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From: playboy@wacsvax.OZ (McIsaac K)
Newsgroups: net.math.symbolic
Subject: Definite Integerals in SMP (Int[Sin[x]/x,{x,-Inf,Inf}])
Message-ID: <245@wacsvax.OZ>
Date: Thu, 14-Nov-85 00:06:00 EST
Article-I.D.: wacsvax.245
Posted: Thu Nov 14 00:06:00 1985
Date-Received: Thu, 12-Dec-85 06:55:36 EST
Organization: Comp Sci, Uni. Western Australia
Lines: 60

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