Quaternions, Evaluation of Integrals and Boundary ...

Computational Methods and Function Theory Volume 7 (2007), No. 2, 443–476

Quaternions, Evaluation of Integrals and Boundary Value Problems Athanassios S. Fokas and Dimitrios A. Pinotsis (Communicated by Fred Brackx) Abstract. In an attempt to enhance the accessibility of the beautiful theory of quaternions, we first revisit this theory emphasising that it provides the proper generalisation of the theory of complex analysis. In particular, we discuss the quaternionic generalisations of the following fundamental complex analytic notions: analytic functions, Cauchy’s Theorem, Cauchy’s integral formula, Taylor series, Laurent series, Residue Theorem, the Pompeiu (or Cauchy-Green, or Dbar) formula, and the Plemelj-Sokhotzki formulae. We then present two applications of the theory of quaternions, which provide generalisations of the analogous complex analytic applications: (a) the solution of certain boundary value problems for the Poisson equation in four dimensions; (b) the explicit computation of certain three dimensional real integrals. Keywords. Quaternions, boundary value problems, complex analysis. 2000 MSC. 32W99, 30E25, 32K99.

1. Introduction The theory of quaternions has a long and illustrious history. There exist several papers where applications of this theory have been presented, see e.g. [13, 14, 3, 15, 16, 20]. However, in our opinion, the analytic component of this theory remains underused in applications. Perhaps, this is partly due to the fact that it is difficult to find an exposition that is useful for the applied community. Indeed, the existing expositions appear to be either restricted to algebraic aspects of the theory and therefore do not provide the necessary armament for applications, or are too abstract and hence inaccessible by scholars interested in applications. The first goal of this paper is to present a didactical introduction to the theory of quaternions, with emphasis on analysis. This is done in Section 3, where we first derive the basic analytic results of the theory of complex variables and then derive the analogous results of the theory of quaternions. These include the quaternionic Received May 29, 2006, in revised versions February 1, 2007, and March 5, 2007. Published online July 25, 2007. c 2007 Heldermann Verlag ISSN 1617-9447/$ 2.50

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generalisations of the following fundamental complex analytic notions: analytic functions, Cauchy’s Theorem, Cauchy’s integral formula, Taylor series, Laurent series, Residue Theorem, the Pompeiu (or Cauchy-Green, or Dbar) formula, and the Plemelj formulae. We recall that in the theory of complex variables, a fundamental role is played by the ∂χ and ∂ χ operators where χ is the usual complex variable. For example, a function f (χ) is analytic iff ∂ χ = 0. We note that due to the non-commutative nature of quaternions each of these operators has two analogues, a left and a right analogue. For example, ∂ χ is generalised to either ∂l or ∂r . Hence, an analytic function is generalised to either a left regular function, which satisfies ∂l f = 0, or to a right regular function, which satisfies ∂r f = 0. The Pompeiu formula, which is given by equation (3.34), is a beautiful generalisation of Cauchy’s integral formula, which appears in a wide range of applications, see e.g. [2, 6, 7, 8, 9, 10, 18, 19, 22, 23]. The quaternionic generalisation of this formula, which is given by equation (3.35), plays a basic role in the theory of quaternions. The second goal of this paper is to present some novel applications of the theory of quaternions. In this respect, we first present some variations of the basic quaternionic formula (3.35). Namely, this basic formula expresses a quaternionvalued function f in terms of ∂l f . In Section 4 we obtain a formula which expresses f in terms of ∆f , see equations (4.2) or (4.7), as well as a formula which expresses f in terms of ∆∂l f , see equation (4.3). The following applications of the theory reviewed in Section 3, as well as of the results obtained in Section 4, are presented: (a) In Section 5 we present a methodology for solving boundary value problems for the Poisson equation in four dimensions. As an illustration of this methodology we solve the Dirichlet as well as the Neumann boundary value problems in the domain x0 > 0, −∞ < xj < ∞, j = 1, 2, 3. Furthermore, using the methodology developed in this section, and employing the quaternionic generalisation of the Plemelj formulae (see equations (3.40)), we show that for the above boundary value problems, it is possible to compute the Neumann boundary value in terms of the Dirichlet boundary value (the Dirichlet to Neumann map) by analysing the so-called global relation. This relation is valid on the boundary of the domain, thus it is possible to construct the Dirichlet to Neumann map without solving the problem in the interior of the given domain. (b) In Section 6 we present a methodology for computing a large class of three dimensional real integrals. This class of integrals includes integrals of the following form Z −2 (1.1) I = f (ξ1 , ξ2 , ξ3 ) (1 − x0 )2 + (ξj − xj )2 (1 + ξj 2 )−(n+2) dξ1 dξ2 dξ3 , R3

n ∈ N, j = 1, 2, 3,

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where f (ξ1 , ξ2 , ξ3 ) is a real polynomial, x0 , x1 , x2 , x3 are real numbers and the Einstein convention of summation over the repeated indices j = 1, 2, 3, is used. In order to make this work accessible to a large audience we have sacrificed some mathematical depth.

2. Basic definitions and notation The complex variable χ is defined by χ = x0 + ix1 , where x0 , x1 are real variables and i2 = −1. The complex conjugate of χ denoted by χ, ¯ is defined by χ¯ = x0 − ix1 . Throughout this paper we will use the Einstein convention of summation over repeated indices. The quaternionic variable z is defined by (2.1)

z = x0 + xj ej ,

j = 1, 2, 3,

where x0 and x1 , x2 , x3 are real variables, and the elements e1 , e2 , e3 satisfy the relations e1 2 = e2 2 = e3 2 e1 e2 e2 e3 e3 e1 ei ej

= = = = =

−1, e3 , e1 , e2 , −ej ei ,

i 6= j.

The conjugate of z is defined by z = x0 − xj ej ,

j = 1, 2, 3.

A quaternion-valued function f (x0 , x1 , x2 , x3 ) is defined by f (x) = f0 (x) + fj (x)ej , where f0 and f1 , f2 , f3 are real-valued functions and we use x to denote the ordered independent variables (x0 , x1 , x2 , x3 ). We will refer to the functions f0 and f1 , f2 , f3 as the components of f . The conjugate of f (x) is defined by f (x) = f0 (x) − fj (x)ej . An important role in complex analysis is played by the derivatives (2.2)

1 ∂χ = (∂x0 − i∂x1 ), 2

1 ∂ χ = (∂x0 + i∂x1 ). 2

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It turns out that, due to the non-commutativity of quaternions, there exist two analogues of each of these derivatives, namely a left and right derivative ∂f ∂f ∂f ∂f (2.3) ∂r f = − ej , ∂l f = − ej , ∂x0 ∂xj ∂x0 ∂xj ∂f ∂f ∂f ∂f (2.4) ∂r f = + ej , ∂l f = + ej , ∂x0 ∂xj ∂x0 ∂xj where the subscripts r and l indicate that ej is to the right and left respectively of ∂f /∂xj . Additional notation. In addition to the quaternionic variable z, we will also use the quaternionic variable ζ defined by (2.5)

ζ = ξ0 + ξj ej ,

where ξ0 and ξ1 , ξ2 , ξ3 are real variables. The following quaternionic 3-form is important in quaternionic analysis: (2.6) J x = dx1 ∧dx2 ∧dx3 −e1 dx2 ∧dx3 ∧dx0 +e2 dx3 ∧dx0 ∧dx1 −e3 dx0 ∧dx1 ∧dx2 , where the superscript x indicates that J x is defined in terms of the independent variables (x0 , x1 , x2 , x3 ). Similarly, J ξ is defined by equation (2.6) with (x0 , x1 , x2 , x3 ) replaced by (ξ0 , ξ1 , ξ2 , ξ3 ). The scalar 4-form dx is defined by (2.7)

dx = dx0 ∧ dx1 ∧ dx2 ∧ dx3 .

Subscripts will denote partial derivatives, for example ∂v ∂u vx = , uy = . ∂x ∂y The main difficulty with the quaternionic algebra is that it is non-commutative. However, certain quaternionic structures do commute: . f f = f f = f0 2 + fj fj = |f |2 , (2.8) . (2.9) ∂l ∂l = ∂l ∂l = ∂x20 + ∂xj ∂xj = ∆, ∂r ∂r = ∂r ∂r = ∆. Equations (2.9), which provide the factorisation of the Laplacian operator in four dimensions in terms of the basic quaternionic derivatives, are the analogues of the factorisation of the Laplacian operator in two dimensions in terms of the basic complex variable derivatives defined in (2.2), 1 ∂χ ∂ χ = (∂x20 + ∂x21 ). 4 Negative powers of z can be written in the standard form using the associativity properties, for example z z¯ z¯ z −1 = z −1 2 = 2 . |z| |z|

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By f (x−1 ) we denote the function obtained from f (x) by replacing x0 and x1 , x2 , x3 by x0 x1 x2 x3 and − 2, ,− 2, − 2. 2 |z| |z| |z| |z| The Poincar´ e-Stokes Lemma. Let D be a compact simply connected domain n in R with smooth boundary ∂D and let W be a real differentiable (n − 1) form. Then Z Z (2.10) W = dW. ∂D

D

Example 2.1. Green’s Theorem in R2 , i.e. the equation, Z Z (2.11) (udx + vdy) = (vx − uy ) dx ∧ dy, ∂D

