Re: A tough Integral
- To: mathgroup at smc.vnet.net
- Subject: [mg28555] Re: A tough Integral
- From: Richard Easther <easther at physics.columbia.edu>
- Date: Fri, 27 Apr 2001 03:56:09 -0400 (EDT)
- Organization: Columbia University
- References: <200104250530.BAA19306@smc.vnet.net> <9c7n4d$sda@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
On 25 Apr 2001, Daniel Lichtblau wrote: >bobbym1953 wrote: >> >> Does anyone know how I can get the following Integral to at least 60 places, >> using Mathematica? >> >> Int(1/(cos(x)+x^2)) between x=0 and x=infinity. Both Integrate and NIntegrate >> seemed helpless. > > 60 places?! I wish you luck. Actually, it can be done. Break the integral into two pieces, I_1 where the integral is integrated from O<x<N, and I_2 where we integrate between N<=x<Infinity. The answer is then simply I_1 + I_2 If N is not too large, the first integral can be done to the rquired precision using NIntegrate. The second integral can be evaluated by noting that the integrand 1/(x^2(1 + Cos[x]/x^2)) can be written as a sum with the general term (-1)^n (Cos[x]/x^2)^n /x^2 For integer n, Mathematica can evaluate the indefinite integral over the general term (although it does not seem to be able to give a general formula) so the series can be integrated term by term. Taking enough terms will give I_2 to arbitrary accuracy. The trick is to choose N large enough so that you do not need a huge number of terms in the series that approximates I_2, but not so large that NIntegrate cannot do the first sub-integral to the desired accuracy. If you worked really hard, and extracted the analytic form of the integral over the general term in I_2, it is just possible you may be able to find an analytic expression for I_2. If you were really lucky you might then be able to analytically continue N back to zero, and thus evalute the integral algebraically. Richard, who after doing this started to wonder why this integral needs to be worked out to 60 significant figures.
- References:
- A tough Integral
- From: bobbym1953@aol.com (bobbym1953)
- A tough Integral