1 /* 2 XLiFE++ is an extended library of finite elements written in C++ 3 Copyright (C) 2014 Lunéville, Eric; Kielbasiewicz, Nicolas; Lafranche, Yvon; Nguyen, Manh-Ha; Chambeyron, Colin 4 5 This program is free software: you can redistribute it and/or modify 6 it under the terms of the GNU General Public License as published by 7 the Free Software Foundation, either version 3 of the License, or 8 (at your option) any later version. 9 This program is distributed in the hope that it will be useful, 10 but WITHOUT ANY WARRANTY; without even the implied warranty of 11 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 12 GNU General Public License for more details. 13 You should have received a copy of the GNU General Public License 14 along with this program. If not, see <http://www.gnu.org/licenses/>. 15 */ 16 17 /*! 18 \file specialFunctions.hpp 19 \author D. Martin 20 \since 10 jan 2003 21 \date 9 apr 2007 22 23 \brief Definition of Bessel, Hankel and Struve functions 24 25 -- Bessel functions 26 - HANDBOOK Of MATHEMATICAL FUNCTIONS, Ed. M.ABRAMOWITZ & I.A. STEGUN 27 - Chap. 9 : Bessel Functions of Integer Order (p.355-456) 28 - http://www.math.sfu.ca/~cbm/aands/page_355.htm 29 - 30 bessel_J0 Bessel function of the first kind and order 0 J_0(x) 31 bessel_J1 Bessel function of the first kind and order 1 J_1(x) 32 bessel_J0N Bessel functions of the first kind and orders 0..N J_0(x)..J_N(x) 33 bessel_Y0 Bessel function of the second kind and order 0 Y_0(x) 34 bessel_Y1 Bessel function of the second kind and order 1 Y_1(x) 35 bessel_Y0N Bessel functions of the second kind and orders 0..N Y_0(x)..Y_N(x) 36 hankel_H10 Hankel function of order 0 (J_0(x) + i * Y_0(x)) H_0^{1}(x) 37 hankel_H11 Hankel function of order 1 (J_1(x) + i * Y_1(x)) H_1^{1}(x) 38 *bessel_I0 Modified Bessel function of the first kind and order 0 I_0(x) 39 *bessel_I1 Modified Bessel function of the first kind and order 1 I_1(x) 40 *bessel_K0 Modified Bessel function of the second kind and order 0 K_0(x) 41 *bessel_K1 Modified Bessel function of the second kind and order 1 K_1(x) 42 43 -- Spherical Bessel functions 44 - HANDBOOK Of MATHEMATICAL FUNCTIONS, Ed. M.ABRAMOWITZ & I.A. STEGUN 45 - Chap. 10 : Bessel Functions of Fractional Order (p.437-494) 46 - http://www.math.sfu.ca/~cbm/aands/page_437.htm 47 - 48 sphericalbessel_j0N Spherical Bessel functions of the first kind and orders 0..N j_0(x)..j_N(x) 49 sphericalbessel_y0N Spherical Bessel functions of the second kind and orders 0..N y_0(x)..y_N(x) 50 51 -- Struve functions 52 - HANDBOOK Of MATHEMATICAL FUNCTIONS , Ed. M.ABRAMOWITZ & I.A. STEGUN 53 - Chap. 12 : Struve Functions and Related Functions (p.495-502) 54 - http://www.math.sfu.ca/~cbm/aands/page_495.htm 55 - MATHEMATICAL FUNCTIONS AND THEIR APPROXIMATIONS, Yudell L.LUKE 56 - Chap. X : Lommel and Struve functions (pp. 416--..) 57 - 58 struveNot_H0 Struve function of order 0 H_0(x) 59 struveNot_H1 Struve function of order 1 H_1(x) 60 struveNot_H01 Nearly a Struve function 61 returning H_0(x) and H_1(x) for |x| < 8 H_0(x),H_1(x) 62 returning H_0(x)-Y_0(x) and H_1(x)-Y_1(x) for |x| > 8 63 */ 64 65 #ifndef SPECIAL_FUNCTIONS_HPP 66 #define SPECIAL_FUNCTIONS_HPP 67 68 #include "config.h" 69 #include "amosWrapper/amosWrapper.hpp" 70 71 namespace xlifepp 72 { 73 74 //-------------------------------------------------------------------------------- 75 // Bessel functions 76 //-------------------------------------------------------------------------------- 77 real_t besselJ0(const real_t x); //!< Bessel function of the first kind and order 0 : J_0(x) 78 complex_t besselJ0(const complex_t z); //!< Bessel function of the first kind and order 0 : J_0(z) (complex case) 79 80 real_t besselJ1(const real_t x); //!< Bessel function of the first kind and order 1 : J_1(x) 81 complex_t besselJ1(const complex_t z); //!< Bessel function of the first kind and order 1 : J_1(z) (complex case) 82 83 std::vector<real_t> besselJ0N(const real_t x, const number_t N); //!