Bill Allombert on Wed, 05 Feb 2014 17:48:42 +0100

[Date Prev] [Date Next] [Thread Prev] [Thread Next] [Date Index] [Thread Index]

Re: How to count the zeros in interval of real or complex function in pari?

On Mon, Feb 03, 2014 at 03:41:47PM +0200, Georgi Guninski wrote:
> I would like to count the zeros in interval of real or complex function
> in pari or some other software.
> As far as I can tell this is related to the argument
> principle and possibly to |intcirc()|.

This only works for complex zeros of meromorphic functions.

> The functions I am interested in are:
> real:
> P(x)=sin(Pi*x)^2+sin(Pi*(gamma(x)+1) /x)^2

This function is meromorphic but has a lot of complex zeros.

> complex:
> Pc(x)=abs(sin(Pi*x))+abs(sin(Pi*(gamma(x)+1) /x))

This function is not meromorphic so it will not work.

If you want to count complex zeros of P:
intcirc is useless for this because it does not work close
to singularities.

But you can use intnumromb:
and then

For example

? \p19
   realprecision = 19 significant digits
? my(f(x)=P'(x)/P(x));intcircromb(3,2.5,f)
%85 = 39.99999999994892927 + 9.474996035627240141 E-12*I

So there is 40 zeros in the disk of center 3 and radius 2.5.