D

where u and v are real-valued differentiable functions of (x, y) ∈ R2 , is a particular case of equation (2.10). Indeed, if W denotes the integrand of the integral in the left hand side of equation (2.11), then dW = dy ∧ (uy dx) + dx ∧ (vx dy) = (vx − uy ) dx ∧ dy. Example 2.2. Gauss’s Theorem in R4 , i.e. the equation, Z f0 dx1 ∧ dx2 ∧ dx3 − f1 dx2 ∧ dx3 ∧ dx0 + ∂D f2 dx3 ∧ dx0 ∧ dx1 − f3 dx0 ∧ dx1 ∧ dx2 Z = f0x0 + f1x1 + f2x2 + f3x3 dx0 ∧ dx1 ∧ dx2 ∧ dx3 , D

where f0 and f1 , f2 , f3 are real-valued differentiable functions of (x0 , x1 , x2 , x3 ) in R4 , is also a particular case of equation (2.10). This equation can be written in the standard vector notation Z (f0 , f1 , f2 , f3 ) · n b dS ∂D Z = (∂x0 , ∂x1 , ∂x2 , ∂x3 ) · (f0 , f1 , f2 , f3 ) dx0 ∧ dx1 ∧ dx2 ∧ dx3 , D

by using the identity Jx = n b dS = (dx1 ∧ dx2 ∧ dx3 , −dx2 ∧ dx3 ∧ dx0 , dx3 ∧ dx0 ∧ dx1 , −dx0 ∧ dx1 ∧ dx2 ) , where n b is the unit quaternion normal to the surface ∂D, dot denotes the usual scalar product, and dS denotes the area element.

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3. Quaternions revisited 3.1. The analogues of analytic functions. We recall that a complex-valued function f (x0 , x1 ), (x0 , x1 ) ∈ R2 , is analytic iff ∂ χ f = 0. Using the operators ∂l and ∂r we can define the following quaternionic analogues of analytic functions. Definition 3.2. Let the left and right quaternionic ∂ derivatives be defined by equations (2.3) and (2.4). A differentiable quaternion-valued function f (x) is called left-regular iff ∂l f = 0. Similarly, f is called right-regular iff ∂r f = 0. We recall that the equation ∂ χ f = 0 implies that the real and imaginary parts of f = u + iv satisfy the Cauchy-Riemann equations: 1 1 i ∂ χ f = (∂x0 + i∂x1 )(u + iv) = (ux0 − vx1 ) + (vx0 + ux1 ) = 0. 2 2 2 Furthermore, the functions u and v are harmonic. Analogous results are valid for regular functions. For economy of presentation, we discuss only left-regular functions. Proposition 3.3. A differentiable quaternion-valued function f (x) = f0 (x) + fj (x)ej is left-regular iff its components satisfy the following equations:

(3.1)

f0x0 − f1x1 − f2x2 f1x0 + f0x1 + f3x2 f2x0 + f0x2 + f1x3 f3x0 + f0x3 + f2x1

− f3x3 − f2x3 − f3x1 − f1x2

= = = =

0, 0, 0, 0.

Furthermore, if these components are twice differentiable, then they satisfy the Laplace equation in four dimensions, (3.2)

∆f0 = ∆f1 = ∆f2 = ∆f3 = 0.

Proof. The equation ∂l f = 0, i.e. the equation ∂ ∂ ∂ ∂ + e1 + e2 + e3 (f0 + e1 f1 + e2 f2 + e3 f3 ) = 0, ∂x0 ∂x1 ∂x2 ∂x3 yields equations (3.1). Differentiating the equations (3.1) with respect to x0 , x1 , x2 , x3 respectively, and adding the resulting equations we find ∆f0 . Similarly with the other equations in (3.2). 3.4. The analogues of the Cauchy Theorem and the Morera Theorem. Let (x0 , x1 ) ∈ D ⊆ R2 and let f (x0 , x1 ) be a differentiable complex-valued function. Let W = f (x0 , x1 )dχ. Then dW = dχ ∧ (fχ dχ),

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and the Poincaré-Stokes Lemma (2.10) yields Z Z (3.3) f dχ = ∂ χ f dχ ∧ dχ. ∂D

D

If f is analytic in D, then ∂ χ f = 0 and equation (3.3) becomes Cauchy’s Theorem, Z (3.4) f dχ = 0. ∂D

Inversely, if f is continuous and satisfies equation (3.4) with ∂D replaced by C, where C is any simple closed contour lying in D, then f is analytic D. This is usually called Morera’s Theorem. In what follows we present the quaternionic analogues of the basic equations (3.3) and (3.4) as well as of Morera’s Theorem. Proposition 3.5. Let D be a bounded simply connected domain in R4 with a smooth boundary ∂D. Let J x and ∂l denote respectively the quaternionic 3-form defined in equation (2.6) and the left quaternionic ∂ derivative defined in (2.4). Let f (x) be a differentiable quaternion-valued function in an open neighborhood of D. Then Z Z x J f (x) = (3.5) ∂l f (x) dx0 ∧ dx1 ∧ dx2 ∧ dx3 . ∂D

D

In particular, if f (x) is a left-regular function in an open neighborhood of D, then Z J x f (x) = 0. (3.6) ∂D

Proof. Equation (3.6) is a direct consequence of the definition of a left-regular function and of equation (3.5). The latter equation is a direct consequence of the Poincaré-Stokes Lemma (2.10). Indeed, letting W = J x f (x), it follows that dW = dx0 ∧ ∂x0 (J x f (x)) + dxj ∧ ∂xj (J x f (x)) = dx0 ∧ dx1 ∧ dx2 ∧ dx3 fx0 − dx1 ∧ dx2 ∧ dx3 ∧ dx0 e1 fx1 +dx2 ∧ dx3 ∧ dx0 ∧ dx1 e2 fx2 − dx3 ∧ dx0 ∧ dx1 ∧ dx2 e3 fx3 = (fx0 + e1 fx1 + e2 fx2 + e3 fx3 ) dx0 ∧ dx1 ∧ dx2 ∧ dx3 , which is the integrand of the right hand side of equation (3.5). In analogy with equations (3.5) and (3.6), the following result is also valid. Proposition 3.6. Let D be a bounded simply connected domain in R4 with a smooth boundary ∂D. Let J x and ∂r denote respectively the quaternionic 3-form defined in equation (2.6) and the right quaternionic ∂ derivative defined in (2.4).

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Let f (x) be a differentiable quaternion-valued function in an open neighborhood of D. Then Z Z x (3.7) f (x)J = ∂r f (x) dx0 ∧ dx1 ∧ dx2 ∧ dx3 . ∂D

D

In particular, if f (x) is a right-regular function in the domain D, then Z f (x)J x = 0. (3.8) ∂D

Equations (3.6) and (3.8) provide generalisations of Cauchy’s Theorem. Due to the non-commutativity of quaternions, there actually exists a further generalisation of Cauchy’s Theorem given by equation (3.10) below. Proposition 3.7. Let D be a bounded simply connected domain in R4 with a smooth boundary ∂D. Let J x , ∂l and ∂r denote respectively the quaternionic 3form defined in equation (2.6) and the left and right quaternionic ∂ derivatives defined in (2.4). Let f (x) and g(x) be differentiable quaternion-valued functions in an open neighborhood of D. Then Z Z x (3.9) g(x)J f (x) = (∂r g(x))f (x) + g(x)(∂l f (x)) dx0 ∧ dx1 ∧ dx2 ∧ dx3 . ∂D

D

In particular, if g(x) is a right-regular function and f (x) is a left-regular function in the domain D, then Z (3.10) g(x)J x f (x) = 0. ∂D

Proof. Equation (3.9) is a direct consequence of the Poincaré-Stokes Lemma (2.10). Indeed, letting W = gJ x f , it follows that dW = dx0 ∧ ∂x0 (gJ x f ) + dxj ∧ ∂xj (gJ x f ) = dx0 ∧ (gx0 J x f + gJ x fx0 ) + dxj ∧ (gxj J x f + gJ x fxj ). Expanding this equation and using the definitions of J x , ∂l and ∂r , we find that dW equals the integrand of the integral of the right hand side of equation (3.9). We note that equations (3.5) and (3.7) are particular cases of equation (3.9). In [25] the following quaternionic analogue of Morera’s Theorem is proved. Proposition 3.8. Let D be a bounded simply connected domain in R4 with a smooth boundary ∂D. Let J x denote the quaternionic 3-form defined in equation (2.6). Let f (x) be a continuous quaternion-valued function in an open neighborhood of D satisfying Z J x f (x) = 0, (3.11) C

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where C is an arbitrary simply connected closed surface lying inside D. Then f (x) is left-regular in D. 3.9. The analogue of the Cauchy integral formula. We recall that if the complex-valued function f (χ) is analytic in D ⊂ R2 , then for χ in the interior of D, it follows that Z 1 f (χ0 ) (3.12) f (χ) = dχ0 . 2πi ∂D χ0 − χ Indeed, using Cauchy’s Theorem, the right hand side of equation (3.12) can be rewritten in the form Z Z 1 1 f (χ0 ) − f (χ) 0 1 0 dχ + dχ f (χ), 0 2πi Cε χ0 − χ 2πi Cε χ − χ where Cε is a circle of radius ε and center χ. The integrand of the first integral in the above equation is analytic in D, thus the first integral vanishes. Furthermore, the second integral can be computed explicitly and hence equation (3.12) follows. In order to generalise equation (3.12) we must first construct the quaternionic analogue of (χ0 − χ)−1 . This turns out to be the function K(ζ − z) defined by (3.13)

K(ζ − z) =

(ζ − z)−1 . |ζ − z|2

It can be verified that the function z −1 /|z|2 is both a left- and a right-regular function for all z 6= 0. Indeed, multiplying this function by zz/|z|2 and then computing its right ∂ derivative, we find ej 4x0 z xj z z z 1 − + 4 6 ej = 0. ∂x0 + ∂xj ej = 4 − |z|4 |z|4 |z| |z|6 |z|4 |z| Similarly, (3.14)

(∂x0 + ej ∂xj )

z |z|4

= 0.

