(* recurrence formulae to differentiate Legandre polinoms to check d^n/dx^n[P_n(x)] = 1*3*5*...*(2*n-1) 1 0.99 1.00 0.99 1.00 0.99 0.00e+0000 0.00e+0000 2 0.99 3.00 2.97 3.00 2.97 0.00e+0000 0.00e+0000 3 0.99 15.00 14.85 15.00 14.85 0.00e+0000 0.00e+0000 4 0.99 105.00 103.95 105.00 103.95 0.00e+0000 0.00e+0000 5 0.99 945.00 935.55 945.00 935.55 0.00e+0000 5.55e-0017 6 0.99 10395.00 10291.05 10395.00 10291.05 0.00e+0000 8.88e-0016 7 0.99 135135.00 133783.65 135135.00 133783.65 0.00e+0000 0.00e+0000 8 0.99 2027025.00 2006754.75 2027025.00 2006754.75 0.00e+0000 1.14e-0013 9 0.99 34459425.00 34114830.75 34459425.00 34114830.75 0.00e+0000 3.64e-0012 10 0.99 654729075.00 648181784.25 654729075.00 648181784.25 0.00e+0000 5.82e-0011 11 0.99 13749310575.00 13611817469.25 13749310575.00 13611817469.25 0.00e+0000 9.31e-0010 12 0.99 316234143225.00 313071801792.75 316234143225.00 313071801792.75 0.00e+0000 2.98e-0008 13 0.99 7905853580625.00 7826795044818.75 7905853580625.00 7826795044818.75 0.00e+0000 9.54e-0007 14 0.99 213458046676875.00 211323466210106.25 213458046676875.00 211323466210106.25 0.00e+0000 3.05e-0005 15 0.99 6190283353629375.00 6128380520093081.25 6190283353629375.00 6128380520093081.25 0.00e+0000 9.77e-0004 16 0.99 191898783962510625.00 189979796122885519.00 191898783962510625.00 189979796122885519.00 0.00e+0000 3.13e-0002 17 0.99 6332659870762850620.00 6269333272055222120.00 6332659870762850620.00 6269333272055222120.00 0.00e+0000 1.00e+0000 18 0.99 2.216430954766998e+0020 2.194266645219328e+0020 2.216430954766998e+0020 2.194266645219328e+0020 0.00e+0000 4.80e+0001 19 0.99 8.200794532637892e+0021 8.118786587311513e+0021 8.200794532637892e+0021 8.118786587311513e+0021 0.00e+0000 1.54e+0003 20 0.99 3.198309867728778e+0023 3.166326769051490e+0023 3.198309867728778e+0023 3.166326769051490e+0023 0.00e+0000 6.55e+0004 21 0.99 1.311307045768799e+0025 1.298193975311111e+0025 1.311307045768799e+0025 1.298193975311111e+0025 0.00e+0000 2.10e+0006 22 0.99 5.638620296805835e+0026 5.582234093837777e+0026 5.638620296805835e+0026 5.582234093837777e+0026 0.00e+0000 1.01e+0008 23 0.99 2.537379133562626e+0028 2.512005342227000e+0028 2.537379133562626e+0028 2.512005342227000e+0028 0.00e+0000 4.29e+0009 24 0.99 1.192568192774434e+0030 1.180642510846690e+0030 1.192568192774434e+0030 1.180642510846690e+0030 0.00e+0000 2.75e+0011 25 0.99 5.843584144594727e+0031 5.785148303148780e+0031 5.843584144594727e+0031 5.785148303148780e+0031 0.00e+0000 1.32e+0013 26 0.99 2.980227913743311e+0033 2.950425634605878e+0033 2.980227913743311e+0033 2.950425634605878e+0033 0.00e+0000 5.63e+0014 27 0.99 1.579520794283955e+0035 1.563725586341115e+0035 1.579520794283955e+0035 1.563725586341115e+0035 0.00e+0000 3.60e+0016 28 0.99 8.687364368561751e+0036 8.600490724876134e+0036 