Bulletin of the American Mathematical Society

MathSciNet review: 1567511
Full text of review: PDF   This review is available free of charge.
Book Information:

Authors: F. Brackx, R. Delanghe and F. Sommen
Title: Clifford analysis
Additional book information: Research Notes in Mathematics, Vol. 76, Pitman Advanced Publishing Program, Boston, 1982, 308 pp., $19.95. ISBN 0-2730-8535-2.

Paul Bosshard, Die Cliffordschen Zahlen, ihre Algebra und ihre Funktionentheorie, University of Zürich, 1940 (German). Thesis. MR 0037823

F. Brackx, On $(k)$-monogenic functions of a quaternion variable, Function theoretic methods in differential equations, Res. Notes in Math., No. 8, Pitman, London, 1976, pp. 22–44. MR 0492321

F. F. Brackx, Non-$(k)$-monogenic points of functions of a quaternion variable, Function theoretic methods for partial differential equations (Proc. Internat. Sympos., Darmstadt, 1976) Lecture Notes in Math., Vol. 561, Springer, Berlin, 1976, pp. 138–149. MR 0466584

Claude C. Chevalley, The algebraic theory of spinors, Columbia University Press, New York, 1954. MR 0060497

Professor Clifford, Applications of Grassmann’s Extensive Algebra, Amer. J. Math. 1 (1878), no. 4, 350–358. MR 1505182, DOI 10.2307/2369379

F. Constantinescu, Analytic properties of nonstrictly localizable fields, J. Mathematical Phys. 12 (1971), 293–298. MR 378617, DOI 10.1063/1.1665590

C. A. Deavours, The quaternion calculus, Amer. Math. Monthly 80 (1973), 995–1008. MR 325986, DOI 10.2307/2318774

Richard Delanghe, On regular-analytic functions with values in a Clifford algebra, Math. Ann. 185 (1970), 91–111. MR 265618, DOI 10.1007/BF01359699

Richard Delanghe, Morera’s theorem for functions with values in a Clifford algebra, Simon Stevin 43 (1969/70), 129–140. MR 271374

Richard Delanghe, On regular points and Liouville’s theorem for functions with values in a Clifford algebra, Simon Stevin 44 (1970/71), 55–66. MR 280727

Richard Delanghe, On the singularities of functions with values in a Clifford algebra, Math. Ann. 196 (1972), 293–319. MR 301213, DOI 10.1007/BF01428219

R. Delanghe and F. Brackx, Hypercomplex function theory and Hilbert modules with reproducing kernel, Proc. London Math. Soc. (3) 37 (1978), no. 3, 545–576. MR 512025, DOI 10.1112/plms/s3-37.3.545

Avron Douglis, A function-theoretic approach to elliptic systems of equations in two variables, Comm. Pure Appl. Math. 6 (1953), 259–289. MR 64277, DOI 10.1002/cpa.3160060205

14.

J. D. Edmonds, Quaternion quantum theory: new physics or number mysticism?, Amer. J. Phys. 42 (1974), 220-223.

Leon Ehrenpreis, Solution of some problems of division. I. Division by a polynomial of derivation, Amer. J. Math. 76 (1954), 883–903. MR 68123, DOI 10.2307/2372662

Leon Ehrenpreis, Solution of some problems of division. II. Division by a punctual distribution, Amer. J. Math. 77 (1955), 286–292. MR 70048, DOI 10.2307/2372532

Leon Ehrenpreis, Solutions of some problems of division. III. Division in the spaces, ${\scr D}’,{\scr H}, {\scr Q}_A,{\scr O}$, Amer. J. Math. 78 (1956), 685–715. MR 83690, DOI 10.2307/2372464

L. Ehrenpreis, Solution of some problems of division. IV. Invertible and elliptic operators, Amer. J. Math. 82 (1960), 522–588. MR 119082, DOI 10.2307/2372971