< Bessel functions of the first kind and order 0..N 84 85 real_t besselJ(const real_t x, const number_t N); //!< Bessel functions of the first kind and order N : J_N(x) 86 complex_t besselJ(const complex_t z, const real_t N); //!< Bessel functions of the first kind and order N : J_N(z) (complex case) 87 88 real_t besselY0withoutSingularity(const real_t x); 89 real_t besselY0(const real_t x); //!< Bessel function of the second kind and order 0 : Y_0(x) 90 complex_t besselY0(const complex_t z); //!< Bessel function of the second kind and order 0 : Y_0(z) (complex case) 91 92 real_t besselY1withoutSingularity(const real_t x); 93 real_t besselY1(const real_t x); //!< Bessel function of the second kind and order 0 : Y_1(x) 94 complex_t besselY1(const complex_t z); //!< Bessel function of the second kind and order 0 : Y_1(z) (complex case) 95 96 std::vector<real_t> besselY0N(const real_t x, const number_t N); //!< Bessel functions of the second kind and order 0..N 97 98 real_t besselY(const real_t x, const number_t N); //!< Bessel functions of the second kind and order N : Y_N(x) 99 complex_t besselY(const complex_t z, const real_t N); //!< Bessel functions of the second kind and order N : Y_N(z) (complex case) 100 101 real_t besselI0(const real_t x); //!< Modified Bessel function of the first kind and order 0 : I_0(x) 102 complex_t besselI0(const complex_t z); //!< Modified Bessel function of the first kind and order 0 : I_0(z) (complex case) 103 104 real_t besselI1(const real_t x); //!< Modified Bessel function of the first kind and order 1 : I_1(x) 105 complex_t besselI1(const complex_t z); //!< Modified Bessel function of the first kind and order 1 : I_1(z) (complex case) 106 107 std::vector<real_t> besselI0N(const real_t x, const number_t n); //!< Modified Bessel functions of the first kind and order 0..N 108 109 real_t besselI(const real_t x, const number_t N); //!< Bessel functions of the first kind and order N : I_N(x) 110 complex_t besselI(const complex_t z, const real_t N); //!< Bessel functions of the first kind and order N : I_N(z) (complex case) 111 112 real_t besselK0(const real_t x); //!< Modified Bessel function of the second kind and order 0 : K_0(x) 113 complex_t besselK0(const complex_t z); //!< Modified Bessel function of the second kind and order 0 : K_0(z) (complex case) 114 115 real_t besselK1(const real_t x); //!< Modified Bessel function of the second kind and order 1 : K_1(x) 116 complex_t besselK1(const complex_t z); //!< Modified Bessel function of the second kind and order 1 : K_1(z) (complex case) 117 118 std::vector<real_t> besselK0N(const real_t x, const number_t n); //!< Modified Bessel functions of the second kind and order 0..N 119 120 real_t besselK(const real_t x, const number_t N); //!< Bessel functions of the second kind and order N : K_N(x) 121 complex_t besselK(const complex_t z, const real_t N); //!< Bessel functions of the second kind and order N : K_N(z) (complex case) 122 123 complex_t hankelH10(const real_t x); //!< Hankel function of the first kind and order 0 : H1_0(x) = J_0(x) + i Y_0(x) 124 complex_t hankelH10(const complex_t z); //!< Hankel function of the first kind and order 0 : H1_0(z) = J_0(z) + i Y_0(z) (complex case) 125 126 complex_t hankelH11(const real_t x); //!< Hankel function of the first kind and order 1 : H1_1(x) = J_1(x) + i Y_1(x) 127 complex_t hankelH11(const complex_t z); //!< Hankel function of the first kind and order 1 : H1_1(z) = J_1(z) + i Y_1(z) (complex case) 128 129 std::vector<complex_t> hankelH10N(const real_t x, const number_t N); //!< Hankel function of the first kind and order 0 ... N : H1_k(x) = J_k(x) + i Y_k(x) 130 131 complex_t hankelH1(const real_t x, const number_t N); //!< Hankel function of the first kind and order N : H1_N(x) = J_N(x) + i Y_N(x) 132 complex_t hankelH1(const complex_t z, const real_t N); //!< Hankel function of the first kind and order N : H1_N(x) = J_N(x) + i Y_N(x) 133 134 complex_t hankelH20(const real_t x); //!< Hankel function of the second kind and order 0 : H2_0(x) 135 complex_t hankelH20(const complex_t z); //!