Using the function K(ζ − z) we can derive the following analogue of equation (3.12). Proposition 3.10. Let D be a bounded simply connected domain in R4 with a smooth boundary ∂D. Let f (x) be a left-regular function in an open neighborhood of D. Let z, ζ, J ξ be defined in equations (2.1), (2.5) and (2.6) with x replaced by ξ, respectively. Then for x in the interior of D, the following formula is valid (3.15)

1 f (x) = 2 2π

Z ∂D

(ζ − z)−1 ξ J f (ξ). |ζ − z|2

Inversely, let f (x) be a continuous quaternion-valued function in D satisfying equation (3.15). Then f (x) is left-regular in D.

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Proof. Let B(z, R) denote the closed ball of radius R and center z in D. Using the analogue of Cauchy’s Theorem, the right hand side of equation (3.15) can be rewritten in the form Z 1 (ζ − z)−1 ξ (3.16) J (f (ξ) − f (x)) 2π 2 B(z,R) |ζ − z|2 Z 1 (ζ − z)−1 ξ + 2 J f (x). 2 2π B(z,R) |ζ − z| The form J x on B(z, R) is (3.17)

Jx =

ζ −z dS, |ζ − z|

where dS denotes the area element. Then, the second integral is Z Z (ζ − z)−1 ξ 1 (3.18) J = dS 2 3 ∂B(z,R) |ζ − z| ∂B(z,R) |ζ − z| Z 1 = 3 dS = 2π 2 . R ∂B(z,R) Because f (x) is continuous |f (ξ) − f (x)| < ε, in a small neighborhood |ζ − z| = R. Hence, Z Z 1 ε (ζ − z)−1 ξ (3.19) J (f (ξ) − f (x)) < 2 3 dS = ε. 2π 2 2 2π R B(z,R) B(z,R) |ζ − z| Equation (3.15) follows by letting ε → 0. Let f (x) be a continuous function satisfying equation (3.15) and let C be an arbitrary simply connected closed surface lying inside D which does not enclose z. Then Z Z Z (ζ − z)−1 ξ x x 1 J f (x) = J J f (ξ) 2π 2 ∂D |ζ − z|2 C C Z Z −1 1 x (ζ − z) = J J ξ f (ξ) = 0, 2π 2 ∂D C |ζ − z|2 since the expression in the bracket vanishes because z is outside the domain with boundary C. The required result follows by an application of Morera’s Theorem (see Proposition 3.8). We now prove a proposition which will be useful for the evaluation of real integrals in Section 6.

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Proposition 3.11. Let D be a bounded simply connected domain in R4 containe be the image of D ing the origin, let ∂D be the smooth boundary of D, and let D −1 under the map z → z . Suppose that the function f (x) is left-regular in D, then e the function z −1 |z|−2 f (x−1 ) is left-regular in D. Proof. Letting z → z −1 in equation (3.15) and then multiplying by z −1 |z|−2 from the left we find Z 1 (ζ − z −1 )−1 ξ −1 −2 −1 (3.20) z |z| f (x ) = 2 z −1 |z|−2 J f (ξ). 2π ∂D |ζ − z −1 |2 Using the identity (3.21)

z −1 |z|−2

(ζ −1 − z)−1 −1 −2 (ζ − z −1 )−1 = − ζ |ζ| , |ζ − z −1 |2 |ζ −1 − z|2

it follows that equation (3.20) becomes Z 1 (ζ −1 − z)−1 −1 −2 ξ −1 −2 −1 (3.22) z |z| f (x ) = − 2 ζ |ζ| J f (ξ). 2π ∂D |ζ −1 − z|2 Letting ζ → ζ −1 in equation (3.22) and noting that under this map ζ −1 |ζ|−2 J ξ → Jeξ ζ −1 |ζ|−2 where Jeξ is obtained from J ξ by the map ζ → ζ −1 , it follows that equation (3.22) yields Z 1 (ζ − z)−1 eξ −1 −2 −1 −1 −2 −1 (3.23) z |z| f (x ) = 2 J ζ |ζ| f (ξ ). 2π ∂ De |ζ − z|2 Equation (3.23) implies that the function z −1 |z|−2 f (x−1 ) is left-regular. 3.12. The analogue of the Taylor series. We recall that the construction of the Taylor series for an analytic function is an immediate consequence of Cauchy’s Integral Formula (3.12). Indeed, suppose that f (χ) is analytic in the disc |χ| < R. Then employing equation (3.12) with ∂D the boundary of this disc, and using −1 X ∞ 1 1 χ χj = 0 1− 0 = , χ0 − χ χ χ χ0 j+1 j=0 we find (3.24)

Z ∞ X f (χ0 ) 0 1 dχ χj , f (χ) = 0 j+1 2πi 0 0 χ |χ |=R j=0

where R0 < R. The coefficients of χj in this equation can be related to the derivatives of f (χ) evaluated at the origin. Indeed, the n-th derivative of equation

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(3.12), where ∂D is the boundary of the disc |χ| = R0 , yields 1 dn 1 f (χ) = n n! dχ 2πi

Z |χ0 |=R0

f (χ0 ) dχ0 , (χ0 − χ)n+1

n ∈ Z+ .

Evaluating this equation at χ = 0, and comparing the resulting equation with equation (3.24), it follows that equation (3.24) can be rewritten in the standard Taylor series form n ∞ X d 1 f (χ) = f (0) χj . n n! dχ j=0 We now discuss the generalisation of the above results to quaternion-valued functions. Let f (x) be left-regular for |z| < R. By analogy with the above construction, we can construct a Taylor series of f (x) by using the quaternionic integral representation (3.15) and the expansion of the function K(ζ − z). This is discussed further in Appendix A, where it is shown that the function f (x) can be expanded in the following series around the origin " (3.25) f (x) = 1 + zj

∂ ∂2 1 ∂2 1 + (z z + z z ) + zj 2 1 2 2 1 ∂xj 2 ∂xj 2 2 ∂x1 ∂x2

# ∂2 1 ∂2 1 + (z3 z1 + z1 z3 ) f (x) z=0 + (z2 z3 + z3 z2 ) 2 ∂x2 ∂x3 2 ∂x1 ∂x3 +··· where the variables z1 , z2 and z3 are defined by the equations (3.26)

z1 = x1 − x0 e1 ,

z2 = x2 − x0 e2 ,

z3 = x3 − x0 e3 .

The generalisation of equation (3.25) to all orders was constructed by Fueter. Proposition 3.13 ([11]). Let f (x) be a left-regular quaternion-valued function in a bounded simply connected domain D in R4 containing the origin, and let ∂D be the smooth boundary of D. Let δ be the distance of the origin to ∂D. Then the function f (x) for z in the hypersphere |z| < δ, can be expanded in the following series, which converges absolutely and uniformly in the compact subsets of |z| < δ, f (x) =

∞ X

X

n=0 n=n1 +n2 +n3

∗

z n1 n2 n3 (x)

∂ n1 +n2 +n3 , f (x) z=0 n n n ∂x1 1 ∂x2 2 ∂x3 3

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∗

where z n1 n2 n3 (x) is the Fueter polynomial of order n = n1 + n2 + n3 , defined as follows:1 , X 1 ∗ z n1 n2 n3 (x) = ζem1 · · · ζemn , ζeml = xml − x0 eml , (3.27) n! (m1 ,...,mn )∈Sn

where each element ml , l = 1, ..., n, of the set (m1 , . . . , mn ) takes the values 1, 2 or 3, n1 is the number of 1’s, n2 is the number of 2’s, n3 is the number of 3’s in the set (m1 , . . . , mn ), and Sn is the symmetric permutation group of n elements. 3.14. The analogues of the Laurent series and of the Residue Theorem. Recall that if a function f (χ) is analytic in the annulus R1 < |χ| < R2 , then Z Z 1 1 f (χ0 ) f (χ0 ) 0 (3.28) f (χ) = dχ − dχ0 , 0 0 2πi |χ0 |=R20 χ − χ 2πi |χ0 |=R10 χ − χ where R10 > R1 and R20 < R2 . In the first integral, |χ|/|χ0 | < 1, thus ∞

X χj 1 = , 0 j+1 χ0 − χ χ j=0 whereas, in the second integral |χ|/|χ0 | > 1, thus ∞ X 1 χ0 j = − . j+1 χ0 − χ χ j=0

Let f (x) be a left-regular function in R1 < |z| < R2 . In order to construct the Laurent series of this function we will use the following analogue of equation (3.28), Z Z (ζ − z)−1 ξ 1 (ζ − z)−1 ξ 1 J f (ξ) − J f (ξ), (3.29) f (x) = 2 2π C2 |ζ − z|2 2π 2 C1 |ζ − z|2 where C2 and C1 denote |ζ| = R20 and |ζ| = R10 respectively. For the first integral we let t = ζ −1 z, thus (ζ − z)−1 = (ζ − ζt)−1 = (1 − t)−1 ζ −1 . Hence, (3.30)

−1 (ζ − z)−1 −1 −1 −1 ζ = (1 − t) (1 − t) (1 − t) , |ζ − z|2 |ζ|2

t = ζ −1 z if ζ ∈ C2 .

