playboy@wacsvax.OZ (McIsaac K) (11/14/85)
The indefinite integral of Sin[x]/x does not have a closed form in terms of elementry functions. As there is no general technique for solving definite integerals with out finding the indefinite integeral the SMP integrator can't evaluate Int[Sin[x]/x,{x,0,Inf}]. The definite integral from 0 to z is the special function Si[z] (known to SMP as Sini[z]). The SMP library file XInt contains subsitutions to evaluate many definite integerals listed in the common texts (still needs extending) (see example below). This is like a table lookup. I suppose this is how MACSYMA does it. To make this file really powerfull requires the addition of code to write definite integrals in a canonical form, ie some replacements like Int[$a,{$x,0,-$end}] -> Int[S[$a,$x->-$x],{$x,0,$end}] Int[$a,{$x,$start,$end}] -> Int[$a,{$x,0,$start}] + Int[$a,{$x,0,$end}] etc. SMP 1.5.0 Thu Nov 14 12:39:21 1985 #I[1]:: <XInt;/*Table of definite integrals*/ #I[2]:: <XSym;/* Include symmetry of Sin*/ #I[3]:: _Int[Sys] : 0; #I[4]:: Int[$a,{$x,0,-$end}] :: Int[S[$a,$x->-$x],{$x,0,$end}]; #I[5]:: Int[$a,{$x,$start_=~P[$start=0],$end}] :: \ Int[$a,{$x,0,$start}] + Int[$a,{$x,0,$end}]; #I[6]:: Int[Sin[a x]/x,{x,0,Inf}] Pi (-(0 > a) + (a > 0)) #O[6]: ----------------------- 2 #I[7]:: Int[Sin[a x]/x,{x,0,-Inf}] Pi (-(0 > a) + (a > 0)) #O[7]: ----------------------- 2 #I[8]:: Int[Sin[a x]/x,{x,-Inf,Inf}] #O[8]: Pi (-(0 > a) + (a > 0)) #I[9]:: <end> -------------------------------------------------------------------------------- Kevin McIsaac ACSnet: playboy@wacsvax Dept. Physics UUCP: seismo!munnari!wacsvax.oz!playboy Uni. of Western Australia decvax!mulga!wacsvax.oz!playboy Nedlands, 6009 AUSTRALIA ARPA: munnari!wacsvax.oz!playboy@seismo.arpa decvax!mulga!wacsvax.oz!playboy@Berkeley