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