DOCSLIB.ORG

Quaternionic Analysis on Riemann Surfaces and Differential Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quaternionic Analysis on Riemann Surfaces and Differential Geometry Doc Math J DMV Quaternionic Analysis on Riemann Surfaces and Differential Geometry 1 2 Franz Pedit and Ulrich Pinkall Abstract We present a new approach to the dierential geometry of 3 4 surfaces in R and R that treats this theory as a quaternionied ver sion of the complex analysis and algebraic geometry of Riemann surfaces Mathematics Sub ject Classication C H Introduction Meromorphic functions Let M b e a Riemann surface Thus M is a twodimensional dierentiable manifold equipp ed with an almost complex structure J ie on each tangent space T M we p 2 have an endomorphism J satisfying J making T M into a onedimensional p complex vector space J induces an op eration on forms dened as X JX A map f M C is called holomorphic if df i df 1 A map f M C fg is called meromorphic if at each p oint either f or f is holomorphic Geometrically a meromorphic function on M is just an orientation 2 preserving p ossibly branched conformal immersion into the plane C R or 1 2 rather the sphere CP S 4 Now consider C as emb edded in the quaternions H R Every immersed 4 4 surface in R can b e descib ed by a conformal immersion f M R where M is a suitable Riemann surface In Section we will show that conformality can again b e expressed by an equation like the CauchyRiemann equations df N df 2 3 where now N M S in R Im H is a map into the purely imaginary quaternions of norm In the imp ortant sp ecial case where f takes values in 1 Research supp orted by NSF grants DMS DMS and SFB at TUBerlin 2 Research supp orted by SFB at TUBerlin Documenta Mathematica Extra Volume ICM II Franz Pedit and Ulrich Pinkall 3 R Im H N is just the unit normal vector for the surface f In the dierential 4 geometry of surfaces in R N is called the left normal vector of f Meromorphic 4 4 1 functions f M R fg S HP are dened as in the complex case To summarize from the quaternionic viewp oint i is just one sp ecial imag inary quaternion of norm one The transition from complex analysis to surface theory is done by i leaving the Riemann surface as it is ii allowing the whole of H fg as the target space of meromorphic functions iii writing the Cauchy Riemann equations with a variable i Line bundles A classical metho d to construct meromorphic functions on a Riemann surface M is to take the quotient of two holomorphic sections of a holomorphic line bundle n over M For example if M is realized as an algebraic curve in CP then the n ane co ordinate functions on CP are quotients of holomorphic sections of the inverse of the tautological bundle over M Another common way to construct meromorphic functions is to take quotients of theta functions which also can b e viewed as sections of certain holomorphic line bundles over M In Section we intro duce the notion of a holomorphic quaternionic line bundle L over M Quotients of holomorphic sections of such bundles are meromorphic conformal maps into H and every conformal map can b e obtained as such a quotient in a unique way Every complex holomorphic line bundle E gives rise to a certain holomor phic quaternionic bundle L E E The deviation of a general holomorphic quaternionic bundle L from just b eing a doubled complex bundle can b e globally measured by a quantity Z 2 W jQj called the Willmore functional of L Here Q is a certain tensor eld the Hopf eld On compact surfaces W is up to a constant the Willmore functional in the 4 usual sense of surface theory of f M H R where f is the quotient of any two holomorphic sections of L Abelian differentials A second classical metho d to construct meromorphic functions on a Riemann surface M is to use Ab elian dierentials ie integrals of meromorphic forms In the quaternionic theory there is no go o d analog of the canonical bundle K On the other hand also in the complex case forms often arise as pro ducts of sections 1 of two line bundles E and KE Notably this is the case in situations where the RiemannRo ch theorem is applied This setup carries over p erfectly to the quaternionic case including the RiemannRo ch theorem itself Documenta Mathematica Extra Volume ICM II Quaternionic Analysis on Riemann Surfaces We show that for each holomorphic quaternionic line bundle L there exists 1 a certain holomorphic quaternionic line bundle KL such that any holomorphic 1 section of L can b e multiplied with any holomorphic section of KL the pro duct b eing a closed H valued form that lo cally integrates to a confor mal map f into H df 1 In the case where KL is isomorphic to L itself we call L a spin bundle If is a nowhere vanishing holomorphic section of a spin bundle then df 3 denes a conformal immersion into R This construction is in fact a more intrinsic