< Hankel function of the second kind and order 0 : H2_0(z) (complex case) 136 137 complex_t hankelH21(const real_t x); //!< Hankel function of the second kind and order 1 : H2_1(x) 138 complex_t hankelH21(const complex_t z); //!< Hankel function of the second kind and order 1 : H2_1(z) (complex case) 139 140 std::vector<complex_t> hankelH20N(const real_t x, const number_t N); //!< Hankel function of the second kind and order 0 ... N : H2_k(x) 141 142 complex_t hankelH2(const real_t x, const number_t N); //!< Hankel function of the second kind and order N : H2_N(x) 143 complex_t hankelH2(const complex_t z, const real_t N); //!< Hankel function of the second kind and order N : H2_N(x) 144 145 /* 146 -------------------------------------------------------------------------------- 147 Struve functions 148 -------------------------------------------------------------------------------- 149 */ 150 151 /*! Struve function of order 0, H_0(x) 152 returns H_0(x) for any |x|<=8, H_0(x)-Y_0(x) for any |x|>8. 153 */ 154 real_t struveNotH0(const real_t x); 155 156 /*! Struve function of order 1, H_1(x) 157 returns H_1(x) for any |x|<=‡8, H_1(x)-Y_1(x) for any |x|>8. 158 */ 159 real_t struveNotH1(const real_t x); 160 161 /*! returns both H_0(x) and H_1(x) for any |x|<8, 162 returns H_0(x)-Y_0(x) and H_1(x)-Y_1(x) for any |x|>8. 163 */ 164 std::pair<real_t, real_t> struveNotH01(const real_t x); 165 166 /* 167 -------------------------------------------------------------------------------- 168 spherical Bessel functions 169 -------------------------------------------------------------------------------- 170 */ 171 //! spherical Bessel function of the first kind and orders 0..N 172 std::vector<real_t> sphericalbesselJ0N(const real_t x, const number_t N); 173 174 //! spherical Bessel function of the second kind and orders 0..N 175 std::vector<real_t> sphericalbesselY0N(const real_t x, const number_t N); 176 177 /* 178 -------------------------------------------------------------------------------- 179 For tests 180 -------------------------------------------------------------------------------- 181 */ 182 void besselJY01Test(std::ostream&); // compare with Table 9.1 of Abramawitz & Stegun 183 /* 184 390 BESSEL FUNCTIONS OF INTEGER ORDER 391 185 186 Table 9.1 BESSEL FUNCTIONS--ORDER 0, 1 AND 2 187 188 x J0(x) J1(x) x Y0(x) Y1(x) 189 190 0.0 1.000000000000e+00 0.000000000000e+00 0.0 -inf -inf 191 0.1 9.975015620660e-01 4.993752603624e-02 0.1 -1.534238651350e+00 -6.458951094702e+00 192 0.2 9.900249722396e-01 9.950083263924e-02 0.2 -1.081105322372e+00 -3.323824988112e+00 193 0.3 9.776262465383e-01 1.483188162731e-01 0.3 -8.072735778045e-01 -2.293105138389e+00 194 0.4 9.603982266596e-01 1.960265779553e-01 0.4 -6.060245684270e-01 -1.780872044270e+00 195 0.5 9.384698072408e-01 2.422684576749e-01 0.5 -4.445187335067e-01 -1.471472392670e+00 196 0.6 9.120048634972e-01 2.867009880639e-01 0.6 -3.085098701156e-01 -1.260391347177e+00 197 0.7 8.812008886074e-01 3.289957415401e-01 0.7 -1.906649293374e-01 -1.103249871908e+00 198 0.8 8.462873527505e-01 3.688420460942e-01 0.8 -8.680227965661e-02 -9.781441766834e-01 199 0.9 8.075237981225e-01 4.059495460788e-01 0.9 5.628306635205e-03 -8.731265824563e-01 200 1.0 7.651976865580e-01 4.400505857449e-01 1.0 8.825696421568e-02 -7.812128213003e-01 201 1.1 7.196220185275e-01 4.709023948663e-01 1.1 1.621632029269e-01 -6.981195600677e-01 202 1.2 6.711327442644e-01 4.982890575672e-01 1.2 2.280835032272e-01 -6.211363797488e-01 203 1.3 6.200859895615e-01 5.220232474147e-01 1.3 2.865353571656e-01 -5.485197299808e-01 204 1.4 5.668551203743e-01 5.419477139309e-01 1.4 3.378951296797e-01 -4.791469742328e-01 205 1.5 5.118276717359e-01 5.579365079101e-01 1.5 3.824489237978e-01 -4.123086269739e-01 206 1.6 4.554021676394e-01 5.698959352617e-01 1.6 4.204268964157e-01 -3.475780082651e-01 207 1.7 3.979848594461e-01 5.777652315290e-01 1.7 4.520270001816e-01 -2.847262450641e-01 208 1.8 3.399864110426e-01 5.815169517312e-01 1.8 4.774317149005e-01 -2.236648681735e-01 