Re: integrate log*sinc

*To*: mathgroup at smc.vnet.net*Subject*: [mg109205] Re: integrate log*sinc*From*: "David Park" <djmpark at comcast.net>*Date*: Sat, 17 Apr 2010 06:03:04 -0400 (EDT)

Well, I'm treading on the edge of my knowledge, but what about doing a series expansion? First, evaluate the indefinite integral and then evaluate the result at the limits. intg0[x_] = Integrate[Log[x] Sin[x]/x, x]; Limit[%, x -> \[Infinity]] - Limit[%, x -> 0] Then do a series expansion of the integral that Mathematica returned. series1 = Series[intg0[x], {x, 0, 7}, Assumptions -> x > 0] // Normal // Expand Then do a series expansion of the initial integrand and integrate term by term. Series[Log[x] Sin[x]/x, {x, 0, 7}] // Normal series2 = Integrate[#, x] & /@ % // Expand The two series match term for term (at least as far as we have gone) so they must represent the same function. David Park djmpark at comcast.net http://home.comcast.net/~djmpark/ From: pimeja [mailto:sed.nivo at gmail.com] Hi All, For Integrate[Log[x] Sin[x]/x, {x, 0, \[Infinity]}] Mathematica returns -EulerGamma \[Pi]. How to proof this analytical? I've tried to use residue theory but it seems not suitable since integrand has pool of second order in zero (for Jordan lema should be first order only). Substitution x=Exp[y] returns even more strange result. Thanks in advance.