version of the Weierstrassrepresentation for general surfaces in space that has received much attention in the recent literature just as when expressed in 4 co ordinates gives a representation for surfaces in R The Hopf eld Q mentioned ab ove can b e identied as the Diracp otential or mean curvature halfdensity of the surface f H jdf j Q Here H is the mean curvature and jdf j is the square ro ot of the induced metric Applications The only geometric application discussed in some detail in this pap er is a rigidity 2 3 theorem for spheres if f g S R are two conformal immersions which are not congruent up to scale but have the same mean curvature halfdensity then Z 2 H This inequality is sharp Many other applications to b e discussed in a more elab orate future pap er will concern the geometry of Willmore surfaces critical p oints of the Willmore 3 4 functional b oth in R and R Moreover rudiments of an algebraic geometry of n holomorphic curves in quaternionic pro jective space HP can b e develop ed Conformal surfaces the standard example 3 Let M b e a Riemann surface and f M R a smo oth map The map f is a conformal immersion if i df v is p erp endicular to df J v for any tangent vector v where J is the complex structure on M and ii jdf v j jdf J v j for v Documenta Mathematica Extra Volume ICM II Franz Pedit and Ulrich Pinkall 2 If N M S is the oriented unit normal to f then the conformality condition can b e rephrased as df J v N df v To see the similarity with complex function theory we rewrite this condition using 3 quaternions H R Im H We will always think of R Im H as the imaginary 3 quaternions If x y R then xy x y x y With the notation the conformality condition for f b ecomes For the rest of the article we will take this to b e the dening equation for conformality also in 4 the case of maps not necessarily immersions into R 4 Definition A map f M R H is conformal if there exists a map 2 N M H such that N and df N df At immersed p oints this is equivalent to the usual notion of conformality and 3 f determines N uniquely If f is R valued then N is the oriented unit normal but otherwise N is not normal to f We will call N the left normal to f Moreover 1 1 if f is conformal so is its Mo ebius inversion f with left normal f N f Thus the ab ove denition is Mo ebius invariant and hence denes conformality of maps 4 1 f M S HP H fg Holomorphic quaternionic line bundles A quaternionic line bund le L over a base manifold is a smo oth rank real vector bundle whose b ers have the structure of dimensional quaternionic right vector spaces varying smo othly over the base Two quaternionic line bundles L and L 1 2 are isomorphic if there exists a smo oth bundle isomorphism A L L that is 1 2 quaternionic linear on each b er We adopt the usual notation Hom L L and H 1 2 End L Hom L L etc for the spaces of quaternionic linear maps H H The zero section of a quaternionic line bundle over an oriented surface has co dimension so that transverse sections have no zeros Thus any quaternionic line bundle over a Riemann surface M is smo othly isomorphic to M H Complex quaternionic line bundles Example Given a conformal map f M H with left normal N the quater nionic line bundle L M H also has a complex structure J L L given by J N for L We make this additional complex structure part of our theory Definition A complex quaternionic line bund le over a base manifold is a pair L J where L is a quaternionic line bundle and J End L is a quaternionic H 2 linear endomorphism such that J Documenta Mathematica Extra Volume ICM II Quaternionic Analysis on Riemann Surfaces Put dierently a complex quaternionic line bundle is a rank two left complex vector bundle whose complex structure is compatible with the right quaternionic structure Two complex quaternionic line bundles are isomorphic if the quater nionic linear isomorphism is also left complex linear The dual of a quaternionic line bundle L 1 L f L H quaternionic linearg has a natural structure of a left quaternionic line bundle via 1 1 for H L and L Using conjugation we can regard L as a right quaternionic line bundle If L has a complex structure then the 1 complex structure on L is given by J J 1 so that L is also complex quaternionic Any complex quaternionic line bundle L can b e tensored on the left by a complex line bundle E yielding the complex quaternionic line bundle EL On a Riemann surface M we have the canonical and anticanonical bundles K and K It is easy to see that KL f TM L J g and KL f TM L J g In this way we have split the quaternionic rank bundle Hom T M L which R has a left complex structure given by as a direct sum KL KL of two complex quaternionic
Load more

Recommended publications
  • Quaternion Zernike Spherical Polynomials