For the second integral we let t = ζz −1 , thus (ζ − z)−1 = (tz − z)−1 = z −1 (t − 1)−1 . 1

The variables z1 , z2 and z3 defined by equation (3.26) are the Fueter polynomials of order 1. ∗ Other examples of Fueter polynomials are presented in Appendix B. In general z n1 n2 n3 (x) is a function of z1 , z2 and z3 .

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Hence, (3.31)

(ζ − z)−1 z −1 = − (1 − t)−1 (1 − t)−1 (1 − t)−1 , |ζ − z|2 |z|2

t = ζz −1

if ζ ∈ C1 .

It is shown in Appendix A that expanding equation (3.30) and substituting the resulting expression in equation (3.29), we find the generalised Taylor series, namely equation (3.25). Similarly, it can be shown that expanding equation (3.31) and substituting the resulting expression in equation (3.29), we find the generalised principal part of the function f (x). Therefore the function f (x) can be expanded in the following series: 1 2 ∂2 ∂ + C.P. + z1 (3.32) f (x) = 1 + z1 + C.P. ∂x1 2 ∂x1 2 ∂2 1 (z1 z2 + z2 z1 ) + + C.P. 2 ∂x1 ∂x2 Z 1 (ζ − z)−1 ξ × 2 J f (ξ) + ··· 2π C2 |ζ − z|2 z=0 Z z −1 1 − 2 2 J ξ f (ξ) |z| 2π C2 −1 −1 ∂ 1z ∂2 z 2 + C.P. + + C.P. z1 − − 4 z1 |z| ∂x1 2 |z|6 ∂x1 2 −1 1z ∂2 + (z1 z2 + z2 z1 ) + C.P. 2 |z|6 ∂x1 ∂x2 Z 1 (ζ −1 − z)−1 ζ −1 ξ × 2 J f (ξ) + ··· 2π C1 |ζ −1 − z|2 |ζ|2 z=0 where C.P. in each curly bracket denotes the two terms obtained from the cyclic permutation {0 → 0, 1 → 2, 2 → 3, 3 → 1} of the term appearing in the curly bracket, and the dots after the first pair of brackets denote the remaining part of the generalised Taylor series, whereas the dots after the last pair of brackets denote the remaining part of the generalised principal part of f (x). We recall that if f (χ) is an analytic function in the annulus R1 < |χ| < R2 , then its integral along any simple closed contour C lying in the annulus is given by Z f (χ)dχ = 2πiC−1 , C

where C−1 is the coefficient of the first term of the principal part of the Laurent expansion of the function f around 0. This result is a direct consequence of the Laurent expansion and is known as the Residue Theorem. In the following we use the expansion (3.32) in order to derive the analogue of the Residue Theorem for quaternion-valued functions.

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Proposition 3.15. Let f (x) be left-regular in the annulus R1 < |z| < R2 . Let B(0, ρ), R1 < ρ < R2 , be a closed ball in four dimensions lying inside the annulus R1 < |z| < R2 . Then Z J ξ f (ξ) = 2π 2 C−1 , ∂B(0,ρ) ξ

where J is defined by equation (2.6) with x replaced by ξ and C−1 is the coefficient of −z −1 /|z|2 in the following expansion of f (x): ∂ (3.33) f (x) = 1 + z1 + C.P. ∂x1 Z 1 (ζ − z)−1 ξ × 2 J f (ξ) + ··· 2 2π ∂B(0,ρ) |ζ − z| z=0 −1 Z z ∂ z −1 1 ξ z1 + C.P. J f (ξ) + − 2 2 |z| 2π ∂B(0,ρ) |z|4 ∂x1 Z (ζ −1 − z)−1 ζ −1 ξ 1 J f (ξ) + ··· , × 2 2π ∂B(0,ρ) |ζ −1 − z|2 |ζ|2 z=0 where C.P. in each curly bracket denotes the two terms obtained from the cyclic permutation {0 → 0, 1 → 2, 2 → 3, 3 → 1} of the term appearing in the curly bracket, and the dots after the first curly bracket denote the remaining part of the generalised Taylor series, whereas the dots after the last curly bracket denote the remaining part of the generalised principal part of f (x). Proof. The expansion (3.33) follows from equation (3.32) after replacing C1 and C2 by ∂B(0, ρ) (which is valid due to equation (3.10)). 3.16. The analogue of the Pompeiu formula. In 1912 Pompeiu in his attempt to answer a certain question raised by Painlevé, obtained the following generalisation of Cauchy’s integral formula: Let D ⊂ R2 be a simply connected bounded domain with smooth boundary ∂D, and let f (χ, χ) be a differentiable function. Then ZZ 1 ∂ χ0 f (χ0 , χ0 ) 0 f (χ, χ) = (3.34) dχ ∧ dχ0 2πi D χ0 − χ Z 1 f (χ0 , χ0 )dχ0 + , χ ∈ D. 2πi ∂D χ0 − χ If f is analytic, then ∂ χ0 f = 0 and equation (3.34) becomes the Cauchy integral formula (3.12). Equation (3.34) can be derived using equation (3.3), see [1]. There exists a quaternionic generalisation of equation (3.34) which can be derived using the quaternionic version of equation (3.3), namely equation (3.9).

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Proposition 3.17 (f in terms of ∂l f ). Let D be a bounded simply connected domain in R4 with a smooth boundary ∂D. Let ∂l and the 3-form J ξ be defined by equations (2.4) and (2.6) with x replaced by ξ, respectively. Let f (x) be a differentiable quaternion-valued function. Then Z 1 (ζ − z)−1 (3.35) f (x) = − 2 ∂l f (ξ) dξ 2π D |ζ − z|2 Z 1 (ζ − z)−1 ξ + 2 J f (ξ), x ∈ D, 2π ∂D |ζ − z|2 where dξ is the scalar 4-form defined by equation (2.7) with x replaced by ξ. If x∈ / D then the left hand side vanishes. Proof. Using the generalisation of Cauchy’s Integral Theorem (3.9) with g(ξ) replaced by (ζ − z)−1 K(ξ) = , |ζ − z|2 and noting that for z ∈ D\B(z, R) ∂ r K(ξ) = 0, we obtain Z (3.36) D\B(z,R)

Z

ξ

Z

KJ f −

K(∂l f ) dξ0 ∧ dξ1 ∧ dξ2 ∧ dξ3 = ∂D

KJ ξ f.

∂B(z,R)

Taking the limit R → 0, the last integral of the right hand side of equation (3.36) equals 2π 2 f (x) (see equation (3.18)), and equation (3.36) yields equation (3.35).

3.18. The analogue of the Plemelj-Sokhotzki formulae. Let the closed smooth curve C divide the complex χ−plane into two regions: the finite region D+ inside C and the infinite region D− outside C. The Cauchy type integral Z 1 φ(τ ) f (χ) = dτ, 2πi C τ − χ where the function φ(τ ) satisfies the Hölder condition, has two distinct limits denoted by f + (t) and f − (t), t on C, as χ approaches C along two different curves lying entirely in D+ and in D− respectively. Let Cε be the part of C that has length 2ε and is centered around t, then the principal value integral is defined by Z Z φ(τ ) φ(τ ) . dτ = lim dτ, t on C. (3.37) PV ε→0 C\C τ − t C τ −t ε

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The functions f + (t) and f − (t) are related with the principal value integral by means of the Plemelj-Sokhotzki formulae Z 1 1 φ(τ ) ± (3.38) f (t) = ± φ(t) + PV dτ, t on C. 2 2πi C τ −t If φ(τ ) is a Hölder function then the derivation of equation (3.38) is complicated [21]. However, the derivation becomes easy in the case that φ(τ ) is C∞ -smooth on ∂D. In what follows we present the generalisation of equations (3.37) and (3.38) for the case that φ(x) is a left-regular function in the neighborhood ∂D, where D is a bounded simply connected domain in R4 . Proposition 3.19. Let D be a bounded simply connected domain in R4 with a smooth boundary ∂D. Let the 3-form J ξ be defined by equation (2.6) with x replaced by ξ. Let φ(x) be a C∞ -smooth function on ∂D. Define the generalised Cauchy type integral by Z 1 (ζ − z)−1 ξ (3.39) f (x) = 2 J φ(ξ). 2π ∂D |ζ − z|2 Then the function φ(x) is left regular in R4 \∂D and decays to zero at infinity. Moreover, the integral (3.39) has the limiting values f + (t) and f − (t), t on ∂D, as z approaches ∂D from inside and outside of D, respectively. These limits are given by the formulae Z 1 1 (ζ − t)−1 ξ ± (3.40) f (t) = ± φ(t) + 2 P V J φ(ξ), t on ∂D, 2 2 2π ∂D |ζ − t| where Z Z (ζ − t)−1 ξ (ζ − t)−1 ξ . PV J φ(ξ) = lim J φ(ξ), t on ∂D, 2 ε→0 ∂D\B(t,ε) |ζ − t|2 ∂D |ζ − t| and B(t, ε) is a ball in R4 with center t and radius ε. Proof. We deform the surface ∂D to two surfaces ∂D\{∂D ∩ Cε } and Cε , where Cε is defined by Cε (t) = {x ∈ R4 , |z − t| = ε} ∩ D. Equation (3.39) then yields Z Z (ζ − z)−1 ξ (ζ − z)−1 ξ 1 1 f (x) = 2 J φ(ξ) + J φ(ξ). 2π ∂D\{∂D∩Cε } |ζ − z|2 2π 2 Cε |ζ − z|2 Considering the limit of the above equation as z tends to ∂D from inside of D, we find Z 1 (ζ − t)−1 ξ + f (t) = lim J φ(ξ) 2 2π 2 ε→0 ∂D\{∂D∩Cε } |ζ − t| Z (ζ − t)−1 ξ + J φ(ξ) , t on ∂D. 2 Cε |ζ − t|

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Since

Z 1 (ζ − t)−1 ξ 1 x ∈ ∂D, J φ(ξ) = φ(t), 2 2 2π Cε |ζ − t| 2 equation (3.40) for f + (t) follows. Similarly for equation (3.40) for f − (t).

