1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 401

which is a generalization of the Laplace equation.

(1) J. D. Abercrombie in [1] uses M. Itoh bicomplex numbers (see

Section 1.19 and reference [15] at the end of Chapter 1) to develop a quater-

nionic function theory. Other approaches to this theory have been proposed

by G. C. Moisil [86] and by R. Fueter [41]. See also C. A. Deavors [32] and

F. F. Brackx [18]. By considering Cayley's octaves (see Section 1.19) as

ordered pairs of quaternions, P. Dentoni and M. See have extended the

theory to the Cayley algebra in [31].

(m) J. Ferrand [40], H. A. Heilbronn [61], R. Isaacs [64], A. C. Allen and

B. H. Murdoch [2], A. Terracini ([116] and [117]), R. J. Duffin [36], and

J. R. Hundhausen [63] have considered classes of functions called preholo-

morphic and preharmonic by Bouligand and Ferrand, discrete analytic and

discrete harmonic by Heilbronn, Duffin, and Hundhausen, and monodriffic

by Terracini and Isaacs. They are also referred to as lattice functions.

(n) For other generalizations of analytic function theory we refer the

reader to the papers by C. G. Costley [26], W. F. Eberlein [38], P. Kra-

jkiewicz [74], G. B. Rizza [103], M. N. Rosculet [106], M. S. K. Sastry

[108], and M. B. Balk and M. P. Zuev [3].

BIBLIOGRAPHY

1. J. D. Abercrombie, Bicomplex quaternionic function theory, University of
   Alabama dissertation, 1970.


2. A. C. Allen and B. H. Murdoch, A note on preharmonic functions, Proc.
   Amer. Math. Soc., 4 (1953), 842-852.


3. M. B. Balk and M. F. Zuev, On polyanalytic functions, Russian Math. Surveys,
   25 (1970), 201-223 (London Math. Soc. translation from Uspekhi Mat. Mekh.
   25 (1970), 203-226.)


4. E. F. Barnett, On a certain class of generalized analytic functions, University
   of Alabama dissertation, 1968.


5. H. S. Bear and G. N. Hile, Gradient characterization of analyticity, Amer.
   Math. Monthly, 85 (1978), 333-337.


6. H. A. von Beckh-Widmanstetter, Lasst sich die Eigenschaft der analytichen
   Funktionen einer gemeinen komplexen Veranderlichen, Monatsh. Math. Phys.,
   23 (1912), 257-260.


7. E. Beltrami, Della variabili complessa sopra una superficie qualunque, Ann.
   Mat. 2a. ser. 1 (1867-68), 329-366.


8. L. Bers, The expa'usion theorem for sigma-monogenic functions, Amer. J.
   Math., 72 (1950), 705-712.