    MATHEMATICS OF COMPUTATION Volume 84, Number 293, May 2015, Pages 1317–1337 S 0025-5718(2014)02888-3 Article electronically published on August 29, 2014 QUATERNION ZERNIKE SPHERICAL POLYNOMIALS J. MORAIS AND I. CAC¸ AO˜ Abstract. Over the past few years considerable attention has been given to the role played by the Zernike polynomials (ZPs) in many different fields of geometrical optics, optical engineering, and astronomy. The ZPs and their applications to corneal surface modeling played a key role in this develop- ment. These polynomials are a complete set of orthogonal functions over the unit circle and are commonly used to describe balanced aberrations. In the present paper we introduce the Zernike spherical polynomials within quater- nionic analysis ((R)QZSPs), which refine and extend the Zernike moments (defined through their polynomial counterparts). In particular, the underlying functions are of three real variables and take on either values in the reduced and full quaternions (identified, respectively, with R3 and R4). (R)QZSPs are orthonormal in the unit ball. The representation of these functions in terms of spherical monogenics over the unit sphere are explicitly given, from which several recurrence formulae for fast computer implementations can be derived. A summary of their fundamental properties and a further second or- der homogeneous differential equation are also discussed. As an application, we provide the reader with plot simulations that demonstrate the effectiveness of our approach. (R)QZSPs are new in literature and have some consequences that are now under investigation. 1. Introduction 1.1. The Zernike spherical polynomials. The complex Zernike polynomials (ZPs) have long been successfully used in many different fields of optics.
    [Show full text]
  • The Riemann Mapping Theorem Christopher J. Bishop
    The Riemann Mapping Theorem Christopher J. Bishop C.J. Bishop, Mathematics Department, SUNY at Stony Brook, Stony Brook, NY 11794-3651 E-mail address: [email protected] 1991 Mathematics Subject Classification. Primary: 30C35, Secondary: 30C85, 30C62 Key words and phrases. numerical conformal mappings, Schwarz-Christoffel formula, hyperbolic 3-manifolds, Sullivan’s theorem, convex hulls, quasiconformal mappings, quasisymmetric mappings, medial axis, CRDT algorithm The author is partially supported by NSF Grant DMS 04-05578. Abstract. These are informal notes based on lectures I am giving in MAT 626 (Topics in Complex Analysis: the Riemann mapping theorem) during Fall 2008 at Stony Brook. We will start with brief introduction to conformal mapping focusing on the Schwarz-Christoffel formula and how to compute the unknown parameters. In later chapters we will fill in some of the details of results and proofs in geometric function theory and survey various numerical methods for computing conformal maps, including a method of my own using ideas from hyperbolic and computational geometry. Contents Chapter 1. Introduction to conformal mapping 1 1. Conformal and holomorphic maps 1 2. M¨obius transformations 16 3. The Schwarz-Christoffel Formula 20 4. Crowding 27 5. Power series of Schwarz-Christoffel maps 29 6. Harmonic measure and Brownian motion 39 7. The quasiconformal distance between polygons 48 8. Schwarz-Christoffel iterations and Davis’s method 56 Chapter 2. The Riemann mapping theorem 67 1. The hyperbolic metric 67 2. Schwarz’s lemma 69 3. The Poisson integral formula 71 4. A proof of Riemann’s theorem 73 5. Koebe’s method 74 6.
    [Show full text]
  • A Generalization of Fueter's Monogenic Functions to Fine Domains
    WSGP 25 Roman Lávička A generalization of Fueter’s monogenic functions to fine domains In: Martin Čadek (ed.): Proceedings of the 25th Winter School "Geometry and Physics". Circolo Matematico di Palermo, Palermo, 2006. Rendiconti del Circolo Matematico di Palermo, Serie II, Supplemento No. 79. pp. [129]--138. Persistent URL: http://dml.cz/dmlcz/701772 Terms of use: © Circolo Matematico di Palermo, 2006 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie II, Suppl. 79 (2006), pp. 129-138 A GENERALIZATION OF FUETER'S MONOGENIC FUNCTIONS TO FINE DOMAINS ROMAN LAVICKA ABSTRACT. The so-called quaternionic analysis is a theory analogous to complex analysis but the complex numbers are replaced by the non-commutative 4-dimensional field H of quaternions. A role of holomorphic functions in quaternionic analysis is played by monogenic functions. In this note we extend the notion of monogeneity to functions defined on fine domains in R4, i.e., domains in the fine topology of classical potential theory In fact, we generalize to some extent Fuglede's theory of finely holomorphic functions for dimension 4. 1. INTRODUCTION At the beginning of the twentieth century E. Borel tried to extend holomorphic functions to more general domains (no longer open) in such a way that the unique continuation property was preserved, see [3].