Rud Fueter, Analytische Funktionen einer Quaternionenvariablen, Comment. Math. Helv. 4 (1932), no. 1, 9–20 (German). MR 1509442, DOI 10.1007/BF01202702

Run Fueter, Die Funktionentheorie der Differentialgleichungen $\Theta u=0$ und $\Theta \Theta u=0$ mit vier reellen Variablen, Comment. Math. Helv. 7 (1934), no. 1, 307–330 (German). MR 1509515, DOI 10.1007/BF01292723

Rud Fueter, Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen, Comment. Math. Helv. 8 (1935), no. 1, 371–378 (German). MR 1509533, DOI 10.1007/BF01199562

Rud Fueter, Zur Theorie der regulären Funktionen einer Quaternionenvariablen, Monatsh. Math. Phys. 43 (1936), no. 1, 69–74 (German). MR 1550511, DOI 10.1007/BF01707588

Robert P. Gilbert (ed.), Plane ellipticity and related problems, Contemporary Mathematics, vol. 11, American Mathematical Society, Providence, R.I., 1982. MR 669227

Robert P. Gilbert and Gerald N. Hile, Generalized hyperanalytic function theory, Bull. Amer. Math. Soc. 78 (1972), 998–1001. MR 306519, DOI 10.1090/S0002-9904-1972-13080-5

Robert P. Gilbert and Gerald Hile, Generalized hypercomplex function theory, Trans. Amer. Math. Soc. 195 (1974), 1–29. MR 450577, DOI 10.1090/S0002-9947-1974-0450577-4

Feza Gürsey and Hsiung Chia Tze, Complex and quaternionic analyticity in chiral and gauge theories. I, Ann. Physics 128 (1980), no. 1, 29–130. MR 587683, DOI 10.1016/0003-4916(80)90056-1

Hans Georg Haefeli, Hyperkomplexe Differentiale, Comment. Math. Helv. 20 (1947), 382–420 (German). MR 25586, DOI 10.1007/BF02568139

William Rowan Hamilton, The mathematical papers of Sir William Rowan Hamilton. Vol. III: Algebra, Cunningham Memoir No. XV, Cambridge University Press, London-New York, 1967. Edited by H. Halberstam and R. E. Ingram. MR 0213217

29.

D. Hestenes, Space-time algebra, Gordon and Breach, New York, 1966.

D. Hestenes, Multivector functions, J. Math. Anal. Appl. 24 (1968), 467–473. MR 234368, DOI 10.1016/0022-247X(68)90002-4

Lars Hörmander, On the division of distributions by polynomials, Ark. Mat. 3 (1958), 555–568. MR 124734, DOI 10.1007/BF02589517

V. Iftimie, Fonctions hypercomplexes, Bull. Math. Soc. Sci. Math. R. S. Roumanie 9(57) (1965), 279–332 (1966) (French). MR 217312

K. Imaeda, A new formulation of classical electrodynamics, Nuovo Cimento B (11) 32 (1976), no. 1, 138–162 (English, with Italian and Russian summaries). MR 408457, DOI 10.1007/BF02726749

M. A. Jafarizadeh, M. Snyder, and H. C. Tze, Quaternionic multi-$S^{4}$ instantons in general covariant $\textrm {SU}(2)$ Yang-Mills and $\textbf {HP}(n)$ $\sigma$-models, Nuclear Phys. B 176 (1980), no. 1, 221–242. MR 596908, DOI 10.1016/0550-3213(80)90074-7

Yu. N. Kafiev, Four-dimensional $\sigma$-models and geometry of Yang-Mills fields, Nuclear Phys. B 178 (1981), no. 1, 177–196. MR 605587, DOI 10.1016/0550-3213(81)90502-2

36.

