The Mathematica Journal Tricks of the Trade

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1 The Mathematica Joural Tricks of the Trade Edited by Paul Abbott This is a colum of programmig tricks ad techiques, most of which, we hope, will be cotributed by our readers, either directly as submissios to The Mathematica Joural or as a edited aswer to a questio posted i the Mathematica ewsgroup, comp.soft-sys.math.mathematica. Sum-Free Set The sumset of two or more subsets of a additive group is the set of all sums formed by takig oe elemet from each set (see plaetmath.org/ecyclopedia/ Sumset.html). The sumset ca be computed usig Tuples. I[]:= SumSet@s ListD := Uio@Plus ûûû Tuples@8s<DD Defie to be SumSet. I[2]:= CirclePlus := SumSet Here is the sumset 8, 2< 8, 3, 5< 82<. I[3]:= 8, 2< 8, 3, 5< 82< Out[3]= 84, 5, 6, 7, 8, 9< A sum-free set S is a set for which the itersectio of S ad the sumset S S is empty (see mathworld.wolfram.com/sum-freeset.html). I[4]:= SumFreeQ@s_ListD := s Hs sl 8< For example, the sum-free subsets of 8, 2, 3< are «ª 8<, 8<, 82<, 83<, 8, 3<, ad 82, 3<. I[5]:= SumFreeQ êû 88<, 8<, 82<, 83<, 8, 3<, 82, 3<< Out[5]= 8True, True, True, True, True, True< Note that 8, 2< is ot sum-free. I[6]:= SumFreeQ@8, 2<D Out[6]= False Here are the sum-free subsets of 8, 3, 5, 7, 8<. I[7]:= Select@Subsets@8, 3, 5, 7, 8<D, SumFreeQD Out[7]= 88<, 8<, 83<, 85<, 87<, 88<, 8, 3<, 8, 5<, 8, 7<, 8, 8<, 83, 5<, 83, 7<, 83, 8<, 85, 7<, 85, 8<, 87, 8<, 8, 3, 5<, 8, 3, 7<, 8, 3, 8<, 8, 5, 7<, 8, 5, 8<, 83, 5, 7<, 83, 7, 8<, 85, 7, 8<, 8, 3, 5, 7<< The Mathematica Joural 0: Wolfram Media, Ic.

2 Tricks of the Trade 463 Sum-free subsets of 8, 2,, < ca be computed recursively as follows. I[8]:= = 88<<; I[9]:= := = - D H# 8< &L êû Select@SumFreeSet@ - D, # H - #L 8< &D The key to this computatio is the use of the test # H - #L 8< & o SumFreeSet@ - D to costruct elemets of SumFreeSet@D. Here are the sum-free subsets for = 0,,, 4. I[0]:= Table@SumFreeSet@D, 8, 0, 4<D êê ColumForm Out[0]= 88<< 88<, 8<< 88<, 8<, 82<< 88<, 8<, 82<, 83<, 8, 3<, 82, 3<< 88<, 8<, 82<, 83<, 84<, 8, 3<, 8, 4<, 82, 3<, 83, 4<< Alteratively, sum-free subsets ca be computed usig NestList, startig from the empty set. I[]:= Module@8 = 0<, NestList@H++; # H# 8< &L êû Select@#, # H - #L 8< &DL &, 88<<, 5D êê ColumFormD Out[]= 88<< 88<, 8<< 88<, 8<, 82<< 88<, 8<, 82<, 83<, 8, 3<, 82, 3<< 88<, 8<, 82<, 83<, 84<, 8, 3<, 8, 4<, 82, 3<, 83, 4<< 88<, 8<, 82<, 83<, 84<, 85<, 8, 3<, 8, 4<, 8, 5<, 82, 3<, 82, 5<, 83, 4<, 83, 5<, 84, 5<, 8, 3, 5<, 83, 4, 5<< The umber of sum-free subsets for each are, 2, 3, 6, 9, 6,. Searchig for this sequece at we fid that it is A I[2]:= Legth êû First@%D Out[2]= 8, 2, 3, 6, 9, 6< Usig Sow ad Reap, here is the umber of sum-free subsets for I[3]:= Module@8 = 0<, First ûlast ûreapûnest@h++; Sow@Legth@#DD; # H# 8< &L êû Select@#, # H - #L 8< &DL &, 88<<, 25DD Out[3]= 8, 2, 3, 6, 9, 6, 24, 42, 6, 08, 5, 253, 369, 607, 847, 400, 954, 339, 4398, 6976, 9583, 5456, 20982, 3286, 4547< The Mathematica Joural 0: Wolfram Media, Ic.

3 464 Edited by Paul Abbott Google Search for Mathematica Chris Chiasso A Google search for keyword site:documets.wolfram.com iurl:mathematica searches for keyword i The Mathematica Book ad i the curret versio of the olie documetatio. For example, a search for NVariatioalBoud returs a lik to documets.wolfram.com/mathematica/add-osliks/stadardpackages/ Calculus/VariatioalMethods.html. Firefox users ca make a bookmark with the URL ematica&sourceid=moilla-search ad the fill i the keyword field for that bookmark as mhelp. Typig mhelp NVariatioalBoud ito the Firefox address bar searches the documetatio for NVariatioalBoud. Asymptotic Expasio ad p Gregory s series (mathworld.wolfram.com/gregoryseries.html) is a slowly coverget formula for p. I[]:= 4 H-L k- k= ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ 2 k - Out[]= p Trucatig the series after 50,000 terms (half a power of 0, i this case 0 5 ) yields a result that is icorrect i the 6 th digit. I[2]:= gregory = 4` H-L k- k= ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ 2 k - Out[2]= I[3]:= pi = N@p, 50D Out[3]= I[4]:= - dlog 0 Hpi - gregorylt Out[4]= 6 However, comparig these two umbers, it is surprisig how may digits they have i commo [, 2]. I[5]:= RealDigits@piD - RealDigits@gregoryD Out[5]= 880, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0,, 0, 0, 0, 0, 0, 0, 0, 0, -2, 7, 8, 0, 0, 0, 0, 0, 0, 3, -3, 7, 0, 0, 0, 0, 0<, 0< The Mathematica Joural 0: Wolfram Media, Ic.

4 Tricks of the Trade 465 Moreover, the idex of the positio of the least sigificat digit of each block of differet digits is a odd multiple of 5. I[6]:= Partitio@Rest û First@%D, 5D Out[6]= i j k y { The differeces ca be computed usig FromDigits. I[7]:= FromDigits êû % Out[7]= 82, 0, -2, 0, 0, 0, -22, 0, 2770< We ca represet the differece betwee p ad Gregory s series trucated after 50,000 terms as _ , _ where umbers above the ceter lie are egative ad those below the lie are positive. Searchig for the sequece of differeces at we fid that they are twice the Euler umbers, E 2 k. I[8]:= Table@8k, 2 E 2 k <, 8k, 0, 4<D Out[8]= i y 3-22 j k { Empirically, we have determied the asymptotic differece betwee p ad the trucated Gregory s series. I[9]:= diff@_, m_d = 2 m k=0 E 2 k ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ 2 k+ ; See [] for a proof of this result. The Mathematica Joural 0: Wolfram Media, Ic.

5 466 Edited by Paul Abbott Addig the asymptotic differece to the trucated Gregory s series ad puttig = 2 µ = 0 5, we ca recover p to (at least) 50 decimal places. I[0]:= p - gregory - diff@0 5, 4D Out[0]= 0. µ 0-50 The asymptotic differece ca be computed directly usig Series. H-L k- I[]:= FullSimplifyASeriesAp - 4 ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ 2 k - Out[]= 3 5 ÄÄÄÄÄÄ 2 k= 7, 8,, 9<E, > 0 Ì ÄÄÄÄÄÄ 4 Œ E 2 ÅÅÅÅÅ - 2 i k y + 0 i { k y - 22 i { k y i { k y + O i j i { kk y { See also [3]. 9 0 y { Hadamard Regulariatio Hadamard regulariatio is a techique for hadlig diverget itegrals (essetially keepig oly the fiite part of the itegral) ad plays a importat role i several braches of mathematical physics (see [4, 5] ad mathworld.wolfram.com/hadamarditegral.html). Cosider evaluatig K f D ª ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ GH-aL f HyL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅ - H - yl a+ i the Hadamard sese, where 0 < a < + ad œ, that is, dat ad f œ C D. Usig itegratio by parts via patter matchig, we ca icrease the expoet of H - yl -p util it is itegrable, that is, - < p 0. I[]:= byparts = H - yl p_ f_@ yd y f H yl H - yl p y - J H - yl p yn y f H yl y; Here is the formal result of itegratig by parts oce. I[2]:= Out[2]= ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ GH-aL f H yl ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ H - yl a+ Ÿ H - yl -a f HyL y ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅ - H - yl-a f H yl ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅ GH - al GH - al y y ê. byparts êê FullSimplify êê Expad The H - yl -a f HyL term is sigular at y = if a > 0. Here is the result of three partial itegratios. The Mathematica Joural 0: Wolfram Media, Ic.

6 Tricks of the Trade 467 I[3]:= CollectANestA# ê. byparts &, 8 f H yl, f H_L H yl<, FullSimplifyE ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ GH-aL f H yl ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ y, 3E, a+ H - yl Out[3]= - f HyL H - yl -a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - f H yl H - yl 2-a f H yl H - yl-a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅ + Ÿ H - yl 2-a f H3L HyL y ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ GH2 - al GH3 - al GH - al GH3 - al Neglectig the sigular terms at y =, we evaluate the partial itegrals at y = -. I[4]:= -% ê. HoldPatter@Itegrate@ DD :> 0 ê. y Æ - Out[4]= 2 -a f H-L ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + 2-a f H-L ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + 22-a f H-L ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ GH - al GH2 - al GH3 - al The patter is clear. Droppig the sigular terms at y =, we obtai k=0 K f D ª ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ GH-aL f HyL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅ - H - yl a+ y 2 k-a ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ f HkL H-L + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ GHk - a + L GH - a + L H - yl -a f H+L HyL y. - As a defiite example, cosider - exph yl ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅ y. a+ H - yl Direct itegratio followed by series expasio about e = 0 reveals the sigular terms. I[5]:= AssumigA > e > 0, Out[5]= HGH-a, el - GH-a, 2LL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ GH-aL I[6]:= Series@%, 8e, 0, <D Out[6]= -e exph yl ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ GH-aL ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ - H - yl a+ HH ÅÅÅÅ + ÅÅÅÅÅÅÅÅÅ e + a -a OHe2 LL e -a + GH-aL - GH-a, 2LL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅ GH-aL Now e -a is sigular at e = 0 for a > 0, ad e -a is either sigular if a > or vaishes if 0 < a <. So both terms are igorable. Hece the osigular part ca be extracted as follows. I[7]:= K = % ê. e -a Æ 0 êê FullSimplify Out[7]= GH-a, 2L - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ GH-aL For example, here is the exact result for a = 3 ê 2. ye The Mathematica Joural 0: Wolfram Media, Ic.

7 468 Edited by Paul Abbott I[8]:= Out[8]= K êê FuctioExpad êê Simplify 3 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅ 4 è!!!!!!! 2 p + erfiè!!! 2 M Alteratively, usig the idetity obtaied usig itegratio by parts, we obtai the same aswer. I[9]:= ModuleA8a = 3 ê 2,, f = Exp<, = Floor@aD; Out[9]= ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄ GH - a + L H - yl -a f H+L H yl ye êê Simplify - 3 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅ 4 è!!!!!!! 2 p + erfiè!!! 2 M k=0 2 k-a ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄ GHk - a + L f HkL H-L + Refereces [] J. M. Borwei, P. B. Borwei, ad K. Dilcher, Pi, Euler Numbers, ad Asymptotic Expasios, America Mathematical Mothly, 96(8), 989 pp [2] G. Almkvist, May Correct Digits of p, Revisited, America Mathematical Mothly, 04(4), 997 pp DOI-Lik: dx.doi.org/0.2307/ [3] S. Matsumoto, Covergece Improvemet of Ifiite Series by Liear Fractios, i Applied Mathematica: Electroic Proceedigs of the Eighth Iteratioal Mathematica Symposium (IMS06), Avigo, Frace (Y. Papegay, ed.), Rocquecourt: INRIA, 2006 ISBN [4] D. Elliott, Three Algorithms for Hadamard Fiite-Part Itegrals ad Fractioal Derivatives, Joural of Computatioal ad Applied Mathematics, 62(3), 995 pp [5] L. Blachet ad G. Faye, Hadamard Regulariatio, Joural of Mathematical Physics, 4, 2000 pp Paul Abbott School of Physics, M03 The Uiversity of Wester Australia 35 Stirlig Highway Crawley WA 6009, Australia tmj@physics.uwa.edu.au physics.uwa.edu.au/~paul The Mathematica Joural 0: Wolfram Media, Ic.

Tricks of the Trade Edited by Paul Abbott

The Mathematica Journal Tricks of the Trade Edited by Paul Abbott This is a column of programming tricks and techniques, most of which, we hope, will be contributed by our readers, either directly as submissions