    [Show full text]
  • The Chazy XII Equation and Schwarz Triangle Functions
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 095, 24 pages The Chazy XII Equation and Schwarz Triangle Functions Oksana BIHUN and Sarbarish CHAKRAVARTY Department of Mathematics, University of Colorado, Colorado Springs, CO 80918, USA E-mail: [email protected], [email protected] Received June 21, 2017, in final form December 12, 2017; Published online December 25, 2017 https://doi.org/10.3842/SIGMA.2017.095 Abstract. Dubrovin [Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120{348] showed that the Chazy XII equation y000 − 2yy00 + 3y02 = K(6y0 − y2)2, K 2 C, is equivalent to a projective-invariant equation for an affine connection on a one-dimensional complex manifold with projective structure. By exploiting this geometric connection it is shown that the Chazy XII solution, for certain values of K, can be expressed as y = a1w1 +a2w2 +a3w3 where wi solve the generalized Darboux{Halphen system. This relationship holds only for certain values of the coefficients (a1; a2; a3) and the Darboux{Halphen parameters (α; β; γ), which are enumerated in Table2. Consequently, the Chazy XII solution y(z) is parametrized by a particular class of Schwarz triangle functions S(α; β; γ; z) which are used to represent the solutions wi of the Darboux{Halphen system. The paper only considers the case where α + β +γ < 1. The associated triangle functions are related among themselves via rational maps that are derived from the classical algebraic transformations of hypergeometric functions. The Chazy XII equation is also shown to be equivalent to a Ramanujan-type differential system for a triple (P;^ Q;^ R^).
    [Show full text]
  • Hypercomplex Analysis Selected Topics
    Hypercomplex Analysis Selected Topics (Habilitation Thesis) Roman Lávička Field: Mathematics — Mathematical Analysis Faculty of Mathematics and Physics Charles University in Prague August 2011 Contents 1 Introduction 5 2 Preliminaries of hypercomplex analysis 11 2.1 Clifford analysis . 11 2.1.1 Euclidean Clifford analysis . 14 2.1.2 Generalized Moisil-Théodoresco systems . 14 2.1.3 Hermitian Clifford analysis . 16 2.2 Quaternionic analysis . 17 3 Complete orthogonal Appell systems 19 3.1 Spherical harmonics . 19 3.1.1 Gelfand-Tsetlin bases for spin modules . 21 3.2 Clifford algebra valued spherical monogenics . 22 3.3 Spinor valued spherical monogenics . 24 3.3.1 The generalized Appell property in dimension 3 . 27 3.4 Hodge-de Rham systems . 29 3.4.1 The Riesz system in dimension 3 . 33 3.5 Hermitian monogenics . 35 4 Finely monogenic functions 39 4.1 Finely holomorphic functions . 39 4.2 Finely monogenic functions . 41 4.3 Finely differentiable monogenic functions . 42 4.4 Open problems . 45 List of reprinted papers 47 Bibliography 49 [L1] Reversible maps in the group of quaternionic Möbius trans- formations 55 [L2] A generalization of monogenic functions to fine domains 71 [L3] A remark on fine differentiability 83 [L4] Finely continuously differentiable functions 91 [L5] The Gelfand-Tsetlin bases for spherical monogenics in di- mension 3 105 3 [L6] Canonical bases for sl(2,C)-modules of spherical monogen- ics in dimension 3 135 [L7] Orthogonal basis for spherical monogenics by step two branch- ing 149 [L8] The Fischer Decomposition for the H-action and Its Appli- cations 177 [L9] The Gelfand-Tsetlin bases for Hodge-de Rham systems in Euclidean spaces 189 [L10] Gelfand-Tsetlin Bases of Orthogonal Polynomials in Her- mitean Clifford Analysis 211 4 Chapter 1 Introduction Without any doubt, complex analysis belongs to the most important areas of mathematics.
