Re: Suppressing the use of special functions

*To*: mathgroup at yoda.physics.unc.edu*Subject*: Re: Suppressing the use of special functions*From*: withoff*Date*: Sat, 24 Oct 92 14:56:15 CDT

> From: BEEZER at ups.edu (Rob Beezer) > Subject: Suppressing the use of special functions > > I would like to have Mathematica demonstrate the result of integrating > expressions like x^n E^(-x) as part of a calculus lesson on integration > by parts. Unfortunately, I get > > Integrate[x^2 E^(-x), x] = -Gamma[3, x] > > when I would prefer to get > > Integrate[x^2 E^(-x), x] = (2 + 2x + x^2) E^(-x). > > Can anyone tell me how to suppress the use of the gamma function for > problems like this? > > Thanks in advance, > > Robert Beezer > University of Puget Sound > beezer at ups.edu There are roughly two ways to do this: add a rule to rewrite Gamma[3, x] in the form you want, or change Integrate to produce that form in the first place. I will describe the second way, which is the way I would recommend. The integral Integrate[x^2 E^(-x), x] is fairly easy and can be evaluated using several methods. The method used in Version 2.1 is to simply look the integral up in a table. If this method is disabled, the integral will go on to another method (the Risch algorithm) which will give the answer you want: In[1]:= Integrate[Exp[-x] x^2, x] Out[1]= -Gamma[3, x] In[2]:= Block[{Integrate`TableMatch}, Integrate[Exp[-x] x^2, x] ] 2 -2 - 2 x - x Out[2]= ------------- x E Using Block[{Integrate`TableMatch}, ...] is one way to temporarily disable the lookup table. A better way is to change the table. The table is defined in a startup package called table.m in the Integrate directory. Near the bottom of that table you should find a rule that looks like this: TableMatch[E^(-X)*X^$a_]:= Condition[ -Gamma[$a+1,X], FreeQ[$a,X]] This is the rule that is used to evaluate this integral. If you remove this rule from table.m and reload the startup packages, you will get the result you want. Unfortunately, doing so also prevents Mathematica from being able to compute Integrate[x^n E^(-x), x], so a better method would be to replace this rule with: TableMatch[E^(-X)*X^$a_]:= Condition[ -Gamma[$a+1,X], FreeQ[$a,X] && !IntegerQ[2 $a]] This is probably the best thing to do if you have a whole classroom full of computers and you want all of them to behave this way whenever they start up. Yet another thing to do, which does not require reloading the initialization packages, is to replace the offending rule. If EditDefinition is implemented on your computer, you can enter EditDefinition[Integrate`TableMatch], find the rule (I think it will be the 49th rule out of 52), and change it. Or, if you want to really mess with the nitty gritty of things you can change the DownValues list for Integrate`TableMatch: In[14]:= Begin["Integrate`"] In[17]:= DownValues[TableMatch][[49]] //InputForm Out[17]//InputForm= Literal[TableMatch[X^($a_)/E^X]] :> -Gamma[$a + 1, X] /; FreeQ[$a, X] In[18]:= DownValues[TableMatch] = ReplacePart[DownValues[TableMatch], Literal[TableMatch[X^($a_)/E^X]] :> -Gamma[$a + 1, X] /; FreeQ[$a, X] && !IntegerQ[2 $a], 49] ; In[19]:= End[] If none of them seem appropriate for your purpose, your could try contacting Wolfram Research Technical Support (support at wri.com). Dave Withoff withoff at wri.com