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