8.687364368561751e+0036 8.600490724876134e+0036 0.00e+0000 1.73e+0018 29 0.99 4.951797690080198e+0038 4.902279713179396e+0038 4.951797690080198e+0038 4.902279713179396e+0038 0.00e+0000 7.38e+0019 30 0.99 2.921560637147317e+0040 2.892345030775844e+0040 2.921560637147317e+0040 2.892345030775844e+0040 0.00e+0000 2.36e+0021 31 0.99 1.782151988659863e+0042 1.764330468773265e+0042 1.782151988659863e+0042 1.764330468773265e+0042 0.00e+0000 1.51e+0023 32 0.99 1.122755752855714e+0044 1.111528195327157e+0044 1.122755752855714e+0044 1.111528195327157e+0044 0.00e+0000 9.67e+0024 33 0.99 7.297912393562140e+0045 7.224933269626519e+0045 7.297912393562140e+0045 7.224933269626519e+0045 0.00e+0000 1.24e+0027 34 0.99 4.889601303686634e+0047 4.840705290649768e+0047 4.889601303686634e+0047 4.840705290649768e+0047 0.00e+0000 7.92e+0028 35 0.99 3.373824899543777e+0049 3.340086650548340e+0049 3.373824899543777e+0049 3.340086650548340e+0049 0.00e+0000 7.61e+0030 36 0.99 2.395415678676082e+0051 2.371461521889321e+0051 2.395415678676082e+0051 2.371461521889321e+0051 0.00e+0000 4.87e+0032 37 0.99 1.748653445433540e+0053 1.731166910979204e+0053 1.748653445433540e+0053 1.731166910979204e+0053 0.00e+0000 3.12e+0034 38 0.99 1.311490084075155e+0055 1.298375183234403e+0055 1.311490084075155e+0055 1.298375183234403e+0055 0.00e+0000 2.66e+0036 39 0.99 1.009847364737869e+0057 9.997488910904906e+0056 1.009847364737869e+0057 9.997488910904906e+0056 0.00e+0000 2.55e+0038 40 0.99 7.977794181429167e+0058 7.898016239614876e+0058 7.977794181429167e+0058 7.898016239614876e+0058 0.00e+0000 2.18e+0040 41 0.99 6.462013286957625e+0060 6.397393154088049e+0060 6.462013286957625e+0060 6.397393154088049e+0060 0.00e+0000 1.74e+0042 42 0.99 5.363471028174829e+0062 5.309836317893081e+0062 5.363471028174829e+0062 5.309836317893081e+0062 0.00e+0000 1.34e+0044 43 0.99 4.558950373948605e+0064 4.513360870209119e+0064 4.558950373948605e+0064 4.513360870209119e+0064 0.00e+0000 8.56e+0045 44 0.99 3.966286825335286e+0066 3.926623957081933e+0066 3.966286825335286e+0066 3.926623957081933e+0066 0.00e+0000 1.10e+0048 45 0.99 3.529995274548405e+0068 3.494695321802921e+0068 3.529995274548405e+0068 3.494695321802921e+0068 0.00e+0000 9.35e+0049 46 0.99 3.212295699839048e+0070 3.180172742840658e+0070 3.212295699839048e+0070 3.180172742840658e+0070 0.00e+0000 5.99e+0051 47 0.99 2.987435000850315e+0072 2.957560650841812e+0072 2.987435000850315e+0072 2.957560650841812e+0072 0.00e+0000 5.75e+0053 48 0.99 2.838063250807799e+0074 2.809682618299721e+0074 2.838063250807799e+0074 2.809682618299721e+0074 0.00e+0000 7.36e+0055 49 0.99 2.752921353283565e+0076 2.725392139750730e+0076 2.752921353283565e+0076 2.725392139750730e+0076 0.00e+0000 6.28e+0057 50 0.99 2.725392139750730e+0078 2.698138218353222e+0078 2.725392139750730e+0078 2.698138218353222e+0078 0.00e+0000 8.03e+0059 51 0.99 2.752646061148237e+0080 2.725119600536754e+0080 2.752646061148237e+0080 2.725119600536754e+0080 0.00e+0000 7.71e+0061 52 0.99 2.835225442982684e+0082 2.806873188552857e+0082 2.835225442982684e+0082 2.806873188552857e+0082 0.00e+0000 9.87e+0063 53 0.99 2.976986715131818e+0084 2.947216847980500e+0084 2.976986715131818e+0084 2.947216847980500e+0084 0.00e+0000 8.42e+0065 54 0.99 3.185375785191045e+0086 3.153522027339135e+0086 3.185375785191045e+0086 3.153522027339135e+0086 0.00e+0000 1.08e+0068 55 0.99 3.472059605858239e+0088 3.437339009799657e+0088 3.472059605858239e+0088 3.437339009799657e+0088 0.00e+0000 1.04e+0070 56 0.99 3.853986162502646e+0090 3.815446300877619e+0090 3.853986162502646e+0090 3.815446300877619e+0090 0.00e+0000 1.33e+0072 57 0.99 4.355004363627990e+0092 4.311454319991710e+0092 4.355004363627990e+0092 4.311454319991710e+0092 0.00e+0000 1.41e+0074 58 0.99 5.008255018172188e+0094 4.958172467990466e+0094 5.008255018172188e+0094 4.958172467990466e+0094 0.00e+0000 1.81e+0076 59 0.99 5.859658371261460e+0096 5.801061787548846e+0096 5.859658371261460e+0096 5.801061787548846e+0096 0.00e+0000 1.85e+0078 60 0.99 6.972993461801138e+0098 6.903263527183126e+0098 6.972993461801138e+0098 6.903263527183126e+0098 0.00e+0000 1.78e+0080 61 0.99 8.437322088779377e+0100 8.352948867891583e+0100 8.437322088779377e+0100 8.352948867891583e+0100 0.00e+0000 2.28e+0082 62 0.99 1.037790616919863e+0103 1.027412710750665e+0103 1.037790616919863e+0103 1.027412710750665e+0103 0.00e+0000 1.94e+0084 63 0.99 1.297238271149829e+0105 1.284265888438331e+0105 1.297238271149829e+0105 1.284265888438331e+0105 0.00e+0000 2.49e+0086 64 0.99 1.647492604360283e+0107 1.631017678316680e+0107 1.647492604360283e+0107 1.631017678316680e+0107 0.00e+0000 4.77e+0088 65 0.99 2.125265459624765e+0109 2.104012805028517e+0109 2.125265459624765e+0109 2.104012805028517e+0109 0.00e+0000 6.11e+0090 66 0.99 2.784097752108442e+0111 2.756256774587358e+0111 2.784097752108442e+0111 2.756256774587358e+0111 0.00e+0000 5.21e+0092 67 0.99 3.702850010304228e+0113 3.665821510201186e+0113 3.702850010304228e+0113 3.665821510201186e+0113 0.00e+0000 1.00e+0095 68 0.99 4.998847513910708e+0115 4.948859038771601e+0115 4.998847513910708e+0115 4.948859038771601e+0115 0.00e+0000 1.28e+0097 69 0.99 6.848421094057670e+0117 6.779936883117093e+0117 6.848421094057670e+0117 6.779936883117093e+0117 0.00e+0000 2.19e+0099 70 0.99 9.519305320740161e+0119 9.424112267532760e+0119 9.519305320740161e+0119 9.424112267532760e+0119 0.00e+0000 2.80e+0101 71 0.99 1.342222050224363e+0122 1.328799829722119e+0122 1.342222050224363e+0122 1.328799829722119e+0122 0.00e+0000 4.48e+0103 72 0.99 1.919377531820839e+0124 1.900183756502630e+0124 1.919377531820839e+0124 