    [Show full text]
  • Quaternionic Analysis
    Quaternionic Analysis A. Sudbery [[email protected]] Department of Mathematics University of York Heslington, York YO1 5DD August 1977 Contents 1 Introduction 2 2 Preliminaries 5 2.1 The Algebra of Quaternions1 . 5 2.2 Quaternionic Differential Forms . 9 3 Regular Functions 11 4 Cauchy’s Theorem and the Integral Formula 17 5 Some Immediate Consequences 25 6 Construction of Regular Functions 26 7 Regular Functions and Conformal Mappings 30 8 Homogeneous Regular Functions 32 9 Regular Power Series 39 1 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 associative 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 century after Hamilton’s discov- ery 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 modern form for the first time in Hamilton’s 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 regular function of a complex variable has interesting consequences when adapted to quater- nions; one is too restrictive, the other not restrictive enough.
    [Show full text]
  • Quaternion Derivatives: the GHR Calculus
    Technical Report, TR-Quat-GHR-Calc-08-01-14/1 Imperial College London Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 Quaternion Derivatives: The GHR Calculus D. P. Xu, C. Jahanchahi, C. C. Took and D. P. Mandic Abstract Quaternion derivatives in the mathematical literature are typically defined only for analytic (regular) functions. However, in engineering problems, functions of interest are often real-valued and thus not analytic, such as the standard cost function. The HR calculus is a convenient way to calculate formal derivatives of both analytic and non-analytic functions of quaternion variables, however, both the HR and other functional calculus in quaternion analysis have encountered an essential technical obstacle, that is, the traditional product rule is invalid due to the non- commutativity of the quaternion algebra. To address this issue, a generalized form of the HR derivative is proposed based on a general orthogonal system. The so introduced generalization, called the generalized HR (GHR) calculus, encompasses not just the left- and right-hand versions of quaternion derivative, but also enables solutions to some long standing problems, such as the novel product rule, the chain rule, the mean-valued theorem and Taylor’s theorem. At the core of the proposed approach is the quaternion rotation, which can naturally be applied to other functional calculi in non-commutative settings. Examples on using the GHR calculus in adaptive signal processing support the analysis. 1. Introduction. Quaternions have become a standard in physics [40], computer graphics [1], and have also been successfully applied to many signal processing and communications problems [9, 10, 11, 20, 37, 38, 39, 43, 45, 46, 47].
    [Show full text]
  • Convergence of Very Weak Solutions to A-Dirac Equations in Clifford Analysis Yueming Lu* and Hui Bi
    Lu and Bi Advances in Difference Equations (2015)2015:219 DOI 10.1186/s13662-015-0555-y R E S E A R C H Open Access Convergence of very weak solutions to A-Dirac equations in Clifford analysis Yueming Lu* and Hui Bi *Correspondence: [email protected] Abstract Department of Applied Mathematics, Harbin University of This paper is concerned with the very weak solutions to A-Dirac equations Science and Technology, Xue Fu DA(x, Du) = 0 with Dirichlet boundary data. By means of the decomposition in a Road, Harbin, 150080, P.R. China Clifford-valued function space, convergence of the very weak solutions to A-Dirac equations is obtained in Clifford analysis. MSC: Primary 35J25; secondary 31A35 Keywords: Dirac equations; very weak solutions; decomposition; convergence 1 Introduction In this paper, we shall consider a nonlinear mapping A : × Cn → Cn such that (N) x → A(x, ξ) is measurable for all ξ ∈ Cn, (N) ξ → A(x, ξ) is continuous for a.e. x ∈ , p– (N) |A(x, ξ)–A(x, ζ )|≤b|ξ – ζ | , p– (N) (A(x, ξ)–A(x, ζ ), ξ – ζ ) ≥ a|ξ – ζ | (|ξ| + |ζ |) , where < a ≤ b < ∞. ,p The exponent p > will determine the natural Sobolev class, denoted by W (, Cn), in which to consider the A-Dirac equations DA(x, Du)=. () ∈ ,p We call u Wloc (, Cn) a weak solution to ()if A(x, Du)Dϕ dx = () ∈ ,p for each ϕ Wloc (, Cn) with compact support. { } ∈ ,s Definition . For s > max , p– , a Clifford-valued function u Wloc(, Cn) is called a s , s–p+ very weak solution of equation ()ifitsatisfies() for all ϕ ∈ W (, Cn)withcompact support.
