREM 8. Every regular polynomial has a primitive, i.e. l maps Un ontoUn1ifn >0.

Proof. Supposef Unis such thatlf= 0. Thenf

t =ei

f

xi= 0.

Thusfcan be regarded as a function on the space P of pure imaginary quaternions.Using vector notation for elements ofP and writingf = f0+ fwithf0 R, f P,the conditionei f/xi= 0becomes

f0+ f= 0, .f= 0.Ifn 0, we can define f(0) so that these hold throughoutP, and so there exists afunction F: P Psuch that

f= F, f0= .F,i.e.

f=eiF

xi.

ThenF is harmonic, i.e. 2F= 0.Let Tn be the right quaternionic vector space of functions F : P

H which

are homogeneous of degree n and satisfy2F = 0; then dim Tn = 2n+ 1. LetKn be the subspace ofTn consisting of functions satisfying eiF/xi = 0; thenKn = ker l Un. The above shows thatei /xi : Tn+1 Tn mapsTn+1 ontoKn; its kernel isKn+1and so

dim Kn+ dim Kn+1= dim Tn+1= 2n + 3.

The solution of this recurrence relation, withdim K0= 1, is dim Kn= n+ 1. But

dim Un dim Un1= 12 (n+ 1)(n+ 2) 12n(n+ 1) =n+ 1.It follows thatl mapsUnontoUn1.

23


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THEOREM 9.Ifn


25/28

Thus

g,Ilf =S

lg(q)q1Dq q1f(q1)

= lg,If.ButIis an isomorphism, the inner product is non-degenerate on Un2, andl mapsUn2onto Un3ifn 3; it follows that l : Un Un1is one-to-one.

In the missing casesn= 1andn= 2, Theorems 8 and 9 are both true trivially,sinceU1= U2= {0}.

7. Regular power series. The power series representing a regular function, and the

Laurent series representing a function with an isolated singularity, are most naturallyexpressed in terms of certain special homogeneous functions.

Letbe an unordered set ofn integers {i1,...,in} with1 ir 3; can alsobe specified by three integers n1, n2, n3 with n1+n2 + n3 = n, wheren1 is thenumber of1s in,n2 the number of 2s andn3 the number of 3s, and we will write = [n1n2n3]. There are

12(n+ 1)(n+ 2) such sets; we will denote the set of all

of them byn. They are to be used as labels; when n = 0, so that=, we use thesuffix0instead of. We writefor thenth order differential operator

= n

xi1 ...xin=

n

xn1yn2zn3.

The functions in question are

G(q) =G(q) (7.1)

and

P(q) = 1

n!

(tei1 xi1 )...(tein xin) (7.2)

where the sum is over all n!/(n1!n2!n3!)different orderings ofn1 1s,n2 2s andn33s. ThenPis homogeneous of degreenandGis homogeneous of degree n 3.

As in the previous section, Un will denote the right quaternionic vector space ofhomogeneous regular functions of degreen.

PROPOSITION 9. The polynomialsP( n)are regular and form a basis forUn.

Proof. (17) Letfbe a regular homogeneous polynomial of degree n. Sincef is

regularf

t +i

eif

xi= 0

and since it is homogeneous,

tf

t +i

xif

xi=nf(q).

Hence

nf(q) =i

(xi tei)fxi

.

25


26/28

Butf/xi is regular and homogeneous of degreen 1, so we can repeat the argu-ment; aftern steps we obtain

f(q) = 1

n!

i1...in

(xi1 tei1)...(xin tein) nf

xi1 ...xin

=n

(1)nP(q)f(q).

Sincef is a polynomial,fis a constant; thus any regular homogeneous polynomialis a linear combination of the

P. Let

Vnbe the right vector space spanned by the

P.

By proposition 6 (iii), the elements ofUnare polynomials, soUn Vn; but

dim Vn 12(n+ 1)(n+ 2) = dim Unby Theorem 7(iii). HenceVn=Un.

The mirror image of this argument proves that thePare also right-regular.Just as for a complex variable, we have

(1 q)1 =n=0

qn

for|q| < 1; the series converges absolutely and uniformly in any ball|q| r withr < 1. This gives rise to an expansion ofG(p q)in powers ofp

1

q; identifying itwith the Taylor series ofGaboutp, we obtain

PROPOSITION 10.The expansions

G(p q) =n=0

n

P(q)G(p)

=

n=0

n

G(p)P(q)

are valid for|q| < |p|; the series converge uniformly in any region {(p, q) : |q| r|p|}ofH2 withr


27/28

COROLLARY.1

22

S

G(q) Dq P(q) =,

whereSis any sphere containing the origin.THEOREM 11 (the Laurent series). Supposefis regular in an open setU except

possibly atq0 U. Then there is a neighbourhoodN ofq0 such that ifq N andq=q0,f(q)can be represented by a series

f(q) =

n=0

n{

P(q

q0)a+ G(q

q0)b

}which converges uniformly in any hollow ball

{q: r |q q0| R}, withr >0, which lies insideN.

The coefficientsaandbare given by

a= 1

22

C

G(q q0) Dq f(q),

b= 1

22

C

P(q q0) Dq f(q),

whereCis any closed 3-chain in U {q0} which is homologous to B for some ballwithq0 B U(so thatChas wrapping number 1 aboutq0).I am grateful for the hospitality of the Department of Applied Mathematics and

Theoretical Physics in the University of Liverpool, where part of this work was done,

and to Drs. P. J. McCarthy and C. J. S. Clarke for many stimulating discussions.

REFERENCES

(1) HAMILTON, W. R.Elements of quaternions (London, Longmans Green, 1866).

(2) TAIT, P. G. An elementary treatise on quaternions (Cambridge University Press,

1867).

(3) JOLY, C. J.A manual of quaternions(London, Macmillan, 1905).

(4) FUETER, R. Die Funktionentheorie der Differentialgleichungenu = 0undu= 0mit vier reellen Variablen. Comment. Math. Helv. 7(1935), 307-330.

(5) FUETER, R. Uber die analytische Darstellung der regularen Funktionen einer

Quaternionenvariablen.Comment. Math. Helv. 8(1936), 371-378.(6) HAEFELI, H. Hyperkomplexe Differentiale.Comment. Math. Helv. 20 (1947),

382-420.

(7) DEAVOURS, C. A. The quaternion calculus. Amer. Math. Monthly 80 (1973),

995-1008.

(8) SCHULER, B. Zur Theorie der regularen Funktionen einer Quaternionen-Variablen.

Comment. Math. Helv. 10 (1937), 327-342.

(9) FUETER, R. Die Singularitaten der eindeutigen regularen Funktionen einer

Quaternionen-variablen.Comment. Math. Helv. 9(1937), 320-335.

(10) PORTEOUS, I. R. Topological geometry (London, Van Nostrand Reinhold,

1969).

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(11) CARTAN, H.Elementary theory of analytic functions of one or several com-

plex variables(London, Addison-Wesley, 1963).

(12) HERVE, M.Several complex variables(Oxford University Press, 1963).

(13) HEINS, M.Complex function theory(London, Academic Press, 1968).

(14) TITCHMARSH, E. C. The theory of functions, 2nd ed. (Oxford University

Press, 1939).

(15) CULLEN, C. G. An integral theorem for analytic intrinsic functions on quater-

nions.Duke Math. J. 32 (1965), 139-148.

(16) SCHOUTEN, J. A.Ricci-calculus, 2nd ed. (Berlin, Springer-Verlag, 1954).

(17) SUGIURA, M.Unitary representations and harmonic analysis (London, Wi-

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quaternionic analysis

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