Partial differential approximants and the elucidation of multisingularities

  • Daniel F. Styer
  • Michael E. Fisher
Critical Phenomena
Part of the Lecture Notes in Mathematics book series (LNM, volume 1105)


A partial differential approximant, or PDA, F(x,y), can accurately approximate a two-variable function, f(x,y), on the basis of its power series expansion even near a multisingular point where the function is intrinsically nonanalytic in both variables. This brief review argues that multisingularities occur frequently in two-variable functions arising in practical situations. Partial differential approximants are defined and it is shown why they can approximate multisingularities. The invariance of PDAs under a change of variables is discussed and new results are presented concerning functions exactly representable by PDAs. Finally, several applications of PDAs are mentioned.


Ising Model Scaling Function Power Series Expansion Singular Locus Common Zero 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    G.A. Baker, Jr., Phys. Rev. 124, 768–774 (1961).MathSciNetCrossRefGoogle Scholar
  2. 2.
    G.A. Baker, Jr., Essentials of Padé Approximants, New York: Academic Press (1975).zbMATHGoogle Scholar
  3. 3.
    G.A. Baker, Jr. and P.R. Graves-Morris, Padé Approximants, parts I & II, Reading, Massachusetts: Addison-Wesley (1981).zbMATHGoogle Scholar
  4. 4.
    N.K. Bose and S. Basu, IEEE Transactions on Automatic Control AC-25, 509–514 (1980).MathSciNetCrossRefGoogle Scholar
  5. 5.
    J.-H. Chen, M.E. Fisher and B.G. Nickel, Phys. Rev. Lett. 48, 630–634 (1982).CrossRefGoogle Scholar
  6. 6.
    J.S.R. Chisholm, Math. Comp. 27, 841–848 (1973).MathSciNetCrossRefGoogle Scholar
  7. 7.
    J.S.R. Chisholm, In Padé and Rational Approximation (eds. E.B. Saff and R.S. Varga), pp. 23–42. New York: Academic Press (1977).CrossRefGoogle Scholar
  8. 8.
    M.E. Fisher, Rev. Mod. Phys. 46, 597–616 (1974).CrossRefGoogle Scholar
  9. 9.
    M.E. Fisher, Amer. Inst. Phys. Conf. Proc. No. 24, Magnetism and Magnetic Materials, 1974, pp. 273–230, A.I.P., New York (1975).Google Scholar
  10. 10.
    M.E. Fisher, Physica B 86–88, 590–592 (1977); also in Statistical Mechanics and Statistical Methods in Theory and Application (ed. U. Landman), pp. 3–31, New York: Plenum Press (1977).CrossRefGoogle Scholar
  11. 11.
    M.E. Fisher and H. Au-Yang, J. Phys. A 12, 1677–1692; 13, 1517 (1979).MathSciNetCrossRefGoogle Scholar
  12. 12.
    M.E. Fisher and J.-H. Chen, In Proceedings 1980 Cargèse Summer Institute on Phase Transitions (ed. M. Lévy, J.-C. Le Guillou and J. Zinn-Justin), pp. 169–216. New York: Plenum Press (1982).CrossRefGoogle Scholar
  13. 13.
    M.E. Fisher, J.-H. Chen and H. Au-Yang, J. Phys. C 13, L459–464 (1980).CrossRefGoogle Scholar
  14. 14.
    M.E. Fisher and R.M. Kerr, Phys. Rev. Lett. 32, 667–670 (1977).CrossRefGoogle Scholar
  15. 15.
    M.E. Fisher and D.F. Styer, Proc. Roy. Soc. A 384, 259–287 (1982).MathSciNetCrossRefGoogle Scholar
  16. 16.
    P.R. Garabedian, Partial Differential Equations. New York: John Wiley & Sons (1964).zbMATHGoogle Scholar
  17. 17.
    P.R. Graves-Morris, In Padé Approximation and its Applications (ed. L. Wuytack), pp. 231–245. Berlin: Springer-Verlag (1979).CrossRefGoogle Scholar
  18. 18.
    D.L. Hunter and G.A. Baker, Jr., Phys. Rev. B 7, 3346–3376, 3377–3392 (1973).CrossRefGoogle Scholar
  19. 19.
    D.L. Hunter and G.A. Baker, Jr., Phys. Rev. B 19, 3808–3821 (1979).CrossRefGoogle Scholar
  20. 20.
    J. Karlsson and H. Wallin, In Padé and Rational Approximation (ed. E.B. Saff and R.S. Varga), pp. 83–100. New York: Academic Press (1977).CrossRefGoogle Scholar
  21. 21.
    A.R. King and H. Rohrer, Phys. Rev. B 19, 5864–5876 (1979).CrossRefGoogle Scholar
  22. 22.
    C.H. Lutterodt, J. Phys. A 7, 1027–1037 (1974).MathSciNetCrossRefGoogle Scholar
  23. 23.
    P. Pfeuty, D. Jasnow and M.E. Fisher, Phys. Rev. B 10, 2088–2112 (1974).CrossRefGoogle Scholar
  24. 24.
    S. Redner and H.E. Stanley, J. Phys. A 12, 1267–1283 (1979).CrossRefGoogle Scholar
  25. 25.
    J.F. Stilck and S.R. Salinas, J. Phys. A 14, 2027–2046 (1981).CrossRefGoogle Scholar
  26. 26.
    D.F. Styer, In Proceedings 1983 Geilo Study Institute on Multicritical Phenomena. (To be published by Plenum Press.)Google Scholar
  27. 27.
    D.F. Styer, Partial Differential Approximants and Applications to Statistical Mechanics. Ph.D. thesis, Cornell University (1984).Google Scholar
  28. 28.
    D.F. Styer, Proc. Roy. Soc. A 390, 321–339 (1983).MathSciNetCrossRefGoogle Scholar
  29. 29.
    D.F. Styer and M.E. Fisher, Proc. Roy. Soc. A 388, 75–102 (1983).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Daniel F. Styer
    • 1
  • Michael E. Fisher
    • 2
  1. 1.Hill Center, Busch CampusRutgers UniversityNew BrunswickUSA
  2. 2.Baker LaboratoryCornell UniversityIthacaUSA

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