Re: An analytical solution to an integral not currently in

*To*: mathgroup at smc.vnet.net*Subject*: [mg131376] Re: An analytical solution to an integral not currently in*From*: Daniel <dosadchy at its.jnj.com>*Date*: Tue, 16 Jul 2013 05:57:05 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-outx@smc.vnet.net*Delivered-to*: mathgroup-newsendx@smc.vnet.net

First I have to say that "Another system's unconfirmed answer" is not a good enough reason for such a topic title. For the math: The "another system answer" is correct only for a=0. And Mathematica's Integrate[] gives the same answer up to a constant. However, for non-zero a, the given analytical expression is not correct, as can be seen by plotting the following: f[a_, b_, x1_, x2_] := NIntegrate[1/Sqrt[Log[x] + a x + b], {x, x1, x2}] g[a_, b_, x_] := -Sqrt[\[Pi]] I Exp[-a x - b] Erf[I Sqrt[Log[x] + a x + b]] Plot[{f[1, 0, 1, x], g[1, 0, x] - g[1, 0, 1]}, {x, 1, 25}] Plotting for a=0 will show identity: Plot[{f[0, 1, 1, x], g[0, 1, x] - g[0, 1, 1]}, {x, 1, 25}] > Question: Integral dx of 1/sqrt(Log[x] + a*x + b) > (sorry if my notation is off; I just used the online > integrator and don't have Mathematica proper, > http://integrals.wolfram.com/index.jsp?expr=1%2Fsqrt%2 > 8Log%5Bx%5D+%2B+a*x+%2B+b%29) > (the online integrator returned this as of the time > of writing this (2013-07-13): "Mathematica could not > find a formula for your integral. Most likely this > means that no formula exists." ) > > > Another system's unconfirmed answer (in that > notation; sorry) (version 5.27.0): > -sqrt(%pi)*%i*%e^(-a*x-b)*erf(%i*sqrt(log(x)+a*x+b)) > > Strangely, the other system only produces this result > when given, say, x(t) in all places for x (including > variable of integration). > > I can't seem to get the other system to verify its > result symbolically, but when I try random numerical > sampling, it does seem to agree, albeit horribly > plagued by floating point errors for large x. > > > Can anyone offer insight, or possibly prove it's > correctness or incorrectness? :) > > > (P.S. I just joined this group, so apologies if it's > the wrong one or I'm not following guidelines) >