209 1.9 2.818185593744e-01 5.811570727134e-01 1.9 4.968199712838e-01 -1.644057723316e-01 210 2.0 2.238907791412e-01 5.767248077569e-01 2.0 5.103756726497e-01 -1.070324315409e-01 211 2.1 1.666069803320e-01 5.682921357570e-01 2.1 5.182937375138e-01 -5.167861213042e-02 212 2.2 1.103622669222e-01 5.559630498191e-01 2.2 5.207842853880e-01 1.487789289764e-03 213 2.3 5.553978444560e-02 5.398725326043e-01 2.3 5.180753962076e-01 5.227731584423e-02 214 2.4 2.507683297243e-03 5.201852681819e-01 2.4 5.104147486657e-01 1.004889383311e-01 215 2.5 -4.838377646820e-02 4.970941024643e-01 2.5 4.980703596152e-01 1.459181379668e-01 216 2.6 -9.680495439704e-02 4.708182665176e-01 2.6 4.813305905870e-01 1.883635443502e-01 217 2.7 -1.424493700460e-01 4.416013791183e-01 2.7 4.605035490754e-01 2.276324458709e-01 218 2.8 -1.850360333644e-01 4.097092468523e-01 2.8 4.359159856397e-01 2.635453936351e-01 219 2.9 -2.243115457920e-01 3.754274818131e-01 2.9 4.079117692362e-01 2.959400546077e-01 220 3.0 -2.600519549019e-01 3.390589585259e-01 3.0 3.768500100128e-01 3.246744247918e-01 221 3.1 -2.920643476507e-01 3.009211331011e-01 3.1 3.431028893545e-01 3.496294822677e-01 222 3.2 -3.201881696571e-01 2.613432487805e-01 3.2 3.070532501324e-01 3.707113384413e-01 223 3.3 -3.442962603989e-01 2.206634529852e-01 3.3 2.690919950545e-01 3.878529310237e-01 224 3.4 -3.642955967620e-01 1.792258516815e-01 3.4 2.296153372096e-01 4.010152921085e-01 225 3.5 -3.801277399873e-01 1.373775273623e-01 3.5 1.890219439208e-01 4.101884178875e-01 226 3.6 -3.917689837008e-01 9.546554717788e-02 3.6 1.477100126112e-01 4.153917621114e-01 227 3.7 -3.992302033712e-01 5.383398774546e-02 3.7 1.060743153204e-01 4.166743726838e-01 228 3.8 -4.025564101786e-01 1.282100292673e-02 3.8 6.450324665499e-02 4.141146893078e-01 229 3.9 -4.018260148876e-01 -2.724403962078e-02 3.9 2.337590819872e-02 4.078200195265e-01 230 4.0 -3.971498098638e-01 -6.604332802355e-02 4.0 -1.694073932507e-02 3.979257105571e-01 231 4.1 -3.886696798359e-01 -1.032732577473e-01 4.1 -5.609462660635e-02 3.845940348189e-01 232 4.2 -3.765570543676e-01 -1.386469421260e-01 4.2 -9.375120131443e-02 3.680128078542e-01 233 4.3 -3.610111172365e-01 -1.718965602215e-01 4.3 -1.295959028909e-01 3.483937583140e-01 234 4.4 -3.422567900039e-01 -2.027755219231e-01 4.4 -1.633364628042e-01 3.259706707535e-01 235 4.5 -3.205425089851e-01 -2.310604319234e-01 4.5 -1.947050086295e-01 3.009973230697e-01 236 4.6 -2.961378165741e-01 -2.565528360974e-01 4.6 -2.234599525536e-01 2.737452414709e-01 237 4.7 -2.693307894198e-01 -2.790807358433e-01 4.7 -2.493876472450e-01 2.445012968471e-01 238 4.8 -2.404253272912e-01 -2.984998580996e-01 4.8 -2.723037944566e-01 2.135651672662e-01 239 4.9 -2.097383275853e-01 -3.146946710152e-01 4.9 -2.920545942440e-01 1.812466920450e-01 240 5.0 -1.775967713143e-01 -3.275791375915e-01 5.0 -3.085176252490e-01 1.478631433912e-01 241 242 392 BESSEL FUNCTIONS OF INTEGER ORDER 393 243 244 Table 9.1 BESSEL FUNCTIONS--ORDER 0, 1 AND 2 245 246 x J0(x) J1(x) x Y0(x) Y1(x) 247 248 5.0 -1.775967713143e-01 -3.275791375915e-01 5.0 -3.085176252490e-01 1.478631433912e-01 249 5.1 -1.443347470605e-01 -3.370972020182e-01 5.1 -3.216024491249e-01 1.137364419775e-01 250 5.2 -1.102904397910e-01 -3.432230058719e-01 5.2 -3.312509348199e-01 7.919034298209e-02 251 5.3 -7.580311158559e-02 -3.459608338012e-01 5.3 -3.374373010937e-01 4.454761908761e-02 252 5.4 -4.121010124499e-02 -3.453447907796e-01 5.4 -3.401678782759e-01 1.012726668250e-02 253 5.5 -6.843869417820e-03 -3.414382154290e-01 5.5 -3.394805928819e-01 -2.375823895639e-02 254 5.6 2.697088468511e-02 -3.343328362910e-01 5.6 -3.354441812453e-01 -5.680561439948e-02 255 5.7 5.992000972404e-02 -3.241476802229e-01 5.7 -3.281571407937e-01 -8.872334050660e-02 256 5.8 9.170256757481e-02 -3.110277443039e-01 5.8 -3.177464299633e-01 -1.192341134563e-01 257 5.9 1.220333545928e-01 -2.951424447290e-01 5.9 -3.043659299972e-01 -1.480771525310e-01 258 6.0 1.506452572510e-01 -2.766838581276e-01 6.0 -2.881946839816e-01 -1.750103443004e-01 259 6.1 1.772914222427e-01 -2.558647725584e-01 6.1 -2.694349304312e-01 -1.998122044893e-01 260 6.2 2.017472229489e-01 -2.329165670732e-01 6.2 -2.483099505160e-01 -2.222836406201e-01 261 6.3 2.238120061322e-01 -2.080869402072e-01 6.3 -2.250617496168e-01 -2.422495004633e-01 262 6.4 2.433106048234e-01 -1.816375090242e-01 6.4 -1.999485952920e-01 -2.595598933986e-01 263 6.5 2.600946055816e-01 -1.538413014100e-01 6.5 -1.732424349190e-01 -2.740912739593e-01 264 6.6 2.740433606241e-01 -1.249801651606e-01 6.6 -1.452262172341e-01 -2.857472790945e-01 265 6.7 2.850647377106e-01 -9.534211804139e-02 6.7 -1.161911427261e-01 -2.944593130029e-01 266 6.8 2.930956031043e-01 -6.521866340169e-02 6.8 -8.643386833578e-02 -3.001868757586e-01 267 6.9 2.981020354048e-01 -3.490209610462e-02 6.9 -5.625369217089e-02 -3.029176343352e-01 268 7.0 3.000792705196e-01 -4.682823482349e-03 7.0 -2.594974396721e-02 -3.026672370242e-01 269 7.1 2.990513805016e-01 2.515327425391e-02 7.1 4.181793191708e-03 -2.994788746010e-01 270 7.2 2.950706914010e-01 5.432742022236e-02 7.2 3.385040483617e-02 -2.934225939199e-01 271 7.3 2.882169476350e-01 8.257043049326e-02 7.3 6.277388637403e-02 -2.845943718681e-01 272 7.4 2.785962326575e-01 1.096250948540e-01 7.4 9.068088018003e-02 -2.731149597811e-01 273 7.5 2.663396578804e-01 1.352484275797e-01 7.5 1.173132861482e-01 -2.591285104861e-01 274 7.6 2.516018338500e-01 1.592137683964e-01 7.6 1.424285246603e-01 -2.428010020793e-01 275 7.7 2.345591395865e-01 1.813127153246e-01 7.7 1.658016324239e-01 -2.243184743430e-01 276 7.8 2.154078077463e-01 2.013568727559e-01 7.8 1.872271733096e-01 -2.038850953513e-01 277 7.9 1.943618448413e-01 2.191793999217e-01 7.9 2.065209481444e-01 -1.817210772806e-01 278 8.0 1.716508071376e-01 2.346363468539e-01 8.0 2.235214893876e-01 -1.580604617313e-01 279 8.1 1.475174540444e-01 2.476077669816e-01 8.1 2.380913287022e-01 -1.331487959525e-01 280 8.2 1.222153017841e-01 2.579985976487e-01 8.2 2.501180276191e-01 -1.072407222508e-01 281 8.3 9.600610089501e-02 2.657393020419e-01 8.3 2.595149637545e-01 -8.059750353385e-02 282 8.4 6.915726165699e-02 2.707862682768e-01 8.4 2.662218673639e-01 -5.348450839742e-02 283 8.5 4.193925184294e-02 2.731219636741e-01 8.5 2.702051053658e-01 -2.616867939854e-02 284 8.6 1.462299127875e-02 2.727548445459e-01 8.6 2.714577123380e-01 1.083991830187e-03 285 8.7 -1.252273244966e-02 2.697190240921e-01 8.7 2.699991703467e-01 2.801095917680e-02 286 8.8 -3.923380317654e-02 2.640737032397e-01 8.8 2.658749417912e-01 5.435556333494e-02 287 8.9 -6.525324685124e-02 2.559023714440e-01 8.9 2.591557617217e-01 7.986939739413e-02 288 9.0 -9.033361118287e-02 2.453117865733e-01 9.0 2.499366982850e-01 1.043145751967e-01 289 9.1 -1.142392326832e-01 2.324307450059e-01 9.1 2.383359920541e-01 1.274658819640e-01 290 9.2 -1.367483707649e-01 2.174086549605e-01 9.2 2.244936869905e-01 1.491127879223e-01 291 9.3 -1.576551899434e-01 2.004139278437e-01 9.3 2.085700676452e-01 1.690613070614e-01 292 9.4 -1.767715727515e-01 1.816322040070e-01 9.4 1.907439189130e-01 1.871356847183e-01 293 9.5 -1.939287476874e-01 1.612644307575e-01 9.5 1.712106262027e-01 2.031798993872e-01 294 9.6 -2.089787183689e-01 1.395248117407e-01 9.6 1.501801352559e-01 2.170589659890e-01 295 9.7 -2.217954820317e-01 1.166386479002e-01 9.7 1.278747920242e-01 2.286600297761e-01 296 9.8 -2.322760275794e-01 9.284009111281e-02 9.8 1.045270839992e-01 2.378932420862e-01 297 9.9 -2.403411055348e-01 6.836983228370e-02 9.9 8.037730516278e-02 2.446924112612e-01 298 10.0 -2.459357644513e-01 4.347274616887e-02 10.0 5.567116728360e-02 2.490154242070e-01 299 300 394 BESSEL FUNCTIONS OF INTEGER ORDER 395 301 302 Table 9.1 BESSEL FUNCTIONS--ORDER 0, 1 AND 2 303 304 x J0(x) J1(x) x Y0(x) Y1(x) 305 306 10.0 -2.459357644513e-01 4.347274616887e-02 10.0 5.567116728360e-02 2.490154242070e-01 307 10.1 -2.490296505809e-01 1.839551545758e-02 10.1 3.065738063332e-02 2.508444362565e-01 308 10.2 -2.496170698541e-01 -6.615743297719e-03 10.2 5.585227315442e-03 2.501858291982e-01 309 10.3 -2.477168134822e-01 -3.131782947600e-02 10.3 -1.929784969275e-02 2.470699395159e-01 310 10.4 -2.433717507142e-01 -5.547276184899e-02 10.4 -4.374861899939e-02 2.415505610460e-01 311 10.5 -2.366481944623e-01 -7.885001422733e-02 10.5 -6.753037249787e-02 2.337042283573e-01 312 10.6 -2.276350476207e-01 -1.012286625856e-01 10.6 -9.041515478014e-02 2.236292891956e-01 313 10.7 -2.164427399238e-01 -1.223994239272e-01 10.7 -1.121858896683e-01 2.114447762747e-01 314 10.8 -2.032019671120e-01 -1.421665682987e-01 10.8 -1.326383843902e-01 1.972890905317e-01 315 10.9 -1.880622459633e-01 -1.603496866808e-01 10.9 -1.515831932230e-01 1.813185096742e-01 316 11.0 -1.711903004072e-01 -1.767852989567e-01 11.0 -1.688473238921e-01 1.637055374149e-01 317 11.1 -1.527682954357e-01 -1.913282877750e-01 11.1 -1.842757716215e-01 1.446371102063e-01 318 11.2 -1.329919368596e-01 -2.038531458647e-01 11.2 -1.977328674821e-01 1.243126795321e-01 319 11.3 -1.120684561098e-01 -2.142550262075e-01 11.3 -2.091034295385e-01 1.029421888862e-01 320 11.4 -9.021450024752e-02 -2.224505864150e-01 11.4 -2.182937072596e-01 8.074396544663e-02 321 11.5 -6.765394811167e-02 -2.283786206653e-01 11.5 -2.252321116912e-01 5.794254714301e-02 322 11.6 -4.461567409444e-02 -2.320004746198e-01 11.6 -2.298697259857e-01 3.476646630071e-02 323 11.7 -2.133128138851e-02 -2.333002408314e-01 11.7 -2.321805930194e-01 1.144601132753e-02 324 11.8 1.967173306734e-03 -2.322847342674e-01 11.8 -2.321617789782e-01 -1.178901201396e-02 325 11.9 2.504944169958e-02 -2.289832496619e-01 11.9 -2.298332139434e-01 -3.471149833402e-02 326 12.0 4.768931079683e-02 -2.234471044906e-01 12.0 -2.252373126344e-01 -5.709921826089e-02 327 12.1 6.966677360680e-02 -2.157489733769e-01 12.1 -2.184383805509e-01 -7.873693145139e-02 328 12.2 9.077012317050e-02 -2.059820216996e-01 12.2 -2.095218127752e-01 -9.941841713893e-02 329 12.3 1.107979503076e-01 -1.942588480406e-01 12.3 -1.985930946350e-01 -1.189484032993e-01 330 12.4 1.295610265175e-01 -1.807102468827e-01 12.4 -1.857766152672e-01 -1.371443765986e-01 331 12.5 1.468840547004e-01 -1.654838046148e-01 12.5 -1.712143068447e-01 -1.538382565375e-01 332 12.6 1.626072717455e-01 -1.487423434220e-01 12.6 -1.550641238173e-01 -1.688779186029e-01 333 12.7 1.765878885615e-01 -1.306622290042e-01 12.7 -1.374983779633e-01 -1.821285527798e-01 334 12.8 1.887013547807e-01 -1.114315592776e-01 12.8 -1.187019463284e-01 -1.934738454315e-01 335 12.9 1.988424371363e-01 -9.124825224995e-02 12.9 -9.887037024150e-02 -2.028169743237e-01 336 13.0 2.069261023771e-01 -7.031805212178e-02 13.0 -7.820786452788e-02 -2.100814084207e-01 337 13.1 2.128881975221e-01 -4.885247333423e-02 13.1 -5.692525678130e-02 -2.152115060050e-01 338 13.2 2.166859222586e-01 -2.706670276479e-02 13.2 -3.523787710230e-02 -2.181729066455e-01 339 13.3 2.182980903193e-01 -5.177480554678e-03 13.3 -1.336341905865e-02 -2.189527145477e-01 340 13.4 2.177251787312e-01 1.659901986400e-02 13.4 8.480207231244e-03 -2.175594728370e-01 341 13.5 2.149891658804e-01 3.804929208599e-02 13.5 3.007700904678e-02 -2.140229303400e-01 342 13.6 2.101331613693e-01 5.896455724873e-02 13.6 5.121501145646e-02 -2.083936044154e-01 343 13.7 2.032208326330e-01 7.914276510011e-02 13.7 7.168830401567e-02 -2.007421453278e-01 344 13.8 1.943356352156e-01 9.839051665835e-02 13.8 9.129901427859e-02 -1.911585095362e-01 345 13.9 1.835798554579e-01 1.165248903691e-01 13.9 1.098591894595e-01 -1.797509510696e-01 346 14.0 1.710734761105e-01 1.333751546988e-01 14.0 1.271925685822e-01 -1.666448418562e-01 347 14.1 1.569528770326e-01 1.487843512974e-01 14.1 1.431362286225e-01 -1.519813334678e-01 348 14.2 1.413693846571e-01 1.626107342002e-01 14.2 1.575420894708e-01 -1.359158741907e-01 349 14.3 1.244876852839e-01 1.747290520132e-01 14.3 1.702782639962e-01 -1.186165966565e-01 350 14.4 1.064841184903e-01 1.850316616146e-01 14.4 1.812302410821e-01 -1.002625924273e-01 351 14.5 8.754486801038e-02 1.934294635960e-01 14.5 1.903018911878e-01 -8.104209092875e-02 352 14.6 6.786406832339e-02 1.998526514474e-01 14.6 1.974162857774e-01 -6.115056094904e-02 353 14.7 4.764184590153e-02 