4. Variations of the quaternionic analogue of the Pompeiu formula Using the identities 1 1 2 z −1 (4.1) ∂r 2 = −2 2 , ∂r = 2 , |z| |z| z |z| it is possible to derive certain variations of the basic equation (3.35), which are given in the following proposition. Proposition 4.1 (f in terms of ∆f and f in terms of ∂l ∆f ). Let D be a bounded simply connected domain in R4 with a smooth boundary ∂D. Let ∂l , ∂l , dξ, ∆ be defined by equations (2.3), (2.4), (2.7) and (2.9), respectively. Let f (x) be a twice differentiable quaternion-valued function. Then the function f (x) admits the following integral representation for x ∈ D: Z ∆f (ξ) 1 (4.2) dξ f (x) = − 2 4π D |ζ − z|2 ) Z ( 1 ξ J ∂ f (ξ) 1 (ζ − z)−1 ξ l + 2 J f (ξ) + 2 . 2π ∂D |ζ − z|2 |ζ − z|2 Furthermore, if f is three times differentiable, then it also admits the integral representation Z 1 (ζ − z)−1 (∂l ∆)f (ξ) dξ (4.3) f (x) = 8π 2 D ) Z ( 1 ξ J ∂ f (ξ) 1 (ζ − z)−1 ξ l J f (ξ) + 2 + 2 2π ∂D |ζ − z|2 |ζ − z|2 Z 1 − 2 (ζ − z)−1 J ξ ∆f (ξ), x ∈ D. 8π ∂D Proof. Let J ξ be the conjugate of J ξ . The conjugate of equation (3.9) yields Z Z (4.4) gJ ξ f = {(∂r g) f + g (∂l f )} dξ. ∂D

D 2

Substituting g = (−1/2)/|ζ − z| in equation (4.4) and using equation (4.1), equation (4.4) becomes Z Z Z (ζ − z)−1 1 J ξ f (ξ) 1 ∂l f (ξ) f (ξ) dξ = − + dξ. (4.5) 2 2 2 ∂D |ζ − z| 2 D |ζ − z|2 D |ζ − z|

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Replacing in this equation f (ξ) by ∂l f (ξ) and using equation (3.35) to replace the left hand side of equation (4.5), we find equation (4.2). Substituting g = (1/2)/(ζ − z) in equation (3.9) and using equation (4.1), it follows that Z Z Z f (ξ) 1 1 −1 ξ (4.6) (ζ − z) J f (ξ) − (ζ − z)−1 ∂l f (ξ) dξ. dξ = 2 |ζ − z| 2 2 D ∂D D Replacing in this equation f (ξ) by ∆f (ξ), and using equation (4.2) in order to replace the left hand side of equation (4.6), we find equation (4.3). In order to use equation (4.2) for the case of f real, it is convenient to rewrite equation (4.2) in an alternative form. Proposition 4.2 (an alternative representation for f in terms of ∆f ). Let D be a bounded simply connected domain in R4 with a smooth boundary ∂D. Let ∂l , ∂l , dξ, ∆ be defined by equations (2.3), (2.4), (2.7) and (2.9), respectively. Let f (x) be a twice differentiable real function. Then the function f (x) admits the following integral representation for x ∈ D: Z 1 ∆f (ξ) f (x) = − 2 (4.7) dξ 4π D |ζ − z|2 h i Z ( (ξ0 − x0 )J ξ + (ξj − xj )J ξ f 0 j 1 + 2 4 2π ∂D |ζ − z| ) ξ ξ 1 (f J + f J ) ξ ξ 0 0 j j +2 , |ζ − z|2 where J ξ = J0ξ + ej Jjξ . Proof. Equation (4.2) can be written as follows Z ( 1 (ξ0 − x0 ) − ej (ξj − xj ) ξ (4.8) f (x) = J f (ξ) 2 2π ∂D [(ξ0 − x0 )2 + (ξj − xj )2 ]2 ) 1 ξ J (f + e f ) ξ j ξ 0 j 2 + (ξ0 − x0 )2 + (ξj − xj )2 Z 4f (ξ) 1 − 2 dξ, x ∈ D. 4π D (ξ0 − x0 )2 + (ξj − xj )2 The numerators of the of the first and second terms appearing in the integrand of the right hand side of equation (4.8) equal h i (ξ0 − x0 )J0ξ + (ξj − xj )Jjξ f n h i o + e1 (ξ0 − x0 )J1ξ − (ξ1 − x1 )J0ξ − (ξ2 − x2 )J3ξ + (ξ3 − x3 )J2ξ f + C.P.

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CMFT

and i o ne h 1 ξ 1 J0 fξ0 + Jjξ fξj + J0ξ fξ1 − J1ξ fξ0 − J2ξ fξ3 + J3ξ fξ2 + C.P. , 2 2 where C.P. denotes the two terms obtained by cyclic permutation {0 → 0, 1 → 2, 2 → 3, 3 → 1} of the term appearing in the curly bracket. It turns out that the derivatives fξj satisfy the identity f 2 2 fξj = (ξ0 − x0 ) + (ξj − xj ) ∂ξj (ξ0 − x0 )2 + (ξl − xl )2 2(ξj − xj )f + , (ξ0 − x0 )2 + (ξl − xl )2 where j is 1 or 2 or 3 and no summation over j is taken. Substituting the derivatives fξj in equation (4.8) we find h i Z ( (ξ0 − x0 )J ξ + (ξj − xj )J ξ f 0 j 1 (4.9) f (x) = 2π 2 ∂D [(ξ0 − x0 )2 + (ξj − xj )2 ]2 i ) h ξ ξ 1 J f + J f ξ ξ 0 0 j j 2 + 2 (ξ0 − x0 ) + (ξj − xj )2 Z 1 4f (ξ) − 2 dξ 4π D (ξ0 − x0 )2 + (ξj − xj )2 Z 1 + e1 2 (J0ξ ∂ξ1 − J1ξ ∂ξ0 − J2ξ ∂ξ3 + J3ξ ∂ξ2 ) 2π ∂D f + C.P. . × (ξ0 − x0 )2 + (ξj − xj )2 The Poincaré-Stokes Lemma (2.10) implies that the terms appearing in the last line of equation (4.9) vanish and hence we obtain equation (4.7). It is also straightforward to derive the analogous formula to equation (4.7) starting from equation (4.3), see [24].

5. Boundary value problems for the Poisson equation in four dimensions In the following we use some of the results of Sections 3 and 4 to solve certain boundary value problems for the Poisson equation in four dimensions. In particular, equation (4.7) immediately implies the following result.

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Proposition 5.1. Let D be a bounded simply connected domain in R4 with a smooth boundary ∂D. Let the real-valued function φ(x) satisfy the Poisson equation in four dimensions (5.1)

∆φ = h,

z ∈ D ⊂ R4 ,

where h(x) is a given real-valued function with sufficient smoothness. Then, for x ∈ D, φ admits the integral representation Z 1 h(ξ) φ(x) = − 2 (5.2) dξ 4π D |ζ − z|2 i h Z ( (ξ0 − x0 )J ξ + (ξj − xj )J ξ φ 0 j 1 + 2 4 2π ∂D |ζ − z| ) (φξ0 J0ξ + φξj Jjξ ) + . 2|ζ − z|2 Furthermore, for x ∈ / D, the boundary values of φ satisfy the global relation Z h(ξ) 1 (5.3) dξ 0=− 2 4π D |ζ − z|2 h i Z ( (ξ0 − x0 )J ξ + (ξj − xj )J ξ φ 0 j 1 + 2 4 2π ∂D |ζ − z| ) (φξ0 J0ξ + φξj Jjξ ) + . 2|ζ − z|2 Proof. Equation (5.2) follows immediately from equation (4.7). Equation (5.3) follows from the derivation of equation (4.7) using the fact that if x ∈ / D, then the function K has ∂r K = 0. The proof of both equations (5.2) and (5.3) can be extended to unbounded domains under appropriate decay conditions. However, it is beyond the scope of this paper to make these conditions precise. In the following, we solve the Dirichlet and Neumann problems for the Poisson equation in the half space. Proposition 5.2 (The Dirichlet and Neumann problems for the Poisson equation in the half space). Let the real-valued function φ(x) satisfy the Poisson equation (5.4)

∆φ = h,

x0 > 0, −∞ < xj < ∞, j = 1, 2, 3,

where h(x) is a given real-valued function with sufficient smoothness and decay. (a) Let φ satisfy the Dirichlet boundary condition φ(0, x1 , x2 , x3 ) = d(x1 , x2 , x3 ),

