Re: Singularities at end point in integrations...

*To*: mathgroup at smc.vnet.net*Subject*: [mg64161] Re: Singularities at end point in integrations...*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Fri, 3 Feb 2006 01:03:48 -0500 (EST)*Organization*: The University of Western Australia*References*: <drq0q6$muq$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

In article <drq0q6$muq$1 at smc.vnet.net>, ashesh <ashesh.cb at gmail.com> wrote: > I Need to perform an integration with poles and zeros in the integrand. > Please let me know if there a way in Mathematican that can be used to > perform the definite integral > > sqrt((x-a)*(x-b)/((x-c)*(x-d))) > > between the limits (c,d), (a,d), (a,b) or (b,c). Each definite integral can be expressed as a combination of elliptic integrals. Note that Mathematica can compute the indefinite integral: Integrate[Sqrt[(x-a) (x-b)/((x-c) (x-d))], x] and from this you can obtain closed-form expressions for each case of interest. From the symmetry of your integrand, there are really only three cases to consider: (c,d) (both poles) (a,b) (both zeros) (a,d) of (b,c) (one zero and one pole) To give a concrete example, suppose that we are interested in computing the integral over (a,b) and that d > c > b > x > a. Note that the definite integral will be pure imaginary since (x-a) (x-b) < 0, so we modify the integrand before computing the indefinite integral: Collect[Integrate[Sqrt[(x-a) (b-x)/((c-x) (d-x))], x], {EllipticPi[a_, b_, c_], EllipticF[d_, e_], EllipticE[f_, g_]}, Simplify[#, d > c > b > x > a] & ] Simplifying this expression, Simplify[%, d > c > b > x > a] and then collecting terms again, intab[a_, b_][c_, d_][x_] = Collect[%, {EllipticPi[a_,b_,c_], EllipticF[d_,e_], EllipticE[f_,g_]}, Simplify[#, d > c > b > x > a] & ] we end up with a moderately complicated exact expression for the indefinite integral. Now, since there are no zeros or poles for a < x < b, we can compute the definite integral over (a,x) (where x < b, because there is a singularity at x = b) by taking the difference of the indefinite integral evaluated at the limits a and x, still in terms of the symbolic parameters a, b, c, and d. defint[a_, b_][c_, d_][x_] = intab[a, b][c, d][x] - intab[a, b][c, d][a] As a check, try {a -> 1, b -> 2, c -> 3, d -> 4}. The plot is reasonable. Plot[Evaluate@Re[defint[1, 2][3, 4][x]], {x, 0, 2}, PlotRange -> All] Comparing defint[1,2][3,4][2-10^(-20`30)] (evaluated with high precision upper limit very near to but just less than 2) with NIntegrate[Sqrt[(x-1) (2-x)/((3-x) (4-x))], {x, 1, 2}] shows excellent agreement. > I have read about the routine in quadpack called "dqawse.f" which can perform > "integration of functions having algebraico-logarithmic end point > singularities". No need to use quadpack here. NIntegrate can handle such integrals -- and many more besides. Cheers, Paul _______________________________________________________________________ Paul Abbott Phone: 61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) AUSTRALIA http://physics.uwa.edu.au/~paul