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

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

• To: pari-dev@pari.math.u-bordeaux.fr
• Subject: Re: How to count the zeros in interval of real or complex function in pari?
• From: Bill Allombert <allomber@math.u-bordeaux.fr>
• Date: Wed, 5 Feb 2014 17:48:40 +0100
• Delivery-date: Wed, 05 Feb 2014 17:48:42 +0100
• User-agent: Mutt/1.5.21 (2010-09-15)

```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:
intcircromb(a,b,f)=intnumromb(t=0,2*Pi,f(a+b*exp(I*t))*exp(I*t))/2/Pi*b
and then
my(f(x)=P'(x)/P(x));intcircromb(10.5,10,f)

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.

Cheers,
Bill.

```