Re: Parametric Numerical Integral

*To*: mathgroup at smc.vnet.net*Subject*: [mg63133] Re: Parametric Numerical Integral*From*: Peter Pein <petsie at dordos.net>*Date*: Thu, 15 Dec 2005 03:06:42 -0500 (EST)*References*: <dnopgm$2hb$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Nikola Letic schrieb: > Hello, > I have the following problem: > I define a parametric integral as follows: > int[p_]:=NIntegrate[If[t == 0, 0,1/(E^((1/2)*(-2 + > 1/t)^2)*(Sqrt[2*Pi]*t))],{t, -p, p}, MaxRecursion -> 50,WorkingPrecision -> > 50] > > it works fine for numerical values of p such as int[10], but when I try > something like this: > > << NumericalMath`NLimit` > > NLimit[int[p], p -> Infinity] > > it ends up with error messages: > > NIntegrate::nlim: t = 1.00000 p is not a valid limit of integration. > > Now I tought that NLimit should also deal only with numerical values, but I > was wrong. Seems that the way I defined this function above has serious > limitations in its further usage. > > Does any of you Mathematica Gurus know an easy workaround for this? > > Thanks in advance > > Nikola Letic > > Hi Nikola, define int[] for numeric arguments only: Needs["NumericalMath`NLimit`"]; f[t_] := 1/(E^((1/2)*(-2 + 1/t)^2)*(Sqrt[2*Pi]*t)); int[p_?NumericQ] := NIntegrate[f[t], {t, -p, 0, p}, WorkingPrecision -> 50, MaxRecursion -> 50] The range {t, -p, 0, p} is a simple way to tell NIntegrate not to evaluate the function for t==0. NLimit[int[p], p -> Infinity] --> 0.639988 If you are just interested in a number, stop reading here. I construct an expression, which combines the negative and the positive arguments of f and do a change of variables: expr = Simplify[(f[t] + f[-t])*Dt[t] /. t -> 1/u /. Dt[u] -> 1] --> -((-1 + E^(4*u))/(E^((1/2)*(2 + u)^2)*(Sqrt[2*Pi]*u))) val = Integrate[-expr, {u, 0, Infinity}] --> (Sqrt[Pi/2]*(I + Erfi[Sqrt[2]]))/E^2 Please, don't ask _me_ where _Mathematica_ found the I (most propably it's the "i" in "Theorem of Residues"). But the real part happens to be seemingly correct: N[Re[val], 20] 0.63998807456540892568 Peter