# Welcome to the Legendre Polynomial Example using a Sphere with Potential # V(theta) = e^(-cos^2theta) # We know that the potential inside the sphere can be written as # Potential(r,theta) = sumn(0, infinity, An*(r^n)*Pn(cos(theta)) # The Integrand of the coefficient formula and the value of the Legendre Poynomial is given by 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0.015707963267949 0.00577982493643006 1 0.00577911189377122 0.999876632481661 0.00577768594040298 0.999629920274299 0.00577554734019714 0.999259909031854 0.00577269648889354 0.998766667223502 0.00576913391401873 0.998150286119577 0.00576486027477749 0.997410879772804 0.00575987636191733 0.996548584994851 0.00575418309756585 0.995563561328201 0.00574778153504122 0.994455991013348 0.00574067285863557 0.993226078951333 0.00573285838337156 0.991874052661619 0.00572433955473205 0.990400162235314 0.00571511794836288 0.988804680283769 0.00570519526974906 0.987087901882527 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-0.0108918681576792 -0.941650920707027 0.0108033910622419 0.934001678430694 -0.0107097704368867 -0.925907754891901 0.010611075297259 0.917375116803156 -0.0105073783634425 -0.908410051142316 0.0103987560002901 0.899019159993885 3.12588469032183 0.00577982493643416 1 -0.00577911189377531 -0.99987663248166 0.00577768594040708 0.999629920274298 -0.00577554734020123 -0.999259909031853 0.00577269648889763 0.998766667223501 -0.00576913391402281 -0.998150286119575 0.00576486027478157 0.997410879772802 -0.0057598763619214 -0.996548584994849 0.00575418309756991 0.995563561328198 -0.00574778153504527 -0.994455991013344 0.00574067285863961 0.993226078951328 -0.00573285838337559 -0.991874052661612 0.00572433955473606 0.990400162235307 -0.00571511794836688 -0.98880468028376 0.00570519526975304 0.987087901882517 -0.00569457335386813 -0.98525014451067 0.00568325416479709 0.983291747985598 -0.00567123979533263 -0.981213074393129 0.00565853246654502 0.979014508013116 -0.00564513452732571 -0.976696455240469 0.00563104845390456 0.974259344501638 3.14159265358978 4.12933300885763e-015 1 -4.12933300885763e-015 -1 4.12933300885763e-015 1 -4.12933300885763e-015 -1 4.12933300885763e-015 1 -4.12933300885763e-015 -1 4.12933300885763e-015 1 -4.12933300885763e-015 -1 4.12933300885763e-015 1 -4.12933300885763e-015 -1 4.12933300885763e-015 1 -4.12933300885763e-015 -1 4.12933300885763e-015 1 -4.12933300885763e-015 -1 4.12933300885763e-015 1 -4.12933300885763e-015 -1 4.12933300885763e-015 1 -4.12933300885763e-015 -1 4.12933300885763e-015 1 -4.12933300885763e-015 -1 4.12933300885763e-015 1 # Calculation of these Integrand points took 1 seconds # The coefficients (An) are given by 0 0.746823904397383 2 -0.446017167324166 4 0.0739229328267532 6 -0.00734431180922668 8 0.000521174470551368 10 -2.77353809482758e-005 12 2.28067287367501e-008 14 1.33730216351664e-006 16 -1.47866781381387e-006 18 1.56752836186327e-006 20 -1.6501892756256e-006 22 1.72884969687573e-006 24 -1.80408472300373e-006 26 1.87630592843496e-006 28 -1.94585076556898e-006 30 2.01299340356952e-006 32 -2.07795740408703e-006 34 2.14094860285472e-006 36 -2.20214394542857e-006 38 2.26168482628781e-006 40 -2.31969913829841e-006 42 2.37629516159224e-006 44 -2.43156728607546e-006 46 2.48561279864649e-006 48 -2.53851325215381e-006 50 2.59033255869189e-006 52 -2.64113462748776e-006 54 2.69097453348136e-006 56 -2.73990413502617e-006 58 2.78798018700052e-006 60 -2.83524543161158e-006 62 2.8817328061088e-006 64 -2.92748029851815e-006 66 2.97252066270968e-006 68 -3.01688739196608e-006 70 3.06061712753774e-006 72 -3.10373301895825e-006 74 3.14625434050125e-006 76 -3.18820709928792e-006 78 3.22961289497791e-006 80 -3.27049514044046e-006 82 3.31087803713286e-006 84 -3.35077385033099e-006 86 3.39019607260963e-006 88 -3.42916488148161e-006 90 3.46769539894442e-006 92 -3.50580434376393e-006 94 3.54350723178908e-006 96 -3.58081018205923e-006 98 3.61772474313963e-006 100 -3.65426770001073e-006 102 3.69044977255551e-006 104 -3.72628293927237e-006 106 3.76177664662516e-006 108 -3.79693400831021e-006 110 3.8317661407646e-006 112 -3.86628697466761e-006 114 3.90050365643942e-006 116 -3.93442485125214e-006 118 3.9680562698143e-006 120 -4.00140022046731e-006 122 4.03446779459989e-006 124 -4.06726969813416e-006 126 4.09980993793999e-006 128 -4.13209491769685e-006 130 4.164128515673e-006 132 -4.19591366891675e-006 134 4.22746124577276e-006 136 -4.25877840594943e-006 138 4.28986654048444e-006 140 -4.32073069156811e-006 142 4.35137407752514e-006 