1.900183756502630e+0124 0.00e+0000 6.88e+0105 73 0.99 2.783097421140216e+0126 2.755266446928814e+0126 2.783097421140216e+0126 2.755266446928814e+0126 0.00e+0000 1.17e+0108 74 0.99 4.091153209076118e+0128 4.050241676985357e+0128 4.091153209076118e+0128 4.050241676985357e+0128 0.00e+0000 1.88e+0110 75 0.99 6.095818281523415e+0130 6.034860098708181e+0130 6.095818281523415e+0130 6.034860098708181e+0130 0.00e+0000 2.89e+0112 76 0.99 9.204685605100357e+0132 9.112638749049354e+0132 9.204685605100357e+0132 9.112638749049354e+0132 0.00e+0000 4.31e+0114 77 0.99 1.408316897580355e+0135 1.394233728604551e+0135 1.408316897580355e+0135 1.394233728604551e+0135 0.00e+0000 7.09e+0116 78 0.99 2.182891191249550e+0137 2.161062279337054e+0137 2.182891191249550e+0137 2.161062279337054e+0137 0.00e+0000 1.21e+0119 79 0.99 3.427139170261793e+0139 3.392867778559175e+0139 3.427139170261793e+0139 3.392867778559175e+0139 0.00e+0000 1.81e+0121 80 0.99 5.449151280716251e+0141 5.394659767909089e+0141 5.449151280716251e+0141 5.394659767909089e+0141 0.00e+0000 2.64e+0123 81 0.99 8.773133561953164e+0143 8.685402226333633e+0143 8.773133561953164e+0143 8.685402226333633e+0143 0.00e+0000 4.23e+0125 82 0.99 1.430020770598366e+0146 1.415720562892382e+0146 1.430020770598366e+0146 1.415720562892382e+0146 0.00e+0000 7.58e+0127 83 0.99 2.359534271487304e+0148 2.335938928772430e+0148 2.359534271487304e+0148 2.335938928772430e+0148 0.00e+0000 1.25e+0130 84 0.99 3.940422233383797e+0150 3.901018011049959e+0150 3.940422233383797e+0150 3.901018011049959e+0150 0.00e+0000 2.13e+0132 85 0.99 6.659313574418617e+0152 6.592720438674431e+0152 6.659313574418617e+0152 6.592720438674431e+0152 0.00e+0000 4.09e+0134 86 0.99 1.138742621225583e+0155 1.127355195013328e+0155 1.138742621225583e+0155 1.127355195013328e+0155 0.00e+0000 6.98e+0136 87 0.99 1.970024734720259e+0157 1.950324487373057e+0157 1.970024734720259e+0157 1.950324487373057e+0157 0.00e+0000 1.19e+0139 88 0.99 3.447543285760454e+0159 3.413067852902849e+0159 3.447543285760454e+0159 3.413067852902849e+0159 0.00e+0000 2.29e+0141 89 0.99 6.102151615796003e+0161 6.041130099638043e+0161 6.102151615796003e+0161 6.041130099638043e+0161 0.00e+0000 3.90e+0143 90 0.99 1.092285139227485e+0164 1.081362287835210e+0164 1.092285139227485e+0164 1.081362287835210e+0164 0.00e+0000 6.87e+0145 91 0.99 1.977036102001747e+0166 1.957265740981730e+0166 1.977036102001747e+0166 1.957265740981730e+0166 0.00e+0000 1.12e+0148 92 0.99 3.617976066663197e+0168 3.581796305996565e+0168 3.617976066663197e+0168 3.581796305996565e+0168 0.00e+0000 2.05e+0150 93 0.99 6.693255723326915e+0170 6.626323166093646e+0170 6.693255723326915e+0170 6.626323166093646e+0170 0.00e+0000 4.19e+0152 94 0.99 1.251638820262133e+0173 1.239122432059512e+0173 1.251638820262133e+0173 