1

Is there a generalized criteria for checking the reality of (all) roots of analytic function ? There are several such existing result for special case of polynomials like Newton's identities or Hermite Sylvester theorem or insights gained from Weierstrass product of a function . Can these methods be generalized ?

  • 3
    When you say "checking the reality of (all) roots " do you mean "checking that all roots of a given function are Real"? If so, you may know about Descartes' Sign Rule which is a a method of determining the maximum number of positive and negative real roots of a polynomial. – NoChance Jul 05 '19 at 16:38
  • 2
    I don't know anything about this, but I do know that being able to show the roots of certain transcendental equations were real was important for some of Fourier's work on heat flow, so this goolge search might lead you to some further ideas for searches. Also, this google search, as I also recall that G. H. Hardy did some work of a similar type. – Dave L. Renfro Jul 05 '19 at 16:48
  • @NoChance yes , I mean checking for all the roots of a given equation are real . –  Jul 05 '19 at 17:56
  • @Dave L. Renfro yes ,sir I know of Hardy's work (Hardy's z - function ) also the Fourier work on Bessel function ( real rootedness by rational approximation ) Also , some recent works of Polya , De bruijn Csordas , Varga , Newmann , Tao and Rodgers , Dimitrov . But they all lack generality ( because it's very hard of course ) . But is there generalized approach toward such problems ? Because I don't know one if there's one –  Jul 05 '19 at 18:01
  • 2
    The zeros of $f$ entire are all real iff the zeros of $f(\frac{z-i}{z+i})$ are all on $|z|=1$ iff the Taylor series of both $\frac{1}{f(\frac{z+i}{z-i})},\frac{1}{f(\frac{z-i}{z+i})}$ have radius of convergence $1$ (replacing $1/f$ with $\log f$ this is Li's criterion for the RH) – reuns Jul 05 '19 at 19:05
  • @reuns thank you sir –  Jul 05 '19 at 19:57
  • Thank you for your reply, it is all to advanced for me. If you know a some simple reference for the this please let me know - Thanks. – NoChance Jul 06 '19 at 10:11
  • Can you please tell me about your mathematical background. I will tell likewise @NoChance –  Jul 07 '19 at 13:43
  • @AmitAkoskar I have a B.Sc. in Statistical Mathematics and Computer Science. This degree requires study of 4 Calculus courses, 2 Linear Algebra courses, Optimization, Operations Research, Real Analysis, among other topics. – NoChance Jul 07 '19 at 22:57
  • @NoChance you can start by studying the work of Poly on the subject ( you can easily get it on Google ) Furthermore you can check Fourier's work on checking reality of roots of Bessel function by rational approximation . Also you can study the following paper by De bruijn : https://www.google.com/url?sa=t&source=web&rct=j&url=https://research.tue.nl/files/1769368/597490.pdf&ved=2ahUKEwjuyL_vu6TjAhUEOI8KHetcDZoQFjACegQIAhAB&usg=AOvVaw3KQGHZNXeK-QoMEY843fwu –  Jul 08 '19 at 04:21
  • @NoChance. Sorry that I don't recommend the references in chronological order but it's a good order of studying the field from my experience –  Jul 08 '19 at 04:24
  • @AmitAkoskar, thank you much for your help. I appreciate it. – NoChance Jul 08 '19 at 09:17

0 Answers0