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CMFT

where the function d(x1 , x2 , x3 ) has appropriate smoothness and decay. Then the function φ(x) is given by Z 1 h(ξ) (5.5) φ(x) = − 2 dξ 4π D (ξ0 − x0 )2 + (ξj − xj )2 Z h(ξ) 1 + 2 dξ 2 4π D (ξ0 + x0 ) + (ξj − xj )2 Z x0 d(ξ1 , ξ2 , ξ3 ) dξ1 dξ2 dξ3 . − 2 π {ξ0 =0} [x0 2 + (ξj − xj )2 ]2 (b) Let φ satisfy the Neumann boundary condition φx0 (0, x1 , x2 , x3 ) = n(x1 , x2 , x3 ), with Z (5.6)

Z n(ξ1 , ξ2 , ξ3 ) dξ1 dξ2 dξ3 =

{ξ0 =0}

h(ξ) dξ, D

and the function n(x1 , x2 , x3 ) has appropriate smoothness and decay. Then the function φ(x) is given by Z 1 h(ξ) φ(x) = − 2 (5.7) dξ 2 4π D (ξ0 − x0 ) + (ξj − xj )2 Z 1 h(ξ) − 2 dξ 2 4π D (ξ0 + x0 ) + (ξj − xj )2 Z 1 n(ξ1 , ξ2 , ξ3 ) dξ1 dξ2 dξ3 . + 2 2 2π {ξ0 =0} x0 + (ξj − xj )2 Proof. The definition of J ξ implies that for ξ0 = 0, J ξ = dξ1 ∧ dξ2 ∧ dξ3 = J0ξ . Hence equation (5.2) implies that for x0 > 0, −∞ < xj < ∞, j = 1, 2, 3, φ(x) is given by the equation Z h(ξ) 1 (5.8) φ(x) = − 2 dξ 2 4π D (ξ0 − x0 ) + (ξj − xj )2 ( Z 1 −x0 φ(0, ξ1 , ξ2 , ξ3 ) + 2 2π {ξ0 =0} [x0 2 + (ξj − xj )2 ]2 ) 1 φ (0, ξ , ξ , ξ ) ξ 1 2 3 0 dξ1 dξ2 dξ3 . +2 2 x0 + (ξj − xj )2 The global relation (5.3) is an equation similar to equation (5.8) where the left hand side is replaced by zero, and which is valid for x0 < 0. Letting in this equation x0 → −x0 we find that for x0 > 0, −∞ < xj < ∞, j = 1, 2, 3, the

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following equation is valid, Z 1 h(ξ) 0=− 2 (5.9) dξ 2 4π D (ξ0 + x0 ) + (ξj − xj )2 ( Z x0 φ(0, ξ1 , ξ2 , ξ3 ) 1 + 2 2π {ξ0 =0} [x0 2 + (ξj − xj )2 ]2 ) 1 φ (0, ξ , ξ , ξ ) ξ 1 2 3 + 2 20 dξ1 dξ2 dξ3 . x0 + (ξj − xj )2 If φ is given, then subtracting equations (5.8) and (5.9), we obtain (5.5). If φξ0 is given, then adding equations (5.8) and (5.9), we obtain (5.7). The condition (5.6) follows by multiplying the global relation by x0 2 and then taking the limit of the resulting equation as x0 → ∞. Remark. The solution of the Dirichlet problem of the Laplace equation is derived in [12] using Green’s functions and the method of images. The simple derivation presented here generalises this solution to the Poisson equation. Furthermore, at the same time it provides the solution of the Neumann problem. Using the generalisation of the Pompeiu formula (3.35) it is possible to obtain an alternative representation for the solution of the Poisson equation. This representation involves a kernel for which one can use the generalisation of the Plemelj-Sokhotzki formulae (3.40). Thus, it also provides a way of constructing the Dirichlet to Neumann map by analysing the global relation on the boundary of the domain without solving the equation in the interior. This representation is given in Proposition 5.3. The Dirichlet to Neumann map is constructed in Proposition 5.4. Proposition 5.3. Let D be a bounded simply connected domain in R4 with a smooth boundary ∂D and ∂l and J ξ be defined by equations (2.3) and (2.6) with x replaced by ξ, respectively. Let the real-valued function φ(x) satisfy the Poisson equation (5.1) where h(x) is as in Proposition 5.1. Then, for x ∈ D, φ admits the integral representation Z Z 1 (ζ − z)−1 1 (ζ − z)−1 ξ (5.10) ∂l φ(x) = − 2 h(ξ) dξ + J ∂l φ(ξ). 2π D |ζ − z|2 2π 2 ∂D |ζ − z|2 Furthermore, for x ∈ / D, the boundary values of φ satisfy the global relation Z Z (ζ − z)−1 1 (ζ − z)−1 ξ 1 (5.11) 0=− 2 h(ξ) dξ + J ∂l φ(ξ). 2π D |ζ − z|2 2π 2 ∂D |ζ − z|2 Proof. Equation (5.10) follows from equation (3.35) by replacing f with ∂l φ. Equation (5.11) follows from the derivation of equation (5.10), using the fact that if x ∈ / D, then the function K has ∂r K = 0.

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CMFT

The result analogous to Proposition 5.3 for the non-homogeneous biharmonic equation in four dimensions is given in [24]. Proposition 5.4 (The Dirichlet to Neumann map for the Poisson equation in the half space). Let the real-valued function φ(ξ) satisfy the Poisson equation (5.4) where h(ξ) is as in Proposition 5.1. Then the Neumann boundary value is given in terms of the Dirichlet boundary value by the equation Z 1 ξ0 h(ξ) (5.12) φx0 = − 2 dξ 2 π D [ξ0 + (ξj − xj )2 ]2 Z −(ξj − xj )φξj 1 + 2PV dξ1 dξ2 dξ3 . 2 2 π {ξ0 =0} [(ξj − xj ) ] Proof. Taking the limit of the global relation (5.11) as z approaches the plane {x0 = 0} from negative x0 , we find Z ξ0 − ej (ξj − xj ) 1 h(ξ) dξ + H − (x), (5.13) 0=− 2 2 2 2 2π D [ξ0 + (ξj − xj ) ] where H − (x) is the limit of the function Z (ζ − z)−1 ξ 1 J ∂l φ(ξ), H(x) = 2 2π ∂D |ζ − z|2 as z approaches the plane {x0 = 0} from negative x0 . Using the analogue of the Plemelj-Sokhotzki formulae (3.40), equation (5.13) becomes Z 1 ξ0 − ej (ξj − xj ) 0=− 2 h(ξ) dξ − (φξ0 − ej φξj ) (5.14) π D [ξ0 2 + (ξj − xj )2 ]2 Z −ej (ξj − xj )(φξ0 − ej φξj ) 1 + 2PV dξ1 ∧ dξ2 ∧ dξ3 . π [(ξj − xj )2 ]2 {ξ0 =0} Writing equation (5.14) in component form we obtain four equations the first of which is (5.12).

6. Evaluation of certain three dimensional integrals We now present a methodology for evaluating a large class of three dimensional integrals. This methodology includes integrals of the form of equation (1.1) and is based on the following result. ∗

Proposition 6.1. Let z n1 n2 n3 (x) be the Fueter polynomial of order n defined by ∗ ∗ equation (3.27), and let z n1 n2 n3 (x−1 ) be the polynomial obtained from z n1 n2 n3 (x)

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by replacing x0 by x0 /|z|2 and xj by −xj /|z|2 , j = 1, 2, 3. Then the following identity is valid Z z −1 ∗ 1 (ζ − z)−1 ζ −1 ∗ −1 z z n1 n2 n3 (ξ −1 ) dξ1 dξ2 dξ3 , (6.1) n1 n2 n3 (x ) = 2 2 2 2 |z| 2π {ξ0 =1} |ζ − z| |ζ| where ζ = ξ0 + ξj ej ,

z = x0 + xj ej , x0 > 1.

Proof. Proposition 3.11 implies that the function z −1 ∗ z n n n (x−1 ) |z|2 1 2 3 is left regular in the complement of the origin, therefore the analogue of Cauchy integral formula (3.15) yields Z z −1 ∗ 1 (ζ − z)−1 ξ ζ −1 ∗ −1 z z n1 n2 n3 (ξ −1 ), J (6.2) (x ) = n n n 1 2 3 2 2 2 2 |z| 2π ∂D |ζ − z| |ζ| where J ξ is defined by equation (2.6) with x replaced by ξ and the domain D is defined by the following equation D = ζ = ξ0 + ξj ej : ξ0 2 + ξj 2 ≤ R, ξ0 ≥ 1, j = 1, 2, 3 , where R is a real number. Taking the limit R → ∞, equation (6.2) can be written as z −1 ∗ z n n n (x−1 ) |z|2 1 2 3 Z (ζ − z)−1 ζ −1 ∗ 1 z n n n (ξ −1 ) dξ1 ∧ dξ2 ∧ dξ3 = 2 2π {ξ0 =1} |ζ − z|2 |ζ|2 1 2 3 Z (ζ − z)−1 ξ ζ −1 ∗ 1 z n1 n2 n3 (ξ −1 ), J + lim 2 R→∞ 2π 2 H |ζ − z|2 |ζ| where H is a hemisphere in D with radius R. We will show that the second integral in the right hand side of equation (6.3) vanishes. Indeed, Z (ζ − z)−1 ξ ζ −1 ∗ −1 z J (ξ ) n n n 2 |ζ|2 1 2 3 H |ζ − z| Z (ζ − z)−1 ξ ζ −1 ∗ −1 ≤ |ζ − z|2 J |ζ|2 z n1 n2 n3 (ξ ) H Z 1 ≤ ds (n+3) → 0, as R → ∞, R H where ds is the volume element of the unit sphere, and we have used 1 ∗ | z n1 n2 n3 (ξ −1 )| ≤ n . R Then, equation (6.3) yields equation (6.1). (6.3)