V. P. Khavin, Spaces of analytic functions, Plenum Press, New York, 1968, pp. 75-167.

Gottfried Köthe, Die Randverteilungen analytischer Funktionen, Math. Z. 57 (1952), 13–33 (German). MR 52032, DOI 10.1007/BF01192913

Hyperfunctions and pseudo-differential equations, Lecture Notes in Mathematics, Vol. 287, Springer-Verlag, Berlin-New York, 1973. Dedicated to the memory of André Martineau; Edited by Hikasaburo Komatsu. MR 0388082

Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234

Bernard Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 271–355 (French). MR 86990

Reinhold Meise, Representation of distributions and ultradistributions by holomorphic functions, Functional analysis: surveys and recent results (Proc. Conf., Paderborn, 1976) North-Holland Math. Studies, Vol. 27; Notas de Mat., No. 63, North-Holland, Amsterdam, 1977, pp. 189–208. MR 0482169

Walter Nef, Die Funktiontheorie der partiellen Differentialgleichungen zweiter Ordnung. (Hyperkomplexe Funktionentheorie), Bull. Soc. Fribourgeoise Sci. Nat. 37 (1942/44), 348–375 (German). MR 22645

Ian R. Porteous, Topological geometry, Van Nostrand Reinhold Co., London-New York-Melbourne, 1969. MR 0254852

Marcel Riesz, Clifford numbers and spinors (Chapters I–IV), Lecture Series, No. 38, University of Maryland, Institute for Fluid Dynamics and Applied Mathematics, College Park, Md., 1958. Lectures delivered October 1957-January 1958. MR 0181951

John Ryan, Clifford analysis with generalized elliptic and quasi elliptic functions, Applicable Anal. 13 (1982), no. 3, 151–171. MR 663771, DOI 10.1080/00036818208839388

John Ryan, Complexified Clifford analysis, Complex Variables Theory Appl. 1 (1982/83), no. 1, 119–149. MR 674364, DOI 10.1080/17476938208814009

Mikio Sato, Theory of hyperfunctions. I, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1959), 139–193. MR 114124

Laurent Schwartz, Théorie des distributions, Publications de l’Institut de Mathématique de l’Université de Strasbourg, IX-X, Hermann, Paris, 1966 (French). Nouvelle édition, entiérement corrigée, refondue et augmentée. MR 0209834

Barry Simon, The $P(\phi )_{2}$ Euclidean (quantum) field theory, Princeton Series in Physics, Princeton University Press, Princeton, N.J., 1974. MR 0489552

F. Sommen, Representations of distributions in $\textbf {R}^{3}$ by regular functions of a quaternion variable, Simon Stevin 53 (1979), no. 1-2, 131–150. MR 536752

F. Sommen, Hypercomplex Fourier and Laplace transforms. I, Illinois J. Math. 26 (1982), no. 2, 332–352. MR 650399

F. Sommen, Some connections between Clifford analysis and complex analysis, Complex Variables Theory Appl. 1 (1982/83), no. 1, 97–118. MR 674363, DOI 10.1080/17476938208814008

F. Sommen, Hypercomplex Fourier and Laplace transforms. II, Complex Variables Theory Appl. 1 (1982/83), no. 2-3, 209–238. MR 690496, DOI 10.1080/17476938308814016

Vladimír Souček, Complex-quaternionic analysis applied to spin-${1\over 2}$ massless fields, Complex Variables Theory Appl. 1 (1983), no. 4, 327–346. MR 706989, DOI 10.1080/17476938308814023

R. F. Streater and A. S. Wightman, PCT, spin and statistics, and all that, W. A. Benjamin, Inc., New York-Amsterdam, 1964. MR 0161603

A. Sudbery, Quaternionic analysis, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 2, 199–224. MR 516081, DOI 10.1017/S0305004100055638

Heinz-Günther Tillmann, Randverteilungen analytischer Funktionen und Distributionen, Math. Z. 59 (1953), 61–83 (German). MR 57345, DOI 10.1007/BF01180242

Heinz Günther Tillmann, Distributionen als Randverteilungen analytischer Funktionen. II, Math. Z. 76 (1961), 5–21 (German). MR 125441, DOI 10.1007/BF01210955

