Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T03:01:06.610Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on a discrete analytic function

Published online by Cambridge University Press:  17 April 2009

C.J. Harman
C.J. Harman
Affiliation:
Department of Supply, Weapons Research Establishment, Salisbury, South Australia, and Department of Mathematics, University of Adelaide, Adelaide, South Australia.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An unsolved problem in discrete analytic function theory has been to find a suitable analogue of the function . An analogue z(α), of the function zα, is found here for discrete analytic functions of the first kind (or monodiffric functions). This function resolves a conjecture of Isaacs in the negative, and at the same time it introduces multi-valued functions into the discrete analytic theory.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 10 , Issue 1 , February 1974 , pp. 123 - 134
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Taussky, Olga, “(1, 2, 4, 8)-sums of squaresand Hadamard matrices”, Combinatorics, 229–233 (Proc. Symposia Pure Math., 19. Amer. Math. Soc., Providence, Rhode Island, 1971).CrossRefGoogle Scholar
[2]Wallis, Jennifer, “Orthogonal (0, 1, -1)-matrices”, Proc. First Austral. Conf. Combinatorial Math., Newcastle, 1972, 61–84 (TUNRA, Newcastle, 1972).Google Scholar
[3]Wallis, W.D., Street, Anne Penfold, Wallis, Jennifer Seberry, Combinatorics: Room squares, sum-free sets, Hadamard matrices (Lecture Notes in Mathematics, 292. Springer-Verlag, Berlin, Heidelberg, New York, 1972).CrossRefGoogle Scholar
[1]Berzsenyi, George, “Line integrals for monodiffric functions”, J. Math. Anal. Appl. 30 (1970), 99–112.CrossRefGoogle Scholar
[2]Berzsenyi, George, “Convolution products of monodiffric functions”, J. Math. Anal. Appl. 37 (1972), 271–287.Google Scholar
[3]Deeter, Charles R., “Problems in discrete function theory and related topics”, Symposium in discrete function theory (Texas Christian University, 1969).Google Scholar
[4]Duffin, R.J., “Basic properties of discrete analytic functions”, Duke Math. J. 23 (1956), 335–363.CrossRefGoogle Scholar
[5]Ferrand, Jacqueline, “Fonctions prharmoniques et fonctions prholomorphes”, Bull. Sci. Math. (2) 68 (1944), 152–180.Google Scholar
[6]Hardy, G.H., Divergent series (Clarendon Press, Oxford, 1949).Google Scholar
[7]Harman, Christopher John, “A discrete analytic theory for geometric difference functions”, PhD thesis, University of Adelaide, Adelaide, 1972. See also the abstract, Bull. Austral. Math. Soc. 9 (1973), 299–300.Google Scholar
[8]Harman., C.J., “A new definition of discrete analytic functions”, Bull. Austral. Math. Soc. 10 (1974), 123–134.CrossRefGoogle Scholar
[9]Isaacs, Rufus Philip, “A finite difference function theory”, Univ. Nac. Tucwmán Rev. Ser. A 2 (1941), 177–201.Google Scholar
[10]Isaacs, Rufus, “Monodiffric functions”, Construction and applications of conformal maps, 257–266 (Proc. Sympos. 1949, Numerical Analysis National Bureau of Standards, Univ. California, Los Angeles. National Bureau of Standards Applied Mathematics Series, 18. United States Department of Commerce; US Government Printing Office, Washington, DC, 1952).Google Scholar
[11]Kurowski, G.J., “Further results in the theory of monodiffric functions”, Pacific J. Math. 18 (1966), 139–147.Google Scholar