1.239122432059512e+0173 0.00e+0000 8.04e+0154 95 0.99 2.365597370295432e+0175 2.341941396592477e+0175 2.365597370295432e+0175 2.341941396592477e+0175 0.00e+0000 1.37e+0157 96 0.99 4.518290977264274e+0177 4.473108067491632e+0177 4.518290977264274e+0177 4.473108067491632e+0177 0.00e+0000 2.64e+0159 97 0.99 8.720301586120049e+0179 8.633098570258849e+0179 8.720301586120049e+0179 8.633098570258849e+0179 0.00e+0000 4.50e+0161 98 0.99 1.700458809293410e+0182 1.683454221200476e+0182 1.700458809293410e+0182 1.683454221200476e+0182 0.00e+0000 7.20e+0163 99 0.99 3.349903854308017e+0184 3.316404815764937e+0184 3.349903854308017e+0184 3.316404815764937e+0184 0.00e+0000 1.66e+0166 100 0.99 6.666308670072954e+0186 6.599645583372224e+0186 6.666308670072954e+0186 6.599645583372224e+0186 0.00e+0000 2.83e+0168 101 0.99 1.339928042684664e+0189 1.326528762257817e+0189 1.339928042684664e+0189 1.326528762257817e+0189 0.00e+0000 4.83e+0170 102 0.99 2.720053926649867e+0191 2.692853387383369e+0191 2.720053926649867e+0191 2.692853387383369e+0191 0.00e+0000 1.08e+0173 103 0.99 5.576110549632228e+0193 5.520349444135906e+0193 5.576110549632228e+0193 5.520349444135906e+0193 0.00e+0000 2.37e+0175 104 0.99 1.154254883773871e+0196 1.142712334936132e+0196 1.154254883773871e+0196 1.142712334936132e+0196 0.00e+0000 4.05e+0177 105 0.99 2.412392707087391e+0198 2.388268780016517e+0198 2.412392707087391e+0198 2.388268780016517e+0198 0.00e+0000 1.04e+0180 106 0.99 5.090148611954395e+0200 5.039247125834851e+0200 5.090148611954395e+0200 5.039247125834851e+0200 0.00e+0000 1.99e+0182 107 0.99 1.084201654346286e+0203 1.073359637802823e+0203 1.084201654346286e+0203 1.073359637802823e+0203 0.00e+0000 4.25e+0184 108 0.99 2.331033556844515e+0205 2.307723221276070e+0205 2.331033556844515e+0205 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5.885517710039842e+1926 5.944967383878628e+1926 5.885517710039842e+1926 0.00e+0000 9.75e+1908 710 0.99 8.435908717723773e+1929 8.351549630546535e+1929 8.435908717723773e+1929 8.351549630546535e+1929 0.00e+0000 1.39e+1912 711 0.99 1.198742628788548e+1933 1.186755202500663e+1933 1.198742628788548e+1933 1.186755202500663e+1933 0.00e+0000 1.87e+1915 712 0.99 1.705810760766104e+1936 1.688752653158443e+1936 1.705810760766104e+1936 1.688752653158443e+1936 0.00e+0000 2.73e+1918 713 0.99 2.430780334091698e+1939 2.406472530750781e+1939 2.430780334091698e+1939 2.406472530750781e+1939 0.00e+0000 3.73e+1921 714 0.99 3.468723536748853e+1942 3.434036301381365e+1942 3.468723536748853e+1942 3.434036301381365e+1942 0.00e+0000 5.34e+1924 715 0.99 4.956805934014111e+1945 4.907237874673970e+1945 4.956805934014111e+1945 4.907237874673970e+1945 0.00e+0000 7.43e+1927 716 0.99 7.093189291574193e+1948 7.022257398658451e+1948 7.093189291574193e+1948 7.022257398658451e+1948 0.00e+0000 