Heinz Günther Tillmann, Darstellung der Schwartzschen Distributionen durch analytische Funktionen, Math. Z. 77 (1961), 106–124 (German). MR 139002, DOI 10.1007/BF01180167

Heinz Günther Tillmann, Darstellung vektorwertiger Distributionen durch holomorphe Funktionen, Math. Ann. 151 (1963), 286–295 (German). MR 172114, DOI 10.1007/BF01470821

Vasiliĭ Sergeevič Vladimirov, Methods of the theory of functions of many complex variables, The M.I.T. Press, Cambridge, Mass.-London, 1966. Translated from the Russian by Scripta Technica, Inc; Translation edited by Leon Ehrenpreis. MR 0201669

V. S. Vladimirov, Obobshchennye funktsii v matematicheskoĭ fizike, Sovremennye Fiziko-Tekhnicheski Problemy. [Contemporary Problems in Physics and Engineering], “Nauka”, Moscow, 1979 (Russian). Second edition, corrected and supplemented. MR 549767

63.

A. I. Zayed, Hardy space theory for functions with values in a Clifford algebra, 1983 (preprint). (See the abstract Boundary behavior of hypercomplex functions with values in a Clifford algebra, Abstracts Amer. Math. Soc. 4 (1983), 71.)

1.

P. Bosshard, Die Cliffordschen Zahlen, ihre Algebra und ihre Funktionentheorie, Thesis, Universitàt, Zürich, 1940. MR 0037823

2.

F. Brackx, On (k)-monogenic functions of a quaternion variable, Function Theoretic Methods in Differential Equations, Pitman, London, 1976, pp. 22-44. MR 492321

3.

F. Brackx, Non (k)-monogenic points of a quaternion variable. Function Theoretic Methods for Partial Differential Equations, Lecture Notes in Math., vol. 561, Springer-Verlag, Berlin, 1976, pp. 138-149. MR 466584

4.

C. C. Chevalley, The algebraic theory of spinors, Columbia Univ. Press, New York, 1954. MR 60497

5.

W. K. Clifford, Mathematical papers, Chelsea, New York, 1968. (See especially pp. 266-276 = Applications of Grassmann's extensive algebra, Amer. J. Math. 1 (1878), 350-358.) MR 1505182

6.

F. Constantinescu, Analytic properties of nonstrictly localizable fields, J. Math. Phys. 12 (1971), 293-298. MR 378617

7.

C. A. Deavours, The quaternion calculus, Amer. Math. Monthly 80 (1973), 995-1008. MR 325986

8.

R. Delanghe, On regular-analytic functions with values in a Clifford algebra, Math. Ann. 185 (1970), 91-111. MR 265618

9.

R. Delanghe, Morera's theorem for functions with values in a Clifford algebra, Simon Stevin 43 (1969-1970), 129-140. MR 271374

10.

R. Delanghe, On regular points and Liouville's theorem for functions with values in a Clifford algebra, Simon Stevin 44 (1970-1971), 55-66. MR 280727

11.

R. Delanghe, On the singularities of functions with values in a Clifford algebra, Math. Ann. 196 (1972), 293-319. MR 301213

12.

R. Delanghe and F. Brackx, Hypercomplex function theory and Hilbert modules with reproducing kernel, Proc. London Math. Soc. 37 (1978), 545-576. MR 512025

13.

A. Douglis, A function-theoretic approach to elliptic systems of equations in two variables, Comm. Pure Appl. Math. 6 (1953), 259-289. MR 64277

14.

J. D. Edmonds, Quaternion quantum theory: new physics or number mysticism?, Amer. J. Phys. 42 (1974), 220-223.

15.

L. Ehrenpreis, Solution of some problems of division. I: Division by a polynomial of derivation, Amer. J. Math. 76 (1954), 883-903. MR 68123

16.

L. Ehrenpreis, Solution of some problems of division. II: Division by a punctual distribution, Amer. J. Math. 77 (1955), 286-292. MR 70048

17.