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Example 6.2. Let the three dimensional real integrals {Ij }41 be defined by Z

(1 − x0 − ξj 2 + xj ξj )A(ξ; x) dξ1 dξ2 dξ3 ,

I1 = ZR

3

ZR

3

(−2ξ1 + x1 + x0 ξ1 − x2 ξ3 + x3 ξ2 )A(ξ; x) dξ1 dξ2 dξ3 ,

I2 =

(−2ξ2 + x2 + x0 ξ2 − x3 ξ1 + x1 ξ3 )A(ξ; x) dξ1 dξ2 dξ3 ,

I3 = R3

Z (−2ξ3 + x3 + x0 ξ3 − x1 ξ2 + x2 ξ1 )A(ξ; x) dξ1 dξ2 dξ3 ,

I4 = R3

where the function A(ξ; x) is defined by −2 A(ξ; x) = (1 − x0 )2 + (ξj − xj )2 (1 + ξj 2 )−2 , and x0 , x1 , x2 , x3 are finite real constants not all of which are zero. Then 2π 2 x0 , (x0 2 + xj 2 )2 (−2π 2 )x2 , I3 = (x0 2 + xj 2 )2 I1 =

(6.4)

(−2π 2 )x1 , (x0 2 + xj 2 )2 (−2π 2 )x3 I4 = . (x0 2 + xj 2 )2 I2 =

Indeed, applying Proposition 6.1 for the case n = 0 we find z −1 1 = 2 2 |z| 2π

Z {ξ0 =1}

(ζ − z)−1 ζ −1 dξ1 ∧ dξ2 ∧ dξ3 . |ζ − z|2 |ζ|2

This equation can be written as Z [1 − x0 − ξj (ξj − xj ) + (x0 − 2)ξj ej + xj ej R3

+{(x2 ξ1 − x1 ξ2 )e3 + C.P.}]A(ξ; x) dξ1 dξ2 dξ3 x0 − xj ej = 2π 2 2 , (x0 + xj 2 )2 where C.P. denotes terms obtained by cyclic permutation of the term appearing in the curly bracket. Writing out the above equation in component form we obtain equations (6.4).

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Example 6.3. Let the three dimensional real integrals {Ij }41 be defined by Z I1 = [ξ1 (3 − 2x0 − ξj 2 + xj ξj ) − x1 − x3 ξ2 + x2 ξ3 ]B(ξ; x) dξ1 dξ2 dξ3 , 3 ZR I2 = [1 − x0 − ξj 2 + xj ξj R3

+ξ1 (−2ξ1 + x1 + x0 ξ1 − x2 ξ3 + x3 ξ2 )]B(ξ; x) dξ1 dξ2 dξ3 , Z [−2ξ3 + x3 + x0 ξ3 − x1 ξ2

I3 = R3

+ξ1 (2(x2 − ξ2 ) + x0 ξ2 − x3 ξ1 + x1 ξ3 )]B(ξ; x) dξ1 dξ2 dξ3 , Z [2ξ2 − x2 − x0 ξ2 − x1 ξ3

I4 = R3

+ξ1 (2(x3 − ξ3 ) + x0 ξ3 − x1 ξ2 + x2 ξ1 )]B(ξ; x) dξ1 dξ2 dξ3 , where the function B(ξ; x) is defined by −2 (6.5) B(ξ; x) = (1 − x0 )2 + (ξj − xj )2 (1 + ξj 2 )−3 , and x0 , x1 , x2 , x3 are finite real constants not all of which are zero. Then 4π 2 x0 x1 , (x0 2 + xj 2 )3 2π 2 (x1 x2 + x0 x3 ) , I3 = (x0 2 + xj 2 )3 I1 = −

(6.6)

2π 2 (x1 2 − x0 2 ) , (x0 2 + xj 2 )3 2π 2 (x1 x3 − x0 x2 ) I4 = . (x0 2 + xj 2 )3 I2 =

Indeed, applying Proposition 6.1 for the case n1 = 1, n2 = n3 = 0 we find z −1 ∗ z 100 (x−1 ) |z|2 Z (ζ − z)−1 ζ −1 ∗ 1 z 100 (ξ −1 ) dξ1 ∧ dξ2 ∧ dξ3 . = 2 2π {ξ0 =1} |ζ − z|2 |ζ|2 This equation can be written as Z [1 − x0 − ξj (ξj − xj ) + (x0 − 2)ξj ej + xj ej R3

+{(x2 ξ1 − x1 ξ2 )e3 + C.P.}][ξ1 + e1 ]B(ξ; x) dξ1 dξ2 dξ3 2x0 x1 + (x0 2 − x1 2 )e1 − (x1 x2 + x0 x3 )e2 + (x0 x2 − x1 x3 )e3 = (−2π 2 ) . (x0 2 + xj 2 )3 Writing the above equation in component form we find equations (6.6).

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CMFT

Example 6.4. Let the three dimensional real integrals {Ij }41 be defined by Z I1 = [ξ2 (3 − 2x0 − ξj 2 + xj ξj ) − x2 − x1 ξ3 + x3 ξ1 ]B(ξ; x) dξ1 dξ2 dξ3 , 3 ZR I2 = [2ξ3 − x2 − x0 ξ3 − x2 ξ1 R3

+ξ2 (2(x1 − ξ1 ) + x0 ξ1 − x2 ξ3 + x3 ξ2 )]B(ξ; x) dξ1 dξ2 dξ3 , Z I3 =

[1 − x0 − ξj 2 + xj ξj

R3

+ξ2 (−2ξ2 + x2 + x0 ξ2 − x3 ξ1 + x1 ξ3 )]B(ξ; x) dξ1 dξ2 dξ3 , Z [−2ξ1 + x1 + x0 ξ1 − x2 ξ3

I4 = R3

+ξ2 (2(x3 − ξ3 ) + x0 ξ3 − x1 ξ2 + x2 ξ1 )]B(ξ; x) dξ1 dξ2 dξ3 , where the function B(ξ; x) is defined by equation (6.5). Then 2π 2 (x1 x2 − x0 x3 ) 4π 2 x0 x2 , I = , 2 (x0 2 + xj 2 )3 (x0 2 + xj 2 )3 (6.7) 2π 2 (x2 2 − x0 2 ) 2π 2 (x0 x1 + x2 x3 ) I3 = , I = . 4 (x0 2 + xj 2 )3 (x0 2 + xj 2 )3 Indeed, these equations follow by applying Proposition 6.1 to the case n2 = 1, n1 = n3 = 0. I1 = −

Using the results of Examples 6.2–6.4 it is possible to compute other three dimensional integrals. Two examples of such integrals are given below. Example 6.5. Without integrating with respect to the real parameters, prove that Z (ξj 2 − ξ1 ) dξ1 dξ2 dξ3 2 2 (6.8) π . 2 2 2 = − 2 25 R3 (ξj − 2ξ1 + 2) (1 + ξj ) Indeed, equations (6.4) with x0 = 2, x1 = 1, x2 = x3 = 0 become Z (1 + ξ1 2 + ξ2 2 + ξ3 2 − ξ1 ) dξ1 dξ2 dξ3 4 2 (6.9) = − π , 2 2 25 (ξj − 2ξ1 + 2)2 (1 + ξj )2 R3 Z 2 2 dξ1 dξ2 dξ3 (6.10) π . 2 2 2 = − 2 25 R3 (ξj − 2ξ1 + 2) (1 + ξj ) Equation (6.8) follows by subtracting equation (6.10) from equation (6.9). Example 6.6. Without integrating with respect to the real parameters, prove that Z π2 (ξj 2 − 4ξ2 + 2) dξ1 dξ2 dξ3 (6.11) = . 2 2 3 2 54 R3 (ξj − 2(ξ1 + ξ2 ) + 3) (1 + ξj )

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Indeed, the second and third equation of (6.7) with x0 = 2, x1 = x2 = 1, x3 = 0 become Z −1 − ξ1 + 2ξ2 − ξ2 ξ3 π2 , dξ dξ dξ = 1 2 3 2 2 3 2 108 R3 (ξj − 2(ξ1 + ξ2 ) + 3) (1 + ξj ) Z −1 − ξj 2 + ξ1 + 2ξ2 + ξ2 ξ3 π2 . dξ dξ dξ = − 1 2 3 2 2 3 2 36 R3 (ξj − 2(ξ1 + ξ2 ) + 3) (1 + ξj ) Adding the above two equations we obtain equation (6.11). Acknowledgement. ASF expresses his gratitude to Professor G. Henkin for suggesting this investigation and for many illuminating discussions. Both authors are grateful to the referee for several important suggestions.