1.08e+1931 717 0.99 1.016454025482582e+1952 1.006289485227756e+1952 1.016454025482582e+1952 1.006289485227756e+1952 0.00e+0000 1.48e+1934 718 0.99 1.458611526567505e+1955 1.444025411301830e+1955 1.458611526567505e+1955 1.444025411301830e+1955 0.00e+0000 2.10e+1937 719 0.99 2.096024763677505e+1958 2.075064516040730e+1958 2.096024763677505e+1958 2.075064516040730e+1958 0.00e+0000 3.09e+1940 720 0.99 3.016179634931929e+1961 2.986017838582610e+1961 3.016179634931929e+1961 2.986017838582610e+1961 0.00e+0000 4.58e+1943 *) { to differentiate P_n(x) 0 <= m <= nmax , m <= n <= nmax but n >=2 d^m(P_n(x))/d(x)^m -> only x as variable } UNIT UnPolFun ; INTERFACE { to check derivatives of Legandre polinoms d^m/dx^m(P_n(x) d^m/dx^m [(1-2*\alpha*x+\alpha^2)^(-1/2)] = Sum (i=0) (i=infinity) [ \alpha^i*[d^m/dx^m P_i(x)] ] } Procedure DerPolLegandre ; { to check recurrence with the values d^(n-1)/dx^(n-1)[P_n(x)] = 1*3*5*...*(2*n-1)*x and d^n/dx^n[P_n(x)] = 1*3*5*...*(2*n-1) } Procedure DerPolegandrem ; IMPLEMENTATION Uses unforprt , { for PrtFile to print } unharstr ; { function FloatWithPower } const maxarr = 777 ; Type TMaxPow = Array[1..maxarr+1] of Extended ; TMaxPowPtr = ^TMaxPow ; { for arrays } Var pl,ul : TMaxPowPtr ; { for Legandre polinoms and r0/r pointer } { for m = 0 initial values for recurrence process P_n(x)=((2n-1)/n)*x*P_(n-1)(x)-(n-1)*P_(n-2)(x) P_n'(x)=n*P_(n-1)(x)+x*P_(n-1)'(x) P_n'(x)=(2n-1)*P_(n-1)(x)+P_(n-2)'(x) } Procedure ToClcInitRecur ( x : extended ) ; { argument x } Var n : word ; q : Extended ; Begin { argument x is known } pl^[1]:=x; { P_1(x)=1 Legandre polinom m=0 } pl^[2]:=1.5*Sqr(x)-0.5; { P_2(x)=3/2*x*x-1/2 Legandre polinom } ul^[1]:=1.0; { derivative P_1'(x)=1 m=0 } ul^[2]:=3.0*x; { derivative P_2'(x)=3*x } For n:=3 To maxarr+1 Do { maxarr array dimension } Begin q:=n; { Legandre polinom and derivative for m=0 } pl^[n]:=((2.0*q-1.0)/q)*x*pl^[n-1]-((q-1.0)/q)*pl^[n-2]; { P_n(x) } ul^[n]:=q*pl^[n-1]+x*ul^[n-1]; { P_n'(x) the first derivative } End; { by using P_n'(x)=n*P_(n-1)(x)+x*P_(n-1)'(x) } End; { to calculate recurrent value of the functions for the next value m we need the value p_(nm)(x)=d^m(P_n(x))/dx^m and p_(nm)'(x)=d^(m+1)(P_n(x))/dx^(m+1) for this calculations it is convenient recurrent formula P_n'(x)=(2n-1)*P_(n-1)(x)+P_(n-2)'(x) the first derivative P_n''(x)=(2n-1)*P_(n-1)'(x)+P_(n-2)''(x) the second derivative P_n'''(x)=(2n-1)*P_(n-1)''(x)+P_(n-2)'''(x) the third derivative ... a value P_n(x) and P_n(x)' we have from initial for 1 <= n <= Nmax the previous values p_(nm) for (m-1) must be in memory } Procedure ToClcRecurP ( m : word ) ; Var k : word ; n : word ; q : extended ; Begin If m < 2 Then k:=2 Else k:=m; For n:=1 To maxarr+1 Do { in pl^[n] are p_(nm)(z/r) } pl^[n]:=ul^[n]; { in ul^[n] are