L. Ehrenpreis, Solutions of some problems of division. III: Division in the spaces 𝓓',𝓗,𝓠,𝓞, O Amer. J. Math. 78 (1956), 685-715. MR 83690

18.

L. Ehrenpreis, Solution of some problems of division. IV: Invertible and elliptic operators, Amer. J. Math. 82 (1960), 522-588. MR 119082

19.

R. Fueter, Analytische Funktionen einer Quaternionenvariablen, Comment. Math. Helv. 4 (1932), 9-20. MR 1509442

20.

R. Fueter, Die Funktionentheorie der Differentialgleichungen Δu = 0 und Δ Δ u = 0 mit vierreellen Variablen, Comment. Math. Helv. 7 (1934/1935), 307-330. MR 1509515

21.

R. Fueter, Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen, Comment. Math. Helv. 8 (1935/1936), 371-378. MR 1509533

22.

R. Fueter, Zur Theorie der regulären Funktionen einer Quaternionenvariablen, Monatsh. Math. 43 (1936), 69-74. MR 1550511

23.

R. P. Gilbert (Editor), Plane ellipticity and related problems, Contemporary Math., Vol. 11, Amer. Math. Soc., Providence, R. I., 1982. MR 669227

24.

R. P. Gilbert and G. N. Hile, Generalized hyperanalytic function theory, Bull. Amer. Math. Soc. 78 (1972), 998-1001. MR 306519

25.

R. P. Gilbert and G. N. Hile, Generalized hypercomplex function theory, Trans. Amer. Math. Soc. 195 (1974), 1-29. MR 450577

26.

F. Gürsey and H. C. Tze, Complex and quaternionic analyticity in chiral and gauge theories. I, Ann. Physics 128 (1980), 29-130. MR 587683

27.

H. Haefeli, Hyperkomplexe Differentiale, Comment. Math. Helv. 20 (1947), 382-420. MR 25586

28.

W. R. Hamilton, Elements of quaternions, Longmans Green, London, 1866; reprinted, Chelsea, New York, 1969, Vols. 1, 2. (Also see: The mathematical papers of Sir William Rowan Hamilton, Vol. III: Algebra (H. Halberstam and R. E. Ingram, eds), Cambridge Univ. Press, Cambridge, 1967). MR 213217

29.

D. Hestenes, Space-time algebra, Gordon and Breach, New York, 1966.

30.

D. Hestenes, Multivector functions, J. Math. Anal. Appl. 24 (1968), 467-473. MR 234368

31.

L. Hörmander, On the division of distributions by polynomials, Ark. Mat. 3 (1958), 555-568. MR 124734

32.

V. Iftimie, Fonctions hypercomplexes, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 9 (1965), 279-332. MR 217312

33.

K. Imaeda, A new formulation of classical electrodynamics, Nuovo Cimento B 32 (1976), 138-162. MR 408457

34.

M. A. Jafarizadeh, M. Snyder and H. C. Tze, Quaternionic multi-S (2) Yang-Mills and HP (n) σ-models, Nuclear Phys. Bull. 176 (1980), 221. MR 596908

35.

Y. N. Kafiev, Four dimensional σ-models and geometry of Yang-Mills fields, Nuclear Phys. Bull. 178 (1981), 177. MR 605587

36.

V. P. Khavin, Spaces of analytic functions, Plenum Press, New York, 1968, pp. 75-167.

37.

G. Köthe, Die Randverteilungen analytischer Funktionen, Math. Z. 57 (1952), 13-33. MR 52032

38.

H. Komatsu (Editor), Hyperfunctions and pseudo-differential equations, Lecture Notes in Math., vol. 287, Springer-Verlag, Berlin, 1973. MR 388082

39.

S. Lang, Algebra, Addison-Wesley, Reading, Mass., 1965. MR 197234

40.

B. Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier (Grenoble) 6 (1955/1956), 271-355. MR 86990

41.

R. Meise, Representation of distributions and ultradistributions by holomorphic functions, Functional Analysis: Surveys and Recent Results (Proc. Paderborn Conf. Functional Analysis), North-Holland, Amsterdam, 1977, pp. 189-208. MR 482169

42.

W. Nef, Die Funktionentheorie der partiellen Differentialgleichungen zweiter Ordnung (Hyperkomplexe Funktionentheorie), Bull. Soc. Firbourgeoise Sci. Nat. 37 (1944), 348-375. (Also see papers on this topic by W. Nef in Comment. Math. Helv., vols. 15, 16 and 17.) MR 22645

43.

I. R. Porteous, Topological geometry, Van Nostrand Reinhold, London, 1969. MR 254852

44.

M. Riesz, Clifford numbers and spinors, Chaps. I-IV, Inst. Fluid Dynamics and Appl. Math., Univ. of Maryland, College Park, Md., 1958. MR 181951

45.

J. Ryan, Clifford analysis with generalized elliptic and quasi elliptic functions, Applicable Anal. 13 (1982), 151-171. MR 663771

46.

J. Ryan, Complexified Clifford analysis, Complex Variables Theory Appl. 1 (1982), 119-149. MR 674364

47.

M. Sato, Theory of hyperfunctions. I and II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 8 (1959/1960), 139-193; 387-436. MR 114124

48.

L. Schwartz, Théorie des distributions, Hermann, Paris, 1966. MR 209834

49.

B. Simon, The $P(\phi )\sb{2}$ Euclidean (quantum) field theory, Princeton Univ. Press, Princeton, N. J.. 1974. MR 489552

50.

F. Sommen, Representation of distributions in R³ by regular functions of a quaternion variable, Simon Stevin 53 (1979), 131-150. MR 536752

51.

F. Sommen, Hypercomplex Fourier and Laplace transforms. I, Illinois J. Math. 26 (1982), 332-352. MR 650399

52.

F. Sommen, Some connections between Clifford analysis and complex analysis, Complex Variables Theory Appl. 1 (1982), 97-118. MR 674363

53.

F. Sommen, Hypercomplex Fourier and Laplace transforms. II, Complex Variables Theory Appl. 1 (1983), 209-238. MR 690496

54.

V. Souček, Complex-quaternionic analysis applied to spin- (1/2) massless fields, Complex Variables Theory Appl. 1 (1983), 327-346. MR 706989

55.

R. F. Streater and A. S. Wightman, PCT, spin and statistics, and all that, Benjamin, New York, 1964. MR 161603

56.

A. Sudbery, Quaternionic analysis, Math. Proc. Cambridge Philos. Soc. 85 (1979), 199-225. MR 516081

57.

H. G. Tillmann, Randverteilungen analytischer Funktionen und Distributionen, Math. Z. 59 (1953), 61-83. MR 57345

58.

H. G. Tillmann, Distributionen als Randverteilungen analytischer Funktionen. II, Math. Z. 76 (1961), 5-21. MR 125441

59.

H. G. Tillmann, Darstellung der Schwartzschen Distributionen durch analytische Funktionen, Math. Z. 77 (1961), 106-124. MR 139002

60.

H. G. Tillmann, Darstellung vektorwertiger Distributionen durch holomorphe Funktionen, Math. Ann. 151 (1963), 286-295. MR 172114

61.

V. S. Vladimirov, Methods of the theory of functions of many complex variables, M. I. T. Press, Cambridge, Mass., 1966. MR 201669

62.

V. S. Vladimirov, Generalized functions in mathematical physics, "Mir", Moscow, 1979. MR 564116

63.

A. I. Zayed, Hardy space theory for functions with values in a Clifford algebra, 1983 (preprint). (See the abstract Boundary behavior of hypercomplex functions with values in a Clifford algebra, Abstracts Amer. Math. Soc. 4 (1983), 71.)