    [Show full text]
  • Conjugacy Classification of Quaternionic M¨Obius Transformations
    CONJUGACY CLASSIFICATION OF QUATERNIONIC MOBIUS¨ TRANSFORMATIONS JOHN R. PARKER AND IAN SHORT Abstract. It is well known that the dynamics and conjugacy class of a complex M¨obius transformation can be determined from a simple rational function of the coef- ficients of the transformation. We study the group of quaternionic M¨obius transforma- tions and identify simple rational functions of the coefficients of the transformations that determine dynamics and conjugacy. 1. Introduction The motivation behind this paper is the classical theory of complex M¨obius trans- formations. A complex M¨obiustransformation f is a conformal map of the extended complex plane of the form f(z) = (az + b)(cz + d)−1, where a, b, c and d are complex numbers such that the quantity σ = ad − bc is not zero (we sometimes write σ = σf in order to avoid ambiguity). There is a homomorphism from SL(2, C) to the group a b of complex M¨obiustransformations which takes the matrix ( c d ) to the map f. This homomorphism is surjective because the coefficients of f can always be scaled so that σ = 1. The collection of all complex M¨obius transformations for which σ takes the value 1 forms a group which can be identified with PSL(2, C). A member f of PSL(2, C) is simple if it is conjugate in PSL(2, C) to an element of PSL(2, R). The map f is k–simple if it may be expressed as the composite of k simple transformations but no fewer. Let τ = a + d (and likewise we write τ = τf where necessary).
    [Show full text]
  • Conformal Mapping Solution of Laplace's Equation on a Polygon
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 14 (1986) 227-249 227 North-Holland Conformal mapping solution of Laplace’s equation on a polygon with oblique derivative boundary conditions Lloyd N. TREFETHEN * Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. Ruth J. WILLIAMS ** Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, U.S.A. Received 6 July 1984 Abstract: We consider Laplace’s equation in a polygonal domain together with the boundary conditions that along each side, the derivative in the direction at a specified oblique angle from the normal should be zero. First we prove that solutions to this problem can always be constructed by taking the real part of an analytic function that maps the domain onto another region with straight sides oriented according to the angles given in the boundary conditions. Then we show that this procedure can be carried out successfully in practice by the numerical calculation of Schwarz-Christoffel transformations. The method is illustrated by application to a Hall effect problem in electronics, and to a reflected Brownian motion problem motivated by queueing theory. Keywor& Laplace equation, conformal mapping, Schwarz-Christoffel map, oblique derivative, Hall effect, Brownian motion, queueing theory. Mathematics Subject CIassi/ication: 3OC30, 35525, 60K25, 65EO5, 65N99. 1. The oblique derivative problem and conformal mapping Let 52 by a polygonal domain in the complex plane C, by which we mean a possibly unbounded simply connected open subset of C whose boundary aa consists of a finite number of straight lines, rays, and line segments.
    [Show full text]
  • Duality in Complex Clifford Analysis
    JOURNAL OF FUNCTIONAL ANALYSIS 61, 117-135 (1985) Duality in Complex Clifford Analysis JOHN RYAN* Department of Mathematics, University of York, Heslington, York YOI SDD, Britain Communicated by L. Gross Received December 6, 1982 The aim of this paper is to characterize the dual and bidual of complex Clifford modules of holomorphic functions which are defined over domains in @“+I and satisfy generalized Cauchy-Riemann equations. In one instance the generalized Cauchy-Riemann equation reduces to a holomorphic extension of Maxwell’s equations in vacua. 0 1985 Academic Press, Inc. 1. INTRODUCTION The development of function theories over certain classes of Clifford algebras has proved increasingly relevant in a number of branches of mathematics and mathematical physics (e.g., [2-4, 6, 14, 15)). In particular in [2] Delanghe and Brackx establish a generalization, to real Clifford algebras, of the classical result of Kijthe [7] which characterizes the dual of the space of holomorphic functions defined over an arbitrary, proper, open subset of the Riemann sphere. In this paper we extend the analysis developed by Delanghe and Brackx [2] and characterize the dual and bidual of complex Clifford modules of holomorphic functions, defined over certain classes of domains lying in C”+ ‘, and satisfying generalized Cauchy-Riemann equations. The paper is a continuation of [9-121 and relies on a number of results and ideas developed in these papers. The analysis developed here will later be applied [ 131 to provide representations of the dual of complex Clifford modules of real analytic functions defined over certain classesof real n-dimensional manifolds lying in F+‘, thereby generalizing recent results obtained on representations of analytic functionals on the unit sphere [14].
    [Show full text]
  • Quaternionic Analysis
    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 Hamilton’s 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 Hamilton’s 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 regular function 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.
    [Show full text]