THE UNIVERSITY OF NOTTINGHAM

NUFFIELD FELLOWSHIP REPORT

Numerical investigations related to problems in mathematical analysis


This page has been prepared by Fangzhao Liu and Prof. J.K. Langley, School of Mathematical Sciences, University of Nottingham, NG7 2RD.

This report covers work done in summer 2011 by Fangzhao Liu, in collaboration with Professor Jim Langley, supported by a

  • Nuffield Foundation Undergraduate Research Bursary

    For the theoretical background to these investigations see:

  • background (PDF)

    Links to MAPLE worksheets:
    The first worksheet concerns the location of zeros of rational functions related to Keldysh functions, which in turn arise in connection with equilibrium points of electrostatic fields (see Section 2 of "background").

  • first worksheet
    The second worksheet is split into five parts, and calculates the non-real zeros of derivatives of functions, each of which is a polynomial multiplied by the exponential of a polynomial (see Section 3 of "background").
  • second worksheet (Part I)
  • second worksheet (Part II)
  • second worksheet (Part III)
  • second worksheet (Part IV)
  • second worksheet (Part V)
    The third worksheet looks at non-real zeros of derivatives of functions, each of which is a trigonometric function multiplied by the exponential of a polynomial.
  • third worksheet
    In the fourth worksheet Fangzhao calculates non-real zeros of derivatives of some real meromorphic functions (see Section 4 of "background"): here each function is the secant of z multiplied by the exponential of a power of z.
  • fourth worksheet
    The final worksheet is related to an approach to derivatives of real meromorphic functions discussed in Section 4 of "background" (following Proposition 4.1). It seems likely that for the reciprocal of a real entire function of finite order with real zeros, the quotient of a derivative of even order and the function itself is eventually positive or infinite on the real axis. For the second derivative this has been proved (Proposition 4.1), and this property is the key to proving the existence of infinitely many non-real zeros of the second derivative. In the fifth worksheet Fangzhao tests this conjecture for certain examples.
  • fifth worksheet

    You can find more information on Jim Langley's research, including on the topics of these investigations, at

  • Jim Langley's research page
    Jim Langley's research is on the value distribution of meromorphic functions, with particular reference to zeros of derivatives, and occasionally to iteration and to differential equations in the complex domain. You can find a list of papers, in most cases with PDF files, at
  • Jim Langley's publications