the previous derivatives } For n:=1 To k-1 Do ul^[n]:=0.0; { to nullo } For n:=k To maxarr+1 Do Begin q:=n; ul^[n]:=(2.0*q-1.0)*pl^[n-1]; If ( n > (k+1) ) Then { in ul^[n] are d(p_(nm)(z/r))/d(z/r) } ul^[n]:=ul^[n]+ul^[n-2]; End; End; { to calculate recurrent value of the functions for the next value m } Procedure ToClcRecurM ( m : word ) ; Begin If m = 0 Then Exit ; { initial values are already } ToClcRecurP(m); { for p_(nm)(x)=d^m(P_n(x))/dx^m } End; { Sum (i=0) (i=infinity) [ \alpha^i*[d^m/dx^m P_i(x)] ] } function vals ( m : word ; a ,f : extended ) : extended ; var i : word ; { count } r : extended ; { \alpha in power } s : extended ; { sum value } b : boolean ; { to compare differencies } begin r:=1.0; { initial for power \alpha } s:=0.0; { initial to sum } b:=False; { initial value } i:=0; While ( ( i < maxarr ) and ( not b ) ) do begin { for the current order } i:=i+1; r:=a*r; { \alpha to the next power } s:=s+r*pl^[i]; { the next term to sum } b:= ( abs((s-f)/f) < 1.0e-6 ) ; { to compare values } if b then { values are closed } if ( ( f < 1.0e13 ) and ( Abs(s) < 1.0e13 ) ) then writeln(prtfile,m:5,i:5,' ',s:35:16,' ', FloatWithPower(((s-f)/f),16), FloatWithPower(pl^[i],12)) else writeln(prtfile,m:5,i:5,' ':15,FloatWithPower(s,16),' ':4, FloatWithPower(((s-f)/f),16), FloatWithPower(pl^[i],12)); end; if ( not b ) then { to maximum number } if ( ( f < 1.0e13 ) and ( Abs(s) < 1.0e13 ) ) then writeln(prtfile,m:5,i:5,' ',s:35:16,' ', FloatWithPower(((s-f)/f),16), FloatWithPower(pl^[i],12)) else writeln(prtfile,m:5,i:5,' ':15,FloatWithPower(s,16),' ':4, FloatWithPower(((s-f)/f),16), FloatWithPower(pl^[i],12)); vals:=s; { our result our sum } end; { d^m/dx^m [(1-2*\alpha*x+\alpha^2)^(-1/2)] = (2*m-1)!! * \alpha^m * (1-2*\alpha*x+\alpha^2)^(-(2*n+1)/2) } function funv ( m : word ; a , x : extended ) : extended ; var i : word ; f : extended ; q : extended ; p : extended ; r : extended ; begin q:=1.0; { initial 1.0 } for i:=1 to m do { for (2*m-1)!! } q:=(2.0*i-1.0)*q; { = 1*3*5*...*(2*m-1) } r:=1.0; { initial 1.0 } for i:=1 to m do { for \alpha^m a for \alpha } r:=a*r; { power by power for \alpha } { for function } p:=1.0/(1.0-2.0*a*x+sqr(a)); { 1/(1-2*\alpha*x+\alpha^2) } f:=sqrt(p); { (1-2*\alpha*x+\alpha^2)^(-1/2) } for i:=1 to m do { to power -(2*n+1)/2 } f:=p*f; { power by power } funv:=q*r*f; { our result } end; { to check derivatives of Legandre polinoms d^m/dx^m(P_n(x) d^m/dx^m [(1-2*\alpha*x+\alpha^2)^(-1/2)] = Sum (i=0) (i=infinity) [ \alpha^i*[d^m/dx^m P_i(x)] ] } Procedure DerPolLegandre ; Var i : word ; m : word ; n : word ; a : extended ; { parameter \alpha < 0.999 } x : extended ; { argument x ( -1 < x < +1 ) } f : extended ; { function } s : extended ; { sum with derivatives } r : extended ; Begin GetMem(pl,SizeOf(TMaxPow)); { for