144 -4.38180074394803e-006 146 4.41202085168355e-006 148 -4.4420381142438e-006 150 4.47185229833615e-006 152 -4.50146789217611e-006 154 4.53088793335664e-006 156 -4.56011727012597e-006 158 4.58916457685509e-006 160 -4.61803071942084e-006 162 4.64671509595134e-006 164 -4.67522243043441e-006 166 4.70355560967539e-006 168 -4.73171950016934e-006 170 4.75972062537949e-006 172 -4.78755773981509e-006 174 4.81523111053089e-006 176 -4.84274609791441e-006 178 4.87010533131786e-006 180 -4.89731301308242e-006 182 4.92437319131431e-006 184 -4.95128339298871e-006 186 4.97804542659206e-006 188 -5.00466520291979e-006 190 5.0311448268899e-006 192 -5.05748755664602e-006 194 5.08369527244016e-006 196 -5.10976519399277e-006 198 5.13570079095713e-006 200 -5.16150786724556e-006 #Calculation of the coefficients took 8255 # # The sum of all the potential values for the 2 approximation is 14264.266820244 # The absolute error is 14262.266820244 and the relative error is 0.999859789498808 # Calculation of the potential for 2 coefficients took 0 # # The sum of all the potential values for the 4 approximation is 14306.2576025191 # The absolute error is 14306.2476025191 and the relative error is 0.999999301005177 # Calculation of the potential for 4 coefficients took 1 # # The sum of all the potential values for the 6 approximation is 14304.1781296723 # The absolute error is 39.9113094282511 and the relative error is 0.00279018543158799 # Calculation of the potential for 6 coefficients took 1 # # The sum of all the potential values for the 8 approximation is 14304.266412413 # The absolute error is -1.99119010615505 and the relative error is -0.000139202532219837 # Calculation of the potential for 8 coefficients took 1 # # The sum of all the potential values for the 10 approximation is 14304.2632850567 # The absolute error is 0.0851553844004229 and the relative error is 5.95314716343223e-006 # Calculation of the potential for 10 coefficients took 2 # # The sum of all the potential values for the 12 approximation is 14304.2632868928 # The absolute error is -0.0031255201647582 and the relative error is -2.1850270105292e-007 # Calculation of the potential for 12 coefficients took 2 # # The sum of all the potential values for the 14 approximation is 14304.263367663 # The absolute error is 8.26062823762186e-005 and the relative error is 5.77494137607695e-009 # Calculation of the potential for 14 coefficients took 2 # # The potential inside the sphere is defined by the following points #theta r Potential 0 0 0.746823904397383 0 0.01 0.746779303419873 0 0.02 0.746645509357653 0 0.03 0.746422548819013 0 0.04 0.746110466142294 0 0.05 0.745709323382668 0 0.06 0.745219200293657 0 0.07 0.744640194303361 0 0.08 0.743972420485444 0 0.09 0.743216011524854 0 0.1 0.742371117678322 0 0.11 0.741437906729624 0 0.12 0.740416563939661 0 0.13 0.739307291991335 0 0.14 0.738110310929278 0 0.15 0.736825858094443 0 0.16 0.735454188053574 0 0.17 0.733995572523608 0 0.18 0.732450300291008 0 0.19 0.73081867712609 0 0.2 0.729101025692351 0 0.21 0.727297685450848 0 0.22 0.725409012559674 0 0.23 0.72343537976854 0 0.24 0.721377176308546 0 0.25 0.719234807777149 0 0.26 0.717008696018393 0 0.27 0.714699278998442 0 0.28 0.712307010676466 0 0.29 0.709832360870932 0 0.3 0.707275815121348 0 0.31 0.704637874545522 0 0.32 0.701919055692381 0 0.33 0.699119890390423 0 0.34 0.696240925591847 0 0.35 0.693282723212425 0 0.36 0.690245859967191 0 0.37 0.687130927201991 0 0.38 0.683938530720977 0 0.39 0.680669290610101 0 0.4 0.677323841056684 0 0.41 0.67390283016513 0 0.42 0.670406919768847 0 0.43 0.666836785238468 0 0.44 0.663193115286417 0 0.45 0.659476611767926 0 0.46 0.655687989478558 0 0.47 0.651827975948322 0 0.48 0.647897311232464 0 0.49 0.643896747698995 0 0.5 0.639827049813065 0 0.51 0.635688993918242 0 0.52 0.631483368014789 0 0.53 0.627210971535019 0 0.54 0.622872615115821 0 0.55 0.618469120368429 0 0.56 0.614001319645533 0 0.57 0.609470055805805 0 0.58 0.604876181975946 0 0.59 0.600220561310318 0 0.6 0.595504066748275 0 0.61 0.590727580769248 0 0.62 0.585891995145715 0 0.63 0.580998210694093 0 0.64 0.576047137023692 0 0.65 0.571039692283772 0 0.66 0.565976802908825 0 0.67 0.560859403362142 0 0.68 0.555688435877769 0 0.69 0.55046485020092 0 0.7 0.545189603326948 0 0.71 0.539863659238926 0 0.72 0.534487988643952 0 0.73 0.529063568708214 0 0.74 0.523591382790931 0 0.75 0.518072420177194 0 0.76 0.512507675809812 0 0.77 0.5068981500202 0 0.78 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