Appendix A. We will derive equation (3.25). If t is a quaternionic variable and |t| < 1, then2 (A.1)

1 = 1 + t + tt + ttt + · · · . 1−t

Using t = ζ −1 z, it follows that (ζ − z)−1 = (ζ − ζt)−1 = [ζ(1 − t)]−1 = (1 − t)−1 ζ −1 . Thus, −1 (ζ − z)−1 (1 − t)−1 ζ −1 −1 −1 −1 ζ = = (1 − t) (1 − t) (1 − t) |ζ − z|2 |1 − t|2 |ζ|2 |ζ|2

= (1 + t + tt + · · · )(1 + t + tt + · · · )(1 + t + tt + · · · )

ζ −1 |ζ|2

ζ −1 = 1 + (2t + t) + (3tt + 2tt + tt) + · · · |ζ|2 ζ −1 = [1 + L1 (t) + L2 (t) + · · · ] 2 , |ζ| where L1 and L2 denote the following linear and quadratic in t terms respectively, L1 = 2ζ −1 z + (ζ −1 z), L2 = 3ζ −1 zζ −1 z + 2|ζ −1 z|2 + (ζ −1 zζ −1 z). 2

The validity of equation (A.1) can be established by multiplying both sides of equation (A.1) with (1 − t).

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CMFT

Computing L1 and L2 we find ξ0 [3x0 + x1 e1 + x2 e2 + x3 e3 ] |ζ|2 ξ1 [3x1 − x0 e1 + x3 e2 − x2 e3 ] + C.P. , + |ζ|2 2ξ0 2 L2 = [3x0 2 − x1 2 − x2 2 − x3 2 + 2x0 (x1 e1 + x2 e2 + x3 e3 )] 4 |ζ| 2 2ξ1 2 2 2 2 + [3x1 − x0 − x2 − x3 + 2x1 (−x0 e1 + x3 e2 − x2 e3 )] + C.P. . |ζ|4 4ξ0 ξ1 + [4x0 x1 + (x1 2 − x0 2 )e1 |ζ|4 +x0 (x3 e2 − x2 e3 ) + x1 (x2 e2 + x3 e3 )] + C.P. 4ξ1 ξ2 [4x1 x2 + (x1 2 − x2 2 )e3 + |ζ|4 −x0 (x2 e1 + x1 e2 ) + x3 (x2 e2 − x1 e1 )] + C.P. , L1 =

where C.P. in each curly bracket denotes the two terms obtained from the cyclic permutation {0 → 0, 1 → 2, 2 → 3, 3 → 1} of the term appearing in the curly bracket. We will denote L1 and L2 by 1 0 j ξ L (x) + ξ L (x) , 0 j 1 1 |ζ|2 1 L2 = 4 2ξ0 2 L02 (x) + 2ξj 2 Lj2 (x) |ζ| L1 =

+4ξ0 ξj L0j 2 (x)

+

4ξ1 ξ2 L12 2 (x)

+

4ξ2 ξ3 L23 2 (x)

+

,

4ξ3 ξ1 L31 2 (x)

where L01 (x) L11 (x) L02 (x) L12 (x) L01 2 (x) L12 2 (x)

= = = = = =

3x0 + x1 e1 + x2 e2 + x3 e3 , 3x1 − x0 e1 + x3 e2 − x2 e3 , 3x0 2 − x1 2 − x2 2 − x3 2 + 2x0 (x1 e1 + x2 e2 + x3 e3 ), 3x1 2 − x0 2 − x2 2 − x3 2 + 2x1 (−x0 e1 + x3 e2 − x2 e3 ), 4x0 x1 + (x1 2 − x0 2 )e1 + x0 (x3 e2 − x2 e3 ) + x1 (x2 e2 + x3 e3 ), 4x1 x2 + (x1 2 − x2 2 )e3 − x0 (x2 e1 + x1 e2 ) + x3 (x2 e2 − x1 e1 ),

03 23 31 1 1 01 where {L21 , L31 }, {L22 , L32 }, {L02 2 , L2 }, {L2 , L2 }, are obtained from L1 , L2 , L2 , 12 L2 respectively by cyclic permutation of the indices {0 → 0, 1 → 2, 2 → 3, 3 → 1}.

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Substituting these expressions in the expansion of K(ζ −z) and then substituting the resulting expression into equation (3.15) we find, Z 1 ζ −1 ξ f (x) = (A.2) J f (ξ) 2π 2 C1 |ζ|2 Z 1 ξ0 ζ −1 ξ 0 +L1 (x) J f (ξ) 2π 2 C1 |ζ|4 Z ξ02 ζ −1 ξ 1 0 J f (ξ) +L2 (x) π 2 C1 |ζ|6 Z ξj ζ −1 ξ 1 j +L1 (x) J f (ξ) 2π 2 C1 |ζ|4 Z ξj 2 ζ −1 ξ 1 j J f (ξ) +L2 (x) π 2 C1 |ζ|6 Z ξ0 ξj ζ −1 ξ 2 0j J f (ξ) +L2 (x) π 2 C1 |ζ|6 Z 2 ξ1 ξ2 ζ −1 ξ 12 +L2 (x) J f (ξ) π 2 C1 |ζ|6 Z 2 ξ2 ξ3 ζ −1 ξ 23 +L2 (x) J f (ξ) π 2 C1 |ζ|6 Z 2 ξ3 ξ1 ζ −1 ξ 31 +L2 (x) J f (ξ) π 2 C1 |ζ|6 +··· where C1 denotes the surface |ζ| = 1. The right hand side of equation (A.2) contains the terms in the Taylor expansion of f (x) which are constant, linear and quadratic in x. Each of these terms is of course a left-regular function. Furthermore, it can be verified that each of these terms can be expressed as a symmetric polynomial of the variables z1 , z2 and z3 defined by equations (3.26). Indeed, L01 L11 L02 L12 L01 2 L12 2

= = = =

zj ej , 3z1 − z2 e3 + z3 e2 , −(z1 2 + z2 2 + z3 2 ), [3z1 2 − z2 2 − z3 2 + (z1 z3 + z3 z1 )e2 − (z1 z2 + z2 z1 )e3 ], 1 = z1 2 e1 + [(z1 z3 + z3 z1 )e3 − (z1 z2 + z2 z1 )e2 ] , 2 1 = (z1 2 − z2 2 )e3 + 2(z1 z2 + z2 z1 ) + [(z2 z3 + z3 z2 )e2 − (z1 z3 + z3 z1 )e1 ] , 2

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CMFT

03 23 31 1 1 01 12 and {L21 , L31 }, {L22 , L32 }, {L02 2 , L2 }, {L2 , L2 }, are obtained from L1 , L2 , L2 , L2 as before.

Thus equation (A.2) can be written in the form (A.3) f (x) = f (0) Z 1 ζ −1 ξ + z1 2 [ξ0 e1 + 3ξ1 + ξ2 e3 − ξ3 e2 ] 4 J f (ξ) + C.P. 2π C1 |ζ| Z z1 2 [ξ0 2 + ξ2 2 + ξ3 2 − 3ξ12 + − 2 π C1 ζ −1 ξ −2ξ0 ξ1 e1 − 2ξ1 ξ2 e3 + 2ξ1 ξ3 e2 ] 6 J f (ξ) + C.P. |ζ| Z (z1 z2 + z2 z1 ) [4ξ1 ξ2 + (ξ2 2 − ξ1 2 )e3 + 2 π C1 ζ −1 ξ +(ξ0 ξ2 + ξ3 ξ1 )e1 − (ξ0 ξ1 + ξ2 ξ3 )e2 ] 6 J f (ξ) + C.P. |ζ| +··· . The brackets appearing in equation (A.3) can be written in terms of derivatives of f (x). Indeed, differentiating the equation zz −1 = 1 with respect to xj we find ∂xj (z −1 ) = −z −1 ej z −1 ,

j = 1, 2, 3.

Using these formulae it can be shown that equation (A.3) can be rewritten in the form (A.4) f (x) = f (0) Z (ζ − z)−1 ξ ∂ 1 + z1 J f (ξ)|z=0 + C.P. ∂x1 2π 2 C1 |ζ − z|2 Z 1 2 ∂2 (ζ − z)−1 ξ 1 + z1 J f (ξ)|z=0 + C.P. 2 ∂x1 ∂x1 2π 2 C1 |ζ − z|2 Z (ζ − z)−1 ξ 1 ∂2 1 + (z1 z2 + z2 z1 ) J f (ξ)|z=0 + C.P. 2 ∂x1 ∂x2 2π 2 C1 |ζ − z|2 +··· .

Equation (3.25) follows from equation (A.4).

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Appendix B. The Fueter polynomials of order n = 3 are defined as follows: ∗

z 300 = ∗

z 030 = ∗

z 003 = ∗

z 210 = ∗

z 120 = ∗

z 012 = ∗

z 021 = ∗

z 201 = ∗

z 102 = ∗

z 111 =

1 3 z1 , 2 1 3 z2 , 2 1 3 z3 , 2 1 z1 z2 z1 + z1 2 z2 + z2 z1 2 , 3! 1 z2 z1 z2 + z2 2 z1 + z1 z2 2 , 3! 1 z3 z2 z3 + z3 2 z2 + z2 z3 2 , 3! 1 z2 z3 z2 + z2 2 z3 + z3 z2 2 , 3! 1 z1 z2 z1 + z1 2 z3 + z3 z1 2 , 3! 1 z3 z1 z3 + z3 2 z1 + z1 z3 2 , 3! 1 (z1 z2 z3 + z3 z1 z2 + z2 z3 z1 + z1 z3 z2 + z3 z2 z1 + z2 z1 z3 ) . 3!

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