P_(n)(x) } GetMem(ul,SizeOf(TMaxPow)); { for P_n'(x) } n:=36; { for the derivatives } a:=+0.90; { parameter \alpha } x:=+0.99; { argument x } writeln(prtfile,'parameter \alpha value',a:10:3); writeln(prtfile,'parameter x value',x:10:3); r:=1.0; { initial for power \alpha } for i:=37 to 47 do begin r:=a*r; writeln(prtfile,i:5,a:10:2,r:16:6); { unharstr } end; ToClcInitRecur(x); { initial values for recurrent formulae , m =0 } m:=1; { the first derivative } Repeat { derivative by derivative } f:=funv(m,a,x); { d^m/dx^m [(1-2*\alpha*x+\alpha^2)^(-1/2)] } if f < 1.0e12 then writeln(prtfile,m:5,' ':5,' ',f:35:16) else writeln(prtfile,m:5,' ':5,' ':15,FloatWithPower(f,16)); ToClcRecurM(m); { recurrent value for the next m result to pl^[..] } s:=vals(m,a,f); { sum [ \alpha^i*[d^m/dx^m P_i(x)] ] } m:=m+1; { the next derivative } Until m > n ; { global uncontyp } FreeMem(ul,SizeOf(TMaxPow)); { for P_n'(x) } FreeMem(pl,SizeOf(TMaxPow)); { for P_(n)(x) } End; { to check recurrence with the values d^(n-1)/dx^(n-1)[P_n(x)] = 1*3*5*...*(2*n-1)*x and d^n/dx^n[P_n(x)] = 1*3*5*...*(2*n-1) } Procedure DerPolegandrem ; Var i : word ; m : word ; n : word ; x : extended ; { argument x ( -1 <= x <= +1 ) } q : extended ; { value temporary } r : extended ; { value temporary } Begin GetMem(pl,SizeOf(TMaxPow)); { for P_(n)(x) } GetMem(ul,SizeOf(TMaxPow)); { for P_n'(x) } n:=720; { for the derivatives } x:=+0.99; { argument x } writeln(prtfile); { global text file from unforprt } writeln(prtfile,'to check d^n/dx^n[P_n(x)] = 1*3*5*...*(2*n-1)'); ToClcInitRecur(x); { initial values for recurrent formulae , m =0 } m:=1; { the first derivative } Repeat { derivative by derivative } q:=1.0; { initial 1.0 } for i:=1 to m do { for (2*m-1)!! } q:=(2.0*i-1.0)*q; { = 1*3*5*...*(2*m-1) } r:=x*q; { = 1*3*5*...*(2*m-1) * x value } if q < 1.0e19 then begin writeln(m:3,x:6:2,' ',q:32:2,r:32:2); writeln(' ':10,ul^[m]:32:2,pl^[m]:32:2); writeln(prtfile,m:3,x:6:2,' ',q:32:2,r:32:2); writeln(prtfile,' ':10,ul^[m]:32:2,pl^[m]:32:2); end else begin writeln(m:3,x:6:2, ' ':6,GloatWithPower(q,25), ' ':5,GloatWithPower(r,25)); writeln(' ':15,GloatWithPower(ul^[m],25), ' ':5,GloatWithPower(pl^[m],25)); writeln(prtfile,m:3,x:6:2, ' ':6,GloatWithPower(q,25), ' ':5,GloatWithPower(r,25)); writeln(prtfile,' ':15,GloatWithPower(ul^[m],25), ' ':5,GloatWithPower(pl^[m],25)); end; writeln(' ':28,GloatWithPower((ul^[m]-q),12), ' ':18,GloatWithPower((pl^[m]-r),12)); writeln(prtfile,' ':28,GloatWithPower((ul^[m]-q),12), ' ':18,GloatWithPower((pl^[m]-r),12)); ToClcRecurM(m); { recurrent value for the next m result to pl^[..] } m:=m+1; { the next derivative } Until m > n ; { global uncontyp } FreeMem(ul,SizeOf(TMaxPow)); { for P_n'(x) } FreeMem(pl,SizeOf(TMaxPow)); { for P_(n)(x) } End; END.