Skip to main content
SpringerLink
Log in

Partial differential approximants and the elucidation of multisingularities

  • Critical Phenomena
  • Conference paper
  • First Online:
Rational Approximation and Interpolation

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1105))

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G.A. Baker, Jr., Phys. Rev. 124, 768–774 (1961).

    Article  MathSciNet  Google Scholar 

  2. G.A. Baker, Jr., Essentials of Padé Approximants, New York: Academic Press (1975).

    MATH  Google Scholar 

  3. G.A. Baker, Jr. and P.R. Graves-Morris, Padé Approximants, parts I & II, Reading, Massachusetts: Addison-Wesley (1981).

    MATH  Google Scholar 

  4. N.K. Bose and S. Basu, IEEE Transactions on Automatic Control AC-25, 509–514 (1980).

    Article  MathSciNet  Google Scholar 

  5. J.-H. Chen, M.E. Fisher and B.G. Nickel, Phys. Rev. Lett. 48, 630–634 (1982).

    Article  Google Scholar 

  6. J.S.R. Chisholm, Math. Comp. 27, 841–848 (1973).

    Article  MathSciNet  Google Scholar 

  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).

    Chapter  Google Scholar 

  8. M.E. Fisher, Rev. Mod. Phys. 46, 597–616 (1974).

    Article  Google Scholar 

  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. 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).

    Article  Google Scholar 

  11. M.E. Fisher and H. Au-Yang, J. Phys. A 12, 1677–1692; 13, 1517 (1979).

    Article  MathSciNet  Google Scholar 

  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).

    Chapter  Google Scholar 

  13. M.E. Fisher, J.-H. Chen and H. Au-Yang, J. Phys. C 13, L459–464 (1980).

    Article  Google Scholar 

  14. M.E. Fisher and R.M. Kerr, Phys. Rev. Lett. 32, 667–670 (1977).

    Article  Google Scholar 

  15. M.E. Fisher and D.F. Styer, Proc. Roy. Soc. A 384, 259–287 (1982).

    Article  MathSciNet  Google Scholar 

  16. P.R. Garabedian, Partial Differential Equations. New York: John Wiley & Sons (1964).

    MATH  Google Scholar 

  17. P.R. Graves-Morris, In Padé Approximation and its Applications (ed. L. Wuytack), pp. 231–245. Berlin: Springer-Verlag (1979).

    Chapter  Google Scholar 

  18. D.L. Hunter and G.A. Baker, Jr., Phys. Rev. B 7, 3346–3376, 3377–3392 (1973).

    Article  Google Scholar 

  19. D.L. Hunter and G.A. Baker, Jr., Phys. Rev. B 19, 3808–3821 (1979).

    Article  Google Scholar 

  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).

    Chapter  Google Scholar 

  21. A.R. King and H. Rohrer, Phys. Rev. B 19, 5864–5876 (1979).

    Article  Google Scholar 

  22. C.H. Lutterodt, J. Phys. A 7, 1027–1037 (1974).

    Article  MathSciNet  Google Scholar 

  23. P. Pfeuty, D. Jasnow and M.E. Fisher, Phys. Rev. B 10, 2088–2112 (1974).

    Article  Google Scholar 

  24. S. Redner and H.E. Stanley, J. Phys. A 12, 1267–1283 (1979).

    Article  Google Scholar 

  25. J.F. Stilck and S.R. Salinas, J. Phys. A 14, 2027–2046 (1981).

    Article  Google Scholar 

  26. D.F. Styer, In Proceedings 1983 Geilo Study Institute on Multicritical Phenomena. (To be published by Plenum Press.)

    Google Scholar 

  27. D.F. Styer, Partial Differential Approximants and Applications to Statistical Mechanics. Ph.D. thesis, Cornell University (1984).

    Google Scholar 

  28. D.F. Styer, Proc. Roy. Soc. A 390, 321–339 (1983).

    Article  MathSciNet  Google Scholar 

  29. D.F. Styer and M.E. Fisher, Proc. Roy. Soc. A 388, 75–102 (1983).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Hill Center, Busch Campus, Rutgers University, 08903, New Brunswick, New Jersey, USA

    Daniel F. Styer

  2. Baker Laboratory, Cornell University, 14853, Ithaca, New York, USA

    Michael E. Fisher

Authors
  1. Daniel F. Styer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael E. Fisher
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Peter Russell Graves-Morris Edward B. Saff Richard S. Varga

Rights and permissions

Reprints and permissions

Copyright information

© 1984 Springer-Verlag

About this paper

Cite this paper

Styer, D.F., Fisher, M.E. (1984). Partial differential approximants and the elucidation of multisingularities. In: Graves-Morris, P.R., Saff, E.B., Varga, R.S. (eds) Rational Approximation and Interpolation. Lecture Notes in Mathematics, vol 1105. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072421

Download citation

  • DOI: https://doi.org/10.1007/BFb0072421

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-13899-0

  • Online ISBN: 978-3-540-39113-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation