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Mathematics Calculus Wolfram Language

Solving a triple integral with assumptions

Sinval Santos

Posted 1 year ago

In a physics problem, I came across the following triple integral: exp = (r Sin[\[Alpha]] (Cos[\[Theta]] Cos[\[Phi]] Sin[\[Alpha]] - Cos[\[Alpha]] Sin[\[Theta]]))/( 4 (-2 + Sqrt[2]) \[Pi]^2 (p^2 + r^2 - 2 p r Cos[\[Alpha]] Cos[\[Theta]] - 2 p r Cos[\[Phi]] Sin[\[Alpha]] Sin[\[Theta]])^(3/2)); Assuming[0 < r < 1 && 0 < \[Alpha] < \[Pi]/4 && \[Alpha] != \[Theta] && 0 < \[Theta] < \[Pi]/4 && 0 < \[Phi] < 2 \[Pi], \!\( \SubsuperscriptBox[\(\[Integral]\), \(0\), \(2 \[Pi]\)]\( \SubsuperscriptBox[\(\[Integral]\), \(0\), \(\[Pi]/4\)]\( \SubsuperscriptBox[\(\[Integral]\), \(0\), \(1\)]exp\ \SuperscriptBox[\(p\), \(2\)] Sin[\[Alpha]]\ \[DifferentialD]p \ \[DifferentialD]\[Alpha] \[DifferentialD]\[Phi]\)\)\)] I tried to perform each integration separately, but got no result. Is there any transformation or procedure, that I am not aware of, to accomplish this task? There is also an attached file. Grateful, Sinval Attachments: Triple integral.nb

In a physics problem, I came across the following triple integral:

exp = (r Sin[\[Alpha]] (Cos[\[Theta]] Cos[\[Phi]] Sin[\[Alpha]] - 
     Cos[\[Alpha]] Sin[\[Theta]]))/(
  4 (-2 + Sqrt[2]) \[Pi]^2 (p^2 + r^2 - 
     2 p r Cos[\[Alpha]] Cos[\[Theta]] - 
     2 p r Cos[\[Phi]] Sin[\[Alpha]] Sin[\[Theta]])^(3/2));
Assuming[0 < r < 1 && 0 < \[Alpha] < \[Pi]/4 && \[Alpha] != \[Theta] &&
   0 < \[Theta] < \[Pi]/4 && 0 < \[Phi] < 2 \[Pi], \!\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(2  \[Pi]\)]\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(\[Pi]/4\)]\(
\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(1\)]exp\ 
\*SuperscriptBox[\(p\), \(2\)] Sin[\[Alpha]]\ \[DifferentialD]p \
\[DifferentialD]\[Alpha] \[DifferentialD]\[Phi]\)\)\)]

I tried to perform each integration separately, but got no result. Is there any transformation or procedure, that I am not aware of, to accomplish this task? There is also an attached file. Grateful, Sinval

POSTED BY: Sinval Santos

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Sinval Santos

Posted 1 year ago

Thanks again for your attention Hans. I also tried it with numerical integration and I got a lot of error messages. Sincerely, sinval

POSTED BY: Sinval Santos

Hans Dolhaine

Hans Dolhaine, retired

Posted 1 year ago

Hi sinval, if ( if ! ) you could do your integration, you would get a function of r and theta. I wondered if it were possible to get an impression of it. So I tried a numerical integration. But there were lots of error messages ( I didn't analyze the integrand for poles and so), and I suppressed them. Perhaps someone else could do it more neatly. This is what I got after a very long calculation - time (and it is by no means sure whether this is near to be correct) exp = (r Sin[\[Alpha]] (Cos[\[Theta]] Cos[\[Phi]] Sin[\[Alpha]] - Cos[\[Alpha]] Sin[\[Theta]]))/(4 (-2 + Sqrt[2]) \[Pi]^2 (p^2 + r^2 - 2 p r Cos[\[Alpha]] Cos[\[Theta]] - 2 p r Cos[\[Phi]] Sin[\[Alpha]] Sin[\[Theta]])^(3/2)) func[rr_, th_] := Module[{}, vals = {r -> rr, \[Theta] -> th}; j1 = NIntegrate[ exp /. vals, {\[Phi], 0, 2 Pi}, {\[Alpha], 0, th - .05}, {p, 0, 1}]; j2 = NIntegrate[ exp /. vals, {\[Phi], 0, 2 Pi}, {\[Alpha], th + .05, .7}, {p, 0, 1}]; j1 + j2 ] Off[NIntegrate::"slwcon", NIntegrate::"eincr", NIntegrate::"ncvb"] funcvals = Table[ {r1, th1, func[r1, th1]}, {r1, .1, .9, .02}, {th1, .1, .7, .05}] ({{{0.1`,0.1`,0.037095407661930334`},{0.1`,0.15000000000000002`,0.\ 13624602801177377`},{0.1`,0.2`,0.24784739689059931`},{0.1`,0.25`,0.\ 36525725834801637`},{0.1`,0.30000000000000004`,0.48636123388319485`},{\ 0.1`,0.35`,0.6103407237487765`},{0.1`,0.4`,0.7368652654451713`},{0.1`,\ 0.45000000000000007`,0.8658240806109885`},{0.1`,0.5`,0.\ 9972192733348225`},{0.1`,0.55`,1.1311180921282638`},{0.1`,0.6`,1.\ 267629578560589`},{0.1`,0.65`,1.4068927899380845`},{0.1`,0.\ 7000000000000001`,1.5490701575336656`}},{{0.12000000000000001`,0.1`,0.\ 030786753976374415`},{0.12000000000000001`,0.15000000000000002`,0.\ 11335477096310927`},{0.12000000000000001`,0.2`,0.20630244398571954`},{\ 0.12000000000000001`,0.25`,0.30409397710826797`},{0.12000000000000001`\ ,0.30000000000000004`,0.40496724242737736`},{0.12000000000000001`,0.\ 35`,0.5082397433610543`},{0.12000000000000001`,0.4`,0.\ 6136357503719297`},{0.12000000000000001`,0.45000000000000007`,0.\ 7210625486097187`},{0.12000000000000001`,0.5`,0.8305215392733643`},{0.\ 12000000000000001`,0.55`,0.9420684509204359`},{0.12000000000000001`,0.\ 6`,1.0557938188630962`},{0.12000000000000001`,0.65`,1.\ 1718134845720622`},{0.12000000000000001`,0.7000000000000001`,1.\ 290262486524428`}},{{0.14`,0.1`,0.026255010523084987`},{0.14`,0.\ 15000000000000002`,0.09696698747895585`},{0.14`,0.2`,0.\ 1765798604140385`},{0.14`,0.25`,0.2603484651436987`},{0.14`,0.\ 30000000000000004`,0.34676207823948035`},{0.14`,0.35`,0.\ 43523538292866737`},{0.14`,0.4`,0.5255315881319881`},{0.14`,0.\ 45000000000000007`,0.617570803864971`},{0.14`,0.5`,0.7113539286886176`\ },{0.14`,0.55`,0.8069283975519177`},{0.14`,0.6`,0.9043715890550458`},{\ 0.14`,0.65`,1.0037824214537276`},{0.14`,0.7000000000000001`,1.\ 1052767902838438`}},{{0.16`,0.1`,0.022832048458461242`},{0.16`,0.\ 15000000000000002`,0.08464114950805689`},{0.16`,0.2`,0.\ 15424294303671027`},{0.16`,0.25`,0.2274850639875588`},{0.16`,0.\ 30000000000000004`,0.30304547261678955`},{0.16`,0.35`,0.\ 3804113633371657`},{0.16`,0.4`,0.4593751894386522`},{0.16`,0.\ 45000000000000007`,0.5398666951715568`},{0.16`,0.5`,0.62188621449204`}\ ,{0.16`,0.55`,0.7054749907186987`},{0.16`,0.6`,0.7907004797278422`},{\ 0.16`,0.65`,0.8776491389265255`},{0.16`,0.7000000000000001`,0.\ 9664221718156949`}},{{0.18`,0.1`,0.020146481667301375`},{0.18`,0.\ 15000000000000002`,0.07502088854316027`},{0.18`,0.2`,0.\ 13682671033592164`},{0.18`,0.25`,0.2018729009309906`},{0.18`,0.\ 30000000000000004`,0.2689836660339534`},{0.18`,0.35`,0.\ 33770287182228104`},{0.18`,0.4`,0.4078455187182168`},{0.18`,0.\ 45000000000000007`,0.47934869521335066`},{0.18`,0.5`,0.\ 5522123872544512`},{0.18`,0.55`,0.626472929557586`},{0.18`,0.6`,0.\ 7021901219730733`},{0.18`,0.65`,0.7794405624264651`},{0.18`,0.\ 7000000000000001`,0.8583144669769459`}},{{0.2`,0.1`,0.\ 017975538470950982`},{0.2`,0.15000000000000002`,0.06729203659498306`},\ {0.2`,0.2`,0.12285204034219327`},{0.2`,0.25`,0.1813329967673245`},{0.\ 2`,0.30000000000000004`,0.241676263443263`},{0.2`,0.35`,0.\ 3034708011126266`},{0.2`,0.4`,0.366549604027008`},{0.2`,0.\ 45000000000000007`,0.4308556319942505`},{0.2`,0.5`,0.4963885020557744`\ },{0.2`,0.55`,0.5631806828143364`},{0.2`,0.6`,0.6312858356555466`},{0.\ 2`,0.65`,0.7007729090952887`},{0.2`,0.7000000000000001`,0.\ 7717228031016744`}},{{0.22`,0.1`,0.01617745298214398`},{0.22`,0.\ 15000000000000002`,0.06093674967404686`},{0.22`,0.2`,0.\ 1113775380506447`},{0.22`,0.25`,0.16447877232754854`},{0.22`,0.\ 30000000000000004`,0.21927730876178725`},{0.22`,0.35`,0.\ 2753991629957458`},{0.22`,0.4`,0.3326917520612254`},{0.22`,0.\ 45000000000000007`,0.3911028466498584`},{0.22`,0.5`,0.\ 45063180286532556`},{0.22`,0.55`,0.5113078955781801`},{0.22`,0.6`,0.\ 5731798387167721`},{0.22`,0.65`,0.6363102781516028`},{0.22`,0.\ 7000000000000001`,0.7007728550892125`}},{{0.24000000000000002`,0.1`,0.\ 0146568155388992`},{0.24000000000000002`,0.15000000000000002`,0.\ 05560942661392806`},{0.24000000000000002`,0.2`,0.10177560656750116`},{\ 0.24000000000000002`,0.25`,0.15038569065000287`},{0.24000000000000002`\ ,0.30000000000000004`,0.20055632298488413`},{0.24000000000000002`,0.\ 35`,0.25194380004204053`},{0.24000000000000002`,0.4`,0.\ 3044079317861367`},{0.24000000000000002`,0.45000000000000007`,0.\ 35790033895947615`},{0.24000000000000002`,0.5`,0.41242012779097903`},{\ 0.24000000000000002`,0.55`,0.46799402177877314`},{0.24000000000000002`\ ,0.6`,0.524666576138427`},{0.24000000000000002`,0.65`,0.\ 5824953680480578`},{0.24000000000000002`,0.7000000000000001`,0.\ 6415478609977913`}},{{0.26`,0.1`,0.013348900839179948`},{0.26`,0.\ 15000000000000002`,0.05107089294638263`},{0.26`,0.2`,0.\ 09361154666111139`},{0.26`,0.25`,0.13841359894251754`},{0.26`,0.\ 30000000000000004`,0.18466087111590007`},{0.26`,0.35`,0.\ 23203552705096195`},{0.26`,0.4`,0.28040747134308636`},{0.26`,0.\ 45000000000000007`,0.32973169155189713`},{0.26`,0.5`,0.\ 38000714793812407`},{0.26`,0.55`,0.4312584078123595`},{0.26`,0.6`,0.\ 4835266473504815`},{0.26`,0.65`,0.5368650571035218`},{0.26`,0.\ 7000000000000001`,0.5913361603998591`}},{{0.28`,0.1`,0.\ 012206526255782953`},{0.28`,0.15000000000000002`,0.047150004075804594`\ },{0.28`,0.2`,0.08657490351654146`},{0.28`,0.25`,0.12810517698725374`}\ ,{0.28`,0.30000000000000004`,0.170982347839135`},{0.28`,0.35`,0.\ 2149104896913902`},{0.28`,0.4`,0.25976830366730785`},{0.28`,0.\ 45000000000000007`,0.30551357913246124`},{0.28`,0.5`,0.\ 35214520417480194`},{0.28`,0.55`,0.39968600008872995`},{0.28`,0.6`,0.\ 4481743180433503`},{0.28`,0.65`,0.49765969989936193`},{0.28`,0.\ 7000000000000001`,0.5482004469116538`}},{{0.30000000000000004`,0.1`,0.\ 011195173295346817`},{0.30000000000000004`,0.15000000000000002`,0.\ 043722156056234696`},{0.30000000000000004`,0.2`,0.08043793414761585`},\ {0.30000000000000004`,0.25`,0.1191249754901639`},{0.30000000000000004`\ ,0.30000000000000004`,0.15907415381331957`},{0.30000000000000004`,0.\ 35`,0.20000842243193384`},{0.30000000000000004`,0.4`,0.\ 24181403657849748`},{0.30000000000000004`,0.45000000000000007`,0.\ 2844513121421092`},{0.30000000000000004`,0.5`,0.32791912035723564`},{\ 0.30000000000000004`,0.55`,0.3722388606996955`},{0.30000000000000004`,\ 0.6`,0.4174465381642965`},{0.30000000000000004`,0.65`,0.\ 4635886547864685`},{0.30000000000000004`,0.7000000000000001`,0.\ 5107191936191435`}},{{0.32`,0.1`,0.010289348566750945`},{0.32`,0.\ 15000000000000002`,0.040692340233826715`},{0.32`,0.2`,0.\ 07502983654698334`},{0.32`,0.25`,0.11122142980892863`},{0.32`,0.\ 30000000000000004`,0.14860134648248635`},{0.32`,0.35`,0.\ 18690901890881817`},{0.32`,0.4`,0.22603727375589638`},{0.32`,0.\ 45000000000000007`,0.2659487037806647`},{0.32`,0.5`,0.\ 30664226454503635`},{0.32`,0.55`,0.3481382475568924`},{0.32`,0.6`,0.\ 3904707075578344`},{0.32`,0.65`,0.43368359472879847`},{0.32`,0.\ 7000000000000001`,0.47782840847473607`}},{{0.33999999999999997`,0.1`,\ 0.009468983403512543`},{0.33999999999999997`,0.15000000000000002`,0.\ 03798918329034114`},{0.33999999999999997`,0.2`,0.07022001058078073`},{\ 0.33999999999999997`,0.25`,0.10420213721311028`},{0.33999999999999997`\ ,0.30000000000000004`,0.13930776984494658`},{0.33999999999999997`,0.\ 35`,0.17529078942243584`},{0.33999999999999997`,0.4`,0.\ 2120498558929178`},{0.33999999999999997`,0.45000000000000007`,0.\ 24954964851691785`},{0.33999999999999997`,0.5`,0.28778935787045307`},{\ 0.33999999999999997`,0.55`,0.3267884172643703`},{0.33999999999999997`,\ 0.6`,0.36657930340244826`},{0.33999999999999997`,0.65`,0.\ 40720381609080614`},{0.33999999999999997`,0.7000000000000001`,0.\ 44871076935047294`}},{{0.36`,0.1`,0.008718899049174889`},{0.36`,0.\ 15000000000000002`,0.03555670339720993`},{0.36`,0.2`,0.\ 06590691487566466`},{0.36`,0.25`,0.09791748030580186`},{0.36`,0.\ 30000000000000004`,0.13099415909496265`},{0.36`,0.35`,0.\ 16490361580211133`},{0.36`,0.4`,0.1995497634235069`},{0.36`,0.\ 45000000000000007`,0.23489922793556767`},{0.36`,0.5`,0.\ 27095161025133535`},{0.36`,0.55`,0.3077257486845272`},{0.36`,0.6`,0.\ 345252876279063`},{0.36`,0.65`,0.38357289449594356`},{0.36`,0.\ 7000000000000001`,0.422732470184697`}},{{0.38`,0.1`,0.\ 008026874579247667`},{0.38`,0.15000000000000002`,0.03335084292808289`}\ ,{0.38`,0.2`,0.062010461187200994`},{0.38`,0.25`,0.09224939586305866`}\ ,{0.38`,0.30000000000000004`,0.12350322273971379`},{0.38`,0.35`,0.\ 15555000872790062`},{0.38`,0.4`,0.18829842538049751`},{0.38`,0.\ 45000000000000007`,0.22171707821260675`},{0.38`,0.5`,0.\ 2558060875778921`},{0.38`,0.55`,0.2905839793363461`},{0.38`,0.6`,0.\ 32608102039500925`},{0.38`,0.65`,0.362335648488212`},{0.38`,0.\ 7000000000000001`,0.39939240350403954`}},{{0.4`,0.1`,0.\ 007383441103911356`},{0.4`,0.15000000000000002`,0.031336442623368474`}\ ,{0.4`,0.2`,0.058466655131822504`},{0.4`,0.25`,0.08710351285850328`},{\ 0.4`,0.30000000000000004`,0.11670915613818397`},{0.4`,0.35`,0.\ 14707192788808487`},{0.4`,0.4`,0.17810492786483667`},{0.4`,0.\ 45000000000000007`,0.2097787525083755`},{0.4`,0.5`,0.\ 24209424826991163`},{0.4`,0.55`,0.2750698666699018`},{0.4`,0.6`,0.\ 3087351952137394`},{0.4`,0.65`,0.3431274475889866`},{0.4`,0.\ 7000000000000001`,0.37828939598725553`}},{{0.42000000000000004`,0.1`,\ 0.006780861156215833`},{0.42000000000000004`,0.15000000000000002`,0.\ 02948516966859021`},{0.42000000000000004`,0.2`,0.055223909666150074`},\ {0.42000000000000004`,0.25`,0.08240355225360244`},{0.\ 42000000000000004`,0.30000000000000004`,0.11051018899547986`},{0.\ 42000000000000004`,0.35`,0.1393414441856753`},{0.42000000000000004`,0.\ 4`,0.16881463854587864`},{0.42000000000000004`,0.45000000000000007`,0.\ 1989024161377033`},{0.42000000000000004`,0.5`,0.2296065822558913`},{0.\ 42000000000000004`,0.55`,0.2609457677825922`},{0.42000000000000004`,0.\ 6`,0.29294914717018705`},{0.42000000000000004`,0.65`,0.\ 3256529577047511`},{0.42000000000000004`,0.7000000000000001`,0.\ 3590984938872998`}},{{0.44000000000000006`,0.1`,0.006212873923227023`}\ ,{0.44000000000000006`,0.15000000000000002`,0.02777398323693666`},{0.\ 44000000000000006`,0.2`,0.052240067786482276`},{0.44000000000000006`,\ 0.25`,0.07808725357197385`},{0.44000000000000006`,0.30000000000000004`\ ,0.10482310476401024`},{0.44000000000000006`,0.35`,0.1322539192887665`\ },{0.44000000000000006`,0.4`,0.16030097566080181`},{0.\ 44000000000000006`,0.45000000000000007`,0.1889391715066503`},{0.\ 44000000000000006`,0.5`,0.21817148704096911`},{0.44000000000000006`,0.\ 55`,0.24801701543206295`},{0.44000000000000006`,0.6`,0.\ 2785047768703247`},{0.44000000000000006`,0.65`,0.3096702548449138`},{\ 0.44000000000000006`,0.7000000000000001`,0.34155344677241495`}},{{0.\ 45999999999999996`,0.1`,0.005674644386413862`},{0.45999999999999996`,\ 0.15000000000000002`,0.026184041314003124`},{0.45999999999999996`,0.2`\ ,0.04948060340559388`},{0.45999999999999996`,0.25`,0.\ 07410336729435911`},{0.45999999999999996`,0.30000000000000004`,0.\ 09957943963541988`},{0.45999999999999996`,0.35`,0.1257229482916581`},{\ 0.45999999999999996`,0.4`,0.15245934297331923`},{0.45999999999999996`,\ 0.45000000000000007`,0.1797658873941002`},{0.45999999999999996`,0.5`,\ 0.20764700784407278`},{0.45999999999999996`,0.55`,0.23612255316815153`\ },{0.45999999999999996`,0.6`,0.26522162165928337`},{0.\ 45999999999999996`,0.65`,0.29497917835930043`},{0.45999999999999996`,\ 0.7000000000000001`,0.3254340969460742`}},{{0.48`,0.1`,0.\ 0051620032097278775`},{0.48`,0.15000000000000002`,0.\ 024699824716662978`},{0.48`,0.2`,0.04691696064987044`},{0.48`,0.25`,0.\ 07040940966519883`},{0.48`,0.30000000000000004`,0.09472207377105077`},\ {0.48`,0.35`,0.11967659691853474`},{0.48`,0.4`,0.14520255893287354`},{\ 0.48`,0.45000000000000007`,0.17127986563071085`},{0.48`,0.5`,0.\ 19791469531368178`},{0.48`,0.55`,0.22512792857538727`},{0.48`,0.6`,0.\ 2529490045563578`},{0.48`,0.65`,0.28141262921922283`},{0.48`,0.\ 7000000000000001`,0.31055665549777806`}},{{0.5`,0.1`,0.\ 004671807703479641`},{0.5`,0.15000000000000002`,0.023308715080640502`}\ ,{0.5`,0.2`,0.044525459446277005`},{0.5`,0.25`,0.066969954695105`},{0.\ 5`,0.30000000000000004`,0.09020336171282939`},{0.5`,0.35`,0.\ 1140545617641953`},{0.5`,0.4`,0.13845740407723847`},{0.5`,0.\ 45000000000000007`,0.16339475027781689`},{0.5`,0.5`,0.\ 1888748663324467`},{0.5`,0.55`,0.21491996483196546`},{0.5`,0.6`,0.\ 2415601217478357`},{0.5`,0.65`,0.2688299221397429`},{0.5`,0.\ 7000000000000001`,0.29676646693970904`}},{{0.52`,0.1`,0.\ 004201600029011534`},{0.52`,0.15000000000000002`,0.02200026237109301`}\ ,{0.52`,0.2`,0.042286447874264645`},{0.52`,0.25`,0.06375533095784687`}\ ,{0.52`,0.30000000000000004`,0.08598316206927321`},{0.52`,0.35`,0.\ 10880593273745046`},{0.52`,0.4`,0.1321619594899347`},{0.52`,0.\ 45000000000000007`,0.15603742110907315`},{0.52`,0.5`,0.\ 1804430259195771`},{0.52`,0.55`,0.20540265981389108`},{0.52`,0.6`,0.\ 23094730865485297`},{0.52`,0.65`,0.25711169505230613`},{0.52`,0.\ 7000000000000001`,0.2839322731885242`}},{{0.54`,0.1`,0.\ 0037494482910936863`},{0.54`,0.15000000000000002`,0.\ 020765835318297406`},{0.54`,0.2`,0.040183403143441994`},{0.54`,0.25`,\ 0.06074060407412084`},{0.54`,0.30000000000000004`,0.08202761344944766`\ },{0.54`,0.35`,0.10388752694076732`},{0.54`,0.4`,0.12626355558143243`}\ ,{0.54`,0.45000000000000007`,0.14914555387509876`},{0.54`,0.5`,0.\ 17254705416953267`},{0.54`,0.55`,0.1964939807711439`},{0.54`,0.6`,0.\ 22101855617376231`},{0.54`,0.65`,0.24615588947473602`},{0.54`,0.\ 7000000000000001`,0.27194194103267666`}},{{0.56`,0.1`,0.\ 0033143244963167885`},{0.56`,0.15000000000000002`,0.\ 019598595342225467`},{0.56`,0.2`,0.03820285036583802`},{0.56`,0.25`,0.\ 057904792175578396`},{0.56`,0.30000000000000004`,0.07830789167404931`}\ ,{0.56`,0.35`,0.09926253252447115`},{0.56`,0.4`,0.1207171485544424`},{\ 0.56`,0.45000000000000007`,0.14266571474213396`},{0.56`,0.5`,0.\ 1651250093245021`},{0.56`,0.55`,0.18812339530418865`},{0.56`,0.6`,0.\ 21169468664561214`},{0.56`,0.65`,0.23587464530853897`},{0.56`,0.\ 7000000000000001`,0.2606989483322136`}},{{0.58`,0.1`,0.\ 002895238204213356`},{0.58`,0.15000000000000002`,0.018493083302953402`\ },{0.58`,0.2`,0.036333464142300424`},{0.58`,0.25`,0.05523022933426621`\ },{0.58`,0.30000000000000004`,0.07479957397249429`},{0.58`,0.35`,0.\ 09489947326342141`},{0.58`,0.4`,0.11548406026363939`},{0.58`,0.\ 45000000000000007`,0.13655187430295565`},{0.58`,0.5`,0.\ 1581233939864294`},{0.58`,0.55`,0.18022985342273193`},{0.58`,0.6`,0.\ 20290707561481444`},{0.58`,0.65`,0.2261918345701003`},{0.58`,0.\ 7000000000000001`,0.25011966625834486`}},{{0.6`,0.1`,0.\ 00249216478623163`},{0.6`,0.15000000000000002`,0.017445133622704966`},\ {0.6`,0.2`,0.03456615061386159`},{0.6`,0.25`,0.05270207722300302`},{0.\ 6`,0.30000000000000004`,0.07148183964978375`},{0.6`,0.35`,0.\ 09077136545893688`},{0.6`,0.4`,0.11053096392693737`},{0.6`,0.\ 45000000000000007`,0.13076484381726317`},{0.6`,0.5`,0.\ 1514957610572852`},{0.6`,0.55`,0.1727602121743483`},{0.6`,0.6`,0.\ 1945958816667462`},{0.6`,0.65`,0.21704096789688376`},{0.6`,0.\ 7000000000000001`,0.2401308119960816`}},{{0.62`,0.1`,0.\ 0021050117027039423`},{0.62`,0.15000000000000002`,0.01645163945469843`\ },{0.62`,0.2`,0.032893349160601854`},{0.62`,0.25`,0.05030791415110741`\ },{0.62`,0.30000000000000004`,0.06833699531326753`},{0.62`,0.35`,0.\ 08685505884462327`},{0.62`,0.4`,0.10582908811686723`},{0.62`,0.\ 45000000000000007`,0.12526816459615386`},{0.62`,0.5`,0.\ 14520161240014454`},{0.62`,0.55`,0.16566275884061896`},{0.62`,0.6`,0.\ 18670860241739126`},{0.62`,0.65`,0.20836367245741302`},{0.62`,0.\ 7000000000000001`,0.23066928802599032`}},{{0.64`,0.1`,0.\ 001734916184875217`},{0.64`,0.15000000000000002`,0.015510642868042249`\ },{0.64`,0.2`,0.03130922718865748`},{0.64`,0.25`,0.048037415311336096`\ },{0.64`,0.30000000000000004`,0.06535003459171984`},{0.64`,0.35`,0.\ 08313069998157724`},{0.64`,0.4`,0.10135356933307618`},{0.64`,0.\ 45000000000000007`,0.12003368246968418`},{0.64`,0.5`,0.\ 1392055143946743`},{0.64`,0.55`,0.15891218320650802`},{0.64`,0.6`,0.\ 17919886997104106`},{0.64`,0.65`,0.20010830379713113`},{0.64`,0.\ 7000000000000001`,0.22167745290415403`}},{{0.66`,0.1`,0.\ 0013831728770735616`},{0.66`,0.15000000000000002`,0.\ 014621204985976065`},{0.66`,0.2`,0.02980927807339331`},{0.66`,0.25`,0.\ 045882076178302464`},{0.66`,0.30000000000000004`,0.06250827498882934`}\ ,{0.66`,0.35`,0.07958131007440407`},{0.66`,0.4`,0.09708295612774145`},\ {0.66`,0.45000000000000007`,0.11503458404013157`},{0.66`,0.5`,0.\ 13347640508292907`},{0.66`,0.55`,0.15245673103705082`},{0.66`,0.6`,0.\ 17202550640702183`},{0.66`,0.65`,0.19222885000350162`},{0.66`,0.\ 7000000000000001`,0.2131051694664612`}},{{0.6799999999999999`,0.1`,0.\ 0010517756523021303`},{0.6799999999999999`,0.15000000000000002`,0.\ 013783163449200941`},{0.6799999999999999`,0.2`,0.028390224497375625`},\ {0.6799999999999999`,0.25`,0.043834995934471134`},{0.6799999999999999`\ ,0.30000000000000004`,0.05980108216568311`},{0.6799999999999999`,0.35`\ ,0.07619243078292157`},{0.6799999999999999`,0.4`,0.0929987729983129`},\ {0.6799999999999999`,0.45000000000000007`,0.11024813259129648`},{0.\ 6799999999999999`,0.5`,0.1279868857290387`},{0.6799999999999999`,0.55`\ ,0.1462695340769692`},{0.6799999999999999`,0.6`,0.1651516721034855`},{\ 0.6799999999999999`,0.65`,0.18468401056640388`},{0.6799999999999999`,\ 0.7000000000000001`,0.20490715225561842`}},{{0.7`,0.1`,0.\ 0007432528463321604`},{0.7`,0.15000000000000002`,0.01299754142757063`}\ ,{0.7`,0.2`,0.027049889390476332`},{0.7`,0.25`,0.04189067443587701`},{\ 0.7`,0.30000000000000004`,0.057219632612867`},{0.7`,0.35`,0.\ 07295184759520719`},{0.7`,0.4`,0.08908523273088516`},{0.7`,0.\ 45000000000000007`,0.1056546161498068`},{0.7`,0.5`,0.\ 12271304965779313`},{0.7`,0.55`,0.1403220492860898`},{0.7`,0.6`,0.\ 15854426103678226`},{0.7`,0.65`,0.17743639847460616`},{0.7`,0.\ 7000000000000001`,0.19704211382260878`}},{{0.72`,0.1`,0.\ 0004610609054012966`},{0.72`,0.15000000000000002`,0.\ 012265860024210786`},{0.72`,0.2`,0.02578707743768357`},{0.72`,0.25`,0.\ 0400448578513182`},{0.72`,0.30000000000000004`,0.05475672610212488`},{\ 0.72`,0.35`,0.06984939723699485`},{0.72`,0.4`,0.0853289474675748`},{0.\ 72`,0.45000000000000007`,0.10123707441233913`},{0.72`,0.5`,0.\ 11763388212388938`},{0.72`,0.55`,0.1345888562273822`},{0.72`,0.6`,0.\ 15217329214935496`},{0.72`,0.65`,0.17045184089924115`},{0.72`,0.\ 7000000000000001`,0.18947326333265874`}},{{0.74`,0.1`,0.\ 00020951261354732603`},{0.74`,0.15000000000000002`,0.\ 01159077526627797`},{0.74`,0.2`,0.024601469813402672`},{0.74`,0.25`,0.\ 03829439853449554`},{0.74`,0.30000000000000004`,0.052406587878837105`}\ ,{0.74`,0.35`,0.06687670213006333`},{0.74`,0.4`,0.08171874468154336`},\ {0.74`,0.45000000000000007`,0.09698108572524702`},{0.74`,0.5`,0.\ 11273109631902041`},{0.74`,0.55`,0.1290472333547455`},{0.74`,0.6`,0.\ 1460114496889197`},{0.74`,0.65`,0.16369886741016973`},{0.74`,0.\ 7000000000000001`,0.18216516091755183`}},{{0.76`,0.1`,-6.\ 0094852931489445`^-6},{0.76`,0.15000000000000002`,0.\ 010975714783619953`},{0.76`,0.2`,0.023493528445236764`},{0.76`,0.25`,\ 0.036637118456408706`},{0.76`,0.30000000000000004`,0.\ 050164759708367895`},{0.76`,0.35`,0.06402709206817836`},{0.76`,0.4`,0.\ 0782455218772854`},{0.76`,0.45000000000000007`,0.09287460184619262`},{\ 0.76`,0.5`,0.10798899002880306`},{0.76`,0.55`,0.12367708473915959`},{\ 0.76`,0.6`,0.140033731671446`},{0.76`,0.65`,0.15714793933802337`},{0.\ 76`,0.7000000000000001`,0.17508550766888642`}},{{0.78`,0.1`,-0.\ 00017884524329898378`},{0.78`,0.15000000000000002`,0.\ 010424968933472743`},{0.78`,0.2`,0.022464405193944756`},{0.78`,0.25`,\ 0.03507171089327005`},{0.78`,0.30000000000000004`,0.04802796377521773`\ },{0.78`,0.35`,0.06129544379125762`},{0.78`,0.4`,0.07490209629181382`}\ ,{0.78`,0.45000000000000007`,0.08890785481933638`},{0.78`,0.5`,0.\ 10339433751513924`},{0.78`,0.55`,0.118460643791526`},{0.78`,0.6`,0.\ 13421713637741367`},{0.78`,0.65`,0.15077116668447402`},{0.78`,0.\ 7000000000000001`,0.1682031938294666`}},{{0.8`,0.1`,-0.\ 0003008384852451994`},{0.8`,0.15000000000000002`,0.009943470293765116`\ },{0.8`,0.2`,0.021515866395617998`},{0.8`,0.25`,0.03359765057065253`},\ {0.8`,0.30000000000000004`,0.045993994669415755`},{0.8`,0.35`,0.\ 05867806038507045`},{0.8`,0.4`,0.07168312614908916`},{0.8`,0.\ 45000000000000007`,0.08507329485033056`},{0.8`,0.5`,0.\ 0989363652147337`},{0.8`,0.55`,0.11338254603386949`},{0.8`,0.6`,0.\ 12854045182821422`},{0.8`,0.65`,0.14454163252156013`},{0.8`,0.\ 7000000000000001`,0.16148793069546577`}},{{0.82`,0.1`,-0.\ 000361892759620357`},{0.82`,0.15000000000000002`,0.009537154758885032`\ },{0.82`,0.2`,0.020650210362479837`},{0.82`,0.25`,0.03221510762050099`\ },{0.82`,0.30000000000000004`,0.04406162963982953`},{0.82`,0.35`,0.\ 056172587099324475`},{0.82`,0.4`,0.0685850102292349`},{0.82`,0.\ 45000000000000007`,0.08136556474511472`},{0.82`,0.5`,0.\ 0946068412016374`},{0.82`,0.55`,0.1084298732532066`},{0.82`,0.6`,0.\ 12298417713810449`},{0.82`,0.65`,0.138432924957512`},{0.82`,0.\ 7000000000000001`,0.15490947080349773`}},{{0.84`,0.1`,-0.\ 00034999441799372054`},{0.84`,0.15000000000000002`,0.\ 009212560297869266`},{0.84`,0.2`,0.01987024375284327`},{0.84`,0.25`,0.\ 030924904395531996`},{0.84`,0.30000000000000004`,0.042230552689613565`\ },{0.84`,0.35`,0.05377789192080406`},{0.84`,0.4`,0.06560580563365609`}\ ,{0.84`,0.45000000000000007`,0.0777814701036196`},{0.84`,0.5`,0.\ 09040018518860668`},{0.84`,0.55`,0.10359244229797879`},{0.84`,0.6`,0.\ 117530589191112`},{0.84`,0.65`,0.1324185718677892`},{0.84`,0.\ 7000000000000001`,0.1484364655875175`}},{{0.86`,0.1`,-0.\ 00025063681195486905`},{0.86`,0.15000000000000002`,0.\ 008976814232688332`},{0.86`,0.2`,0.019179211126440263`},{0.86`,0.25`,\ 0.02972847724244816`},{0.86`,0.30000000000000004`,0.04050127433397144`\ },{0.86`,0.35`,0.05149399288351781`},{0.86`,0.4`,0.06274513148783536`}\ ,{0.86`,0.45000000000000007`,0.0743199789101869`},{0.86`,0.5`,0.\ 08631363046511527`},{0.86`,0.55`,0.09886321012663177`},{0.86`,0.6`,0.\ 11216409734883441`},{0.86`,0.65`,0.12647143229965271`},{0.86`,0.\ 7000000000000001`,0.14203529458733996`}},{{0.88`,0.1`,-0.\ 00004684879336533127`},{0.88`,0.15000000000000002`,0.\ 008837677246599002`},{0.88`,0.2`,0.01858081428465247`},{0.88`,0.25`,0.\ 02862785930884268`},{0.88`,0.30000000000000004`,0.03887511598438441`},\ {0.88`,0.35`,0.049321936348749706`},{0.88`,0.4`,0.06000405206233255`},\ {0.88`,0.45000000000000007`,0.07098213430825215`},{0.88`,0.5`,0.\ 0823474076165073`},{0.88`,0.55`,0.09423887891952622`},{0.88`,0.6`,0.\ 10687196039472273`},{0.88`,0.65`,0.12056308125527966`},{0.88`,0.\ 7000000000000001`,0.1356679101411853`}},{{0.9`,0.1`,0.\ 000280556356522587`},{0.9`,0.15000000000000002`,0.008803230346079426`}\ ,{0.9`,0.2`,0.018079159010494113`},{0.9`,0.25`,0.027680962`},\ {0.9`,0.30000000000000004`,0.03735411808007316`},{0.9`,0.35`,0.\ 04726369376414208`},{0.9`,0.4`,0.05738491676380775`},{0.9`,0.\ 45000000000000007`,0.06777094322233586`},{0.9`,0.5`,0.\ 07850481926145882`},{0.9`,0.55`,0.08972062475522789`},{0.9`,0.6`,0.\ 10164569083242705`},{0.9`,0.65`,0.11466335104925238`},{0.9`,0.\ 7000000000000001`,0.1292883380413513`}}}*) ListPlot3D[Flatten[funcvals, 1], PlotRange -> {0, 1}, AxesLabel -> {r, th, f}]

Hi sinval,

if ( if ! ) you could do your integration, you would get a function of r and theta. I wondered if it were possible to get an impression of it. So I tried a numerical integration.

But there were lots of error messages ( I didn't analyze the integrand for poles and so), and I suppressed them. Perhaps someone else could do it more neatly. This is what I got after a very long calculation - time (and it is by no means sure whether this is near to be correct)

exp = (r Sin[\[Alpha]] (Cos[\[Theta]] Cos[\[Phi]] Sin[\[Alpha]] - 
      Cos[\[Alpha]] Sin[\[Theta]]))/(4 (-2 + 
      Sqrt[2]) \[Pi]^2 (p^2 + r^2 - 
       2 p r Cos[\[Alpha]] Cos[\[Theta]] - 
       2 p r Cos[\[Phi]] Sin[\[Alpha]] Sin[\[Theta]])^(3/2))

func[rr_, th_] := Module[{},
  vals = {r -> rr, \[Theta] -> th};
  j1 = NIntegrate[
    exp /. vals, {\[Phi], 0, 2 Pi}, {\[Alpha], 0, th - .05}, {p, 0, 
     1}];
  j2 = NIntegrate[
    exp /. vals, {\[Phi], 0, 2 Pi}, {\[Alpha], th + .05, .7}, {p, 0, 
     1}];
  j1 + j2
  ]

Off[NIntegrate::"slwcon", NIntegrate::"eincr", NIntegrate::"ncvb"]
funcvals = Table[
  {r1, th1, func[r1, th1]},
  {r1, .1, .9, .02}, {th1, .1, .7, .05}]

(*{{{0.1`,0.1`,0.037095407661930334`},{0.1`,0.15000000000000002`,0.\
13624602801177377`},{0.1`,0.2`,0.24784739689059931`},{0.1`,0.25`,0.\
36525725834801637`},{0.1`,0.30000000000000004`,0.48636123388319485`},{\
0.1`,0.35`,0.6103407237487765`},{0.1`,0.4`,0.7368652654451713`},{0.1`,\
0.45000000000000007`,0.8658240806109885`},{0.1`,0.5`,0.\
9972192733348225`},{0.1`,0.55`,1.1311180921282638`},{0.1`,0.6`,1.\
267629578560589`},{0.1`,0.65`,1.4068927899380845`},{0.1`,0.\
7000000000000001`,1.5490701575336656`}},{{0.12000000000000001`,0.1`,0.\
030786753976374415`},{0.12000000000000001`,0.15000000000000002`,0.\
11335477096310927`},{0.12000000000000001`,0.2`,0.20630244398571954`},{\
0.12000000000000001`,0.25`,0.30409397710826797`},{0.12000000000000001`\
,0.30000000000000004`,0.40496724242737736`},{0.12000000000000001`,0.\
35`,0.5082397433610543`},{0.12000000000000001`,0.4`,0.\
6136357503719297`},{0.12000000000000001`,0.45000000000000007`,0.\
7210625486097187`},{0.12000000000000001`,0.5`,0.8305215392733643`},{0.\
12000000000000001`,0.55`,0.9420684509204359`},{0.12000000000000001`,0.\
6`,1.0557938188630962`},{0.12000000000000001`,0.65`,1.\
1718134845720622`},{0.12000000000000001`,0.7000000000000001`,1.\
290262486524428`}},{{0.14`,0.1`,0.026255010523084987`},{0.14`,0.\
15000000000000002`,0.09696698747895585`},{0.14`,0.2`,0.\
1765798604140385`},{0.14`,0.25`,0.2603484651436987`},{0.14`,0.\
30000000000000004`,0.34676207823948035`},{0.14`,0.35`,0.\
43523538292866737`},{0.14`,0.4`,0.5255315881319881`},{0.14`,0.\
45000000000000007`,0.617570803864971`},{0.14`,0.5`,0.7113539286886176`\
},{0.14`,0.55`,0.8069283975519177`},{0.14`,0.6`,0.9043715890550458`},{\
0.14`,0.65`,1.0037824214537276`},{0.14`,0.7000000000000001`,1.\
1052767902838438`}},{{0.16`,0.1`,0.022832048458461242`},{0.16`,0.\
15000000000000002`,0.08464114950805689`},{0.16`,0.2`,0.\
15424294303671027`},{0.16`,0.25`,0.2274850639875588`},{0.16`,0.\
30000000000000004`,0.30304547261678955`},{0.16`,0.35`,0.\
3804113633371657`},{0.16`,0.4`,0.4593751894386522`},{0.16`,0.\
45000000000000007`,0.5398666951715568`},{0.16`,0.5`,0.62188621449204`}\
,{0.16`,0.55`,0.7054749907186987`},{0.16`,0.6`,0.7907004797278422`},{\
0.16`,0.65`,0.8776491389265255`},{0.16`,0.7000000000000001`,0.\
9664221718156949`}},{{0.18`,0.1`,0.020146481667301375`},{0.18`,0.\
15000000000000002`,0.07502088854316027`},{0.18`,0.2`,0.\
13682671033592164`},{0.18`,0.25`,0.2018729009309906`},{0.18`,0.\
30000000000000004`,0.2689836660339534`},{0.18`,0.35`,0.\
33770287182228104`},{0.18`,0.4`,0.4078455187182168`},{0.18`,0.\
45000000000000007`,0.47934869521335066`},{0.18`,0.5`,0.\
5522123872544512`},{0.18`,0.55`,0.626472929557586`},{0.18`,0.6`,0.\
7021901219730733`},{0.18`,0.65`,0.7794405624264651`},{0.18`,0.\
7000000000000001`,0.8583144669769459`}},{{0.2`,0.1`,0.\
017975538470950982`},{0.2`,0.15000000000000002`,0.06729203659498306`},\
{0.2`,0.2`,0.12285204034219327`},{0.2`,0.25`,0.1813329967673245`},{0.\
2`,0.30000000000000004`,0.241676263443263`},{0.2`,0.35`,0.\
3034708011126266`},{0.2`,0.4`,0.366549604027008`},{0.2`,0.\
45000000000000007`,0.4308556319942505`},{0.2`,0.5`,0.4963885020557744`\
},{0.2`,0.55`,0.5631806828143364`},{0.2`,0.6`,0.6312858356555466`},{0.\
2`,0.65`,0.7007729090952887`},{0.2`,0.7000000000000001`,0.\
7717228031016744`}},{{0.22`,0.1`,0.01617745298214398`},{0.22`,0.\
15000000000000002`,0.06093674967404686`},{0.22`,0.2`,0.\
1113775380506447`},{0.22`,0.25`,0.16447877232754854`},{0.22`,0.\
30000000000000004`,0.21927730876178725`},{0.22`,0.35`,0.\
2753991629957458`},{0.22`,0.4`,0.3326917520612254`},{0.22`,0.\
45000000000000007`,0.3911028466498584`},{0.22`,0.5`,0.\
45063180286532556`},{0.22`,0.55`,0.5113078955781801`},{0.22`,0.6`,0.\
5731798387167721`},{0.22`,0.65`,0.6363102781516028`},{0.22`,0.\
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},{0.84`,0.35`,0.05377789192080406`},{0.84`,0.4`,0.06560580563365609`}\
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,{0.86`,0.45000000000000007`,0.0743199789101869`},{0.86`,0.5`,0.\
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{0.88`,0.35`,0.049321936348749706`},{0.88`,0.4`,0.06000405206233255`},\
{0.88`,0.45000000000000007`,0.07098213430825215`},{0.88`,0.5`,0.\
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,{0.9`,0.2`,0.018079159010494113`},{0.9`,0.25`,0.027680962`},\
{0.9`,0.30000000000000004`,0.03735411808007316`},{0.9`,0.35`,0.\
04726369376414208`},{0.9`,0.4`,0.05738491676380775`},{0.9`,0.\
45000000000000007`,0.06777094322233586`},{0.9`,0.5`,0.\
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7000000000000001`,0.1292883380413513`}}}*)

ListPlot3D[Flatten[funcvals, 1], PlotRange -> {0, 1}, 
 AxesLabel -> {r, th, f}]

POSTED BY: Hans Dolhaine

Sinval Santos

Posted 1 year ago

Hello, really this problem is almost impossible, and thank you for your contribution Hans. Regards, Sinval

POSTED BY: Sinval Santos

Hans Dolhaine

Hans Dolhaine, retired

Posted 1 year ago

Hello, I am afraid your problem is next to impossible. Here is a remark, which is unfortunately essentially useless, as you do not integrate over theta. Your integrand exp = (r Sin[\[Alpha]] (Cos[\[Theta]] Cos[\[Phi]] Sin[\[Alpha]] - Cos[\[Alpha]] Sin[\[Theta]]))/( 4 (-2 + Sqrt[2]) \[Pi]^2 (p^2 + r^2 - 2 p r Cos[\[Alpha]] Cos[\[Theta]] - 2 p r Cos[\[Phi]] Sin[\[Alpha]] Sin[\[Theta]])^(3/2)); a bit modified e1 = 4 Pi^2 (Sqrt[2] - 2) exp p^2 Sin[\[Alpha]] Now let hh = p^2 + r^2 - 2 p r Cos[\[Alpha]] Cos[\[Theta]] - 2 p r Cos[\[Phi]] Sin[\[Alpha]] Sin[\[Theta]] j1 = Sin[\[Alpha]]/(4 (Sqrt[2] - 2) p Pi^2 ) D[hh^(-1/2), \[Theta]] // Simplify j1 is your integrand as derivative of theta. exp - j1 // Simplify But as already mentioned integration over theta is no issue and this is not a step to the solution of your problem.

Hello, I am afraid your problem is next to impossible.

Here is a remark, which is unfortunately essentially useless, as you do not integrate over theta.

Your integrand

exp = (r Sin[\[Alpha]] (Cos[\[Theta]] Cos[\[Phi]] Sin[\[Alpha]] - 
            Cos[\[Alpha]] Sin[\[Theta]]))/(
      4 (-2 + Sqrt[2]) \[Pi]^2 (p^2 + r^2 - 
             2 p r Cos[\[Alpha]] Cos[\[Theta]] - 
             2 p r Cos[\[Phi]] Sin[\[Alpha]] Sin[\[Theta]])^(3/2));

a bit modified

e1 = 4 Pi^2 (Sqrt[2] - 2) exp p^2 Sin[\[Alpha]]

Now let

hh = p^2 + r^2 - 2 p r Cos[\[Alpha]] Cos[\[Theta]] -  2 p r Cos[\[Phi]] Sin[\[Alpha]] Sin[\[Theta]]

j1 = Sin[\[Alpha]]/(4 (Sqrt[2] - 2) p Pi^2  ) D[hh^(-1/2), \[Theta]] // Simplify

j1 is your integrand as derivative of theta.

exp - j1 // Simplify

But as already mentioned integration over theta is no issue and this is not a step to the solution of your problem.

POSTED BY: Hans Dolhaine

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