Numerical analisis and complex integrals

[eljose]

I have a dobut,can a complex integral be evaluated by using numerical analisis?..for example the integral LnR(s)/R(s) where R(s) is Riemann Zeta function with the limits (c+i8,c-i8) i would use the change of variable s=c+iu so the integral becomes a real integral with the limit (-8,8) now how could i evaluate it?..thanks.

 

[shmoe]

Yes complex integrals can be evaluated (should say "approximated") using numerical methods. The time it takes to achieve a certain level of precision depends on a few factors. Like if you're integrating over an infinite contour, how fast does the integral converge? The slower it converges, the farther out you'll have to go to get a certain level of accuracy and the more time it will take you.

There's another problem if you're trying to approximate an integral involving Zeta numerically- you'll have to approximate Zeta itself numerically. The more accurate you want your integral, the more accurate you'll need to approximate zeta and the more points you'll need to approximate Zeta at.

You've often claimed how your integrals can be evaluated numerically in a very cavalier way. Are you beginning to see some of the difficulties? I'm no wiz at numerical methods, you might want to do some research yourself and try to get at least some understanding of the time it would take to calculate your integrals to even a few decimal places of accuracy, let alone the precision you'd need to calculate pi(x) when x is a mere 4 or 5 digits.

 

[eljose]

Thanks for replying shmoe,but let,s suppose we have that Pi(x) is equal to a triple integral then would be enoguh to know the integral with an accuracy of 01. for example.

Another question i have...if f(x) is equal to a complex integral does the time to evaluate an integral depend on x?..for example if we want to evaluate numerically the inverse Laplace transform given by the integral ds1/R(s)exp(st) does the tiem emplyed to eavluate this integral depend on the value of t?...

 

[matt grime]

You're triple integral actually evaluted pi(x)/x^4 I seem to recall. So, to work out pi(10)^n you'd need to evaluate the integral to an error of no more than 10^{-4n} units. (i've no idea wat accuracy of 01. means)

Is the x inside your integral? (even as a free variable)? if so that'll affect the cost of the integral.

 

[eljose]

with 0.1 i meant the error in the sense [Ireal-Iapprox]<0.1.

the x in the integral appears in the form (x^s)f(s) where the integration is made in s across the real line Re(s)=c

Using the integral transform Int(1,8)f(x)x^-(s+1) i have managed to get a triple integral representation for Pi(x) but it involves calculating the poles for R(4-q) and LnR(ns) that,s why i have decided to search for a numerical approach the integrand is F(n,q,s)x^s/s but i don,t know if the error in approximating it will depend on the value x

 

[matt grime]

Of course it will depend on the x, since it has an x^s inside the integral, so naturally it will have some effect on the approximation, and an exponentially bad one at a guess.