2.042512683299e-01 14.7 2.025163238107e-01 -4.078875357125e-02 354 14.8 2.708231458588e-02 2.065955671798e-01 14.8 2.055651604029e-01 -2.016070586319e-02 355 14.9 6.391544890861e-03 2.068761718099e-01 14.9 2.065464347070e-01 5.282750764142e-04 356 15.0 -1.422447282677e-02 2.051040386135e-01 15.0 2.054642960389e-01 2.107362803687e-02 357 358 396 BESSEL FUNCTIONS OF INTEGER ORDER 397 359 360 Table 9.1 BESSEL FUNCTIONS--ORDER 0, 1 AND 2 361 362 x J0(x) J1(x) x Y0(x) Y1(x) 363 364 15.0 -1.422447282677e-02 2.051040386135e-01 15.0 2.054642960389e-01 2.107362803687e-02 365 15.1 -3.456185145556e-02 2.013102204085e-01 15.1 2.023432292287e-01 4.127353400948e-02 366 15.2 -5.442079684403e-02 1.955454358660e-01 15.2 1.972276821250e-01 6.093087362448e-02 367 15.3 -7.360754495112e-02 1.878794498324e-01 15.3 1.901815000860e-01 7.985512692091e-02 368 15.4 -9.193622786231e-02 1.784002716552e-01 15.4 1.812871741344e-01 9.786419732092e-02 369 15.5 -1.092306509000e-01 1.672131803518e-01 15.5 1.706449112294e-01 1.147861425133e-01 370 15.6 -1.253259640225e-01 1.544395870857e-01 15.6 1.583715367894e-01 1.304607959476e-01 371 15.7 -1.400702118290e-01 1.402157469423e-01 15.7 1.445992411700e-01 1.447412637878e-01 372 15.8 -1.533257477607e-01 1.246913333884e-01 15.8 1.294741832575e-01 1.574952834674e-01 373 15.9 -1.649704994857e-01 1.080278900631e-01 15.9 1.131549656518e-01 1.686064314007e-01 374 16.0 -1.748990739836e-01 9.039717566131e-02 16.0 9.581099708072e-02 1.779751689394e-01 375 16.1 -1.830236924653e-01 7.197941862246e-02 16.1 7.762075870139e-02 1.855197172915e-01 376 16.2 -1.892749469779e-01 5.296149912684e-02 16.2 5.876999178350e-02 1.911767538284e-01 377 16.3 -1.936023723284e-01 3.353507651579e-02 16.3 3.944982494208e-02 1.949019239839e-01 378 16.4 -1.959748287910e-01 1.389468068626e-02 16.4 1.985485957386e-02 1.966701647698e-01 379 16.5 -1.963806929369e-01 -5.764213735624e-03 16.5 1.812324575481e-04 1.964758377859e-01 380 16.6 -1.948278558056e-01 -2.524711156840e-02 16.6 -1.937532540317e-02 1.943326714661e-01 381 16.7 -1.913435295252e-01 -4.436240083665e-02 16.7 -3.862141468187e-02 1.902735141580e-01 382 16.8 -1.859738653476e-01 -6.292321765586e-02 16.8 -5.736785961672e-02 1.843499014676e-01 383 16.9 -1.787833878912e-01 -8.074925425014e-02 16.9 -7.543154755580e-02 1.766314430901e-01 384 17.0 -1.698542521512e-01 -9.766849275778e-02 17.0 -9.263719844232e-02 1.672050360772e-01 385 17.1 -1.592853315323e-01 -1.135188482914e-01 17.1 -1.088190473004e-01 1.561739131484e-01 386 17.2 -1.471911467660e-01 -1.281497056872e-01 17.2 -1.238224237217e-01 1.436565362143e-01 387 17.3 -1.337006470758e-01 -1.414233354920e-01 17.3 -1.375052134435e-01 1.297853467391e-01 388 17.4 -1.189558563364e-01 -1.532161759880e-01 17.4 -1.497391883399e-01 1.147053859034e-01 389 17.5 -1.031103982287e-01 -1.634199694258e-01 17.5 -1.604111925050e-01 9.857279873422e-02 390 17.6 -8.632791549801e-02 -1.719427421176e-01 17.6 -1.694241735779e-01 8.155323742984e-02 391 17.7 -6.878039938214e-02 -1.787096196184e-01 17.7 -1.766980500287e-01 6.382018001233e-02 392 17.8 -5.064644607112e-02 -1.836634698715e-01 17.8 -1.821704067768e-01 4.555318118908e-02 393 17.9 -3.210945768656e-02 -1.867653689135e-01 17.9 -1.857970132314e-01 2.693607288059e-02 394 18.0 -1.335580572199e-02 -1.879948854881e-01 18.0 -1.875521596114e-01 8.155132278224e-03 395 18.1 5.427024838490e-03 -1.873501827064e-01 18.1 -1.874288092000e-01 -1.060276447553e-02 396 18.2 2.405229240961e-02 -1.848479366856e-01 18.2 -1.854385660060e-01 -2.915198483965e-02 397 18.3 4.233583847140e-02 -1.805230738865e-01 18.3 -1.816114591091e-01 -4.730996064035e-02 398 18.4 6.009789204225e-02 -1.744283306318e-01 18.4 -1.759955467582e-01 -6.489896933080e-02 399 18.5 7.716482142255e-02 -1.666336400100e-01 18.5 -1.686563450403e-01 -8.174785849681e-02 400 18.6 9.337081627746e-02 -1.572253530294e-01 18.6 -1.596760876317e-01 -9.769369699823e-02 401 18.7 1.085594838932e-01 -1.463053024741e-01 18.7 -1.491528247654e-01 -1.125833369325e-01 402 18.8 1.225853443627e-01 -1.339897194140e-01 18.8 -1.371993710854e-01 -1.262748715543e-01 403 18.9 1.353152105223e-01 -1.204080137110e-01 18.9 -1.239421134900e-01 -1.386389753721e-01 404 19.0 1.466294396597e-01 -1.057014311424e-01 19.0 -1.095196913853e-01 -1.495601138627e-01 405 19.1 1.564230453336e-01 -9.002160090899e-02 19.1 -9.408156295987e-02 -1.589376115760e-01 406 19.2 1.646066590768e-01 -7.352898830426e-02 19.2 -7.778647214352e-02 -1.666865688437e-01 407 19.3 1.711073332708e-01 -5.639126817890e-02 19.3 -6.080083181602e-02 -1.727386188313e-01 408 19.4 1.758691780863e-01 -3.878163553798e-02 19.4 -4.329703957743e-02 -1.770425182737e-01 409 19.5 1.788538270402e-01 -2.087707014810e-02 19.5 -2.545174297615e-02 -1.795645668963e-01 410 19.6 1.800407274325e-01 -2.856572403404e-03 19.6 -7.444071505223e-03 -1.802888522206e-01 411 19.7 1.794272536588e-01 1.510061209776e-02 19.7 1.054614707836e-02 -1.792173181821e-01 412 19.8 1.770286431451e-01 3.281676178388e-02 19.8 2.834016848615e-02 -1.763696577181e-01 413 19.9 1.728777563926e-01 5.011742480738e-02 19.9 4.576209415939e-02 -1.717830312105e-01 414 20.0 1.670246643406e-01 6.683312417585e-02 20.0 6.264059680939e-02 -1.655116143625e-01 415 */ 416 void sphericalbesselJ0NTest(std::ostream&); // compare with Table 10.1 of Abramawitz & Stegun 417 /* 418 457 BESSEL FUNCTIONS OF FRACTIONAL ORDER 419 420 Table 10.1 SPHERICAL BESSEL FUNCTIONS--ORDER 0, 1 AND 2 421 422 x j0(x) j1(x) j2(x) 423 424 0.0 1.000000000000e+00 0.000000000000e+00 0.000000000000e+00 425 0.1 9.983341664683e-01 3.330001190256e-02 6.661906084456e-04 426 0.2 9.933466539753e-01 6.640038067032e-02 2.659056079527e-03 427 0.3 9.850673555378e-01 9.910288804064e-02 5.961524868620e-03 428 0.4 9.735458557716e-01 1.312121544219e-01 1.054530239227e-02 429 0.5 9.588510772084e-01 1.625370306361e-01 1.637110660799e-02 430 0.6 9.410707889917e-01 1.928919568034e-01 2.338899502534e-02 431 0.7 9.203109817681e-01 2.220982778338e-01 3.153878037661e-02 432 0.8 8.966951136244e-01 2.499855053465e-01 4.075053142515e-02 433 0.9 8.703632329194e-01 2.763925162764e-01 5.094515466858e-02 434 1.0 8.414709848079e-01 3.011686789398e-01 6.203505201137e-02 435 1.1 8.101885091468e-01 3.241748979283e-01 7.392484883964e-02 436 1.2 7.766992383060e-01 3.452845698578e-01 8.651218633845e-02 437 1.3 7.411986041671e-01 3.643844427250e-01 9.968857135213e-02 438 1.4 7.038926642775e-01 3.813753724123e-01 1.133402766060e-01 439 1.5 6.649966577360e-01 3.961729707122e-01 1.273492836884e-01 440 1.6 6.247335019009e-01 4.087081401264e-01 1.415942608360e-01 441 1.7 5.833322414426e-01 4.189274916107e-01 1.559515672821e-01 442 1.8 5.410264615990e-01 4.267936423845e-01 1.702962757085e-01 443 1.9 4.980526777302e-01 4.322853918914e-01 1.845032042036e-01 444 2.0 4.546487134128e-01 4.353977749800e-01 1.984479490571e-01 445 2.1 4.110520793566e-01 4.361419923602e-01 2.120079097294e-01 446 2.2 3.674983653725e-01 4.345452193763e-01 2.250632974133e-01 447 2.3 3.242196574681e-01 4.306502951078e-01 2.374981187594e-01 448 2.4 2.814429918963e-01 4.245152947656e-01 2.492011265607e-01 449 2.5 2.393888576416e-01 4.162129892754e-01 2.600667294889e-01 450 2.6 1.982697583929e-01 4.058301968315e-01 2.699958533357e-01 451 2.7 1.582888445310e-01 3.934670320549e-01 2.788967466410e-01 452 2.8 1.196386250557e-01 3.792360591873e-01 2.866857240735e-01 453 2.9 8.249976869448e-02 3.632613564980e-01 2.932878414758e-01 454 3.0 4.704000268662e-02 3.456774997624e-01 2.986374970757e-01 455 3.1 1.341311691396e-02 3.266284732862e-01 3.026789540082e-01 456 3.2 -1.824191982112e-02 3.062665174918e-01 3.053667799696e-01 457 3.3 -4.780172549795e-02 2.847509225488e-01 3.066662005423e-01 458 3.4 -7.515914765495e-02 2.622467779190e-01 3.065533634658e-01 459 3.5 -1.002237793399e-01 2.389236879860e-01 3.050155118993e-01 460 3.6 -1.229223453597e-01 2.149544641596e-01 3.020510654927e-01 461 3.7 -1.431989570023e-01 1.905138039752e-01 2.976696088741e-01 462 3.8 -1.610152344586e-01 1.657769677515e-01 2.918917879467e-01 463 3.9 -1.763502972267e-01 1.409184633265e-01 2.847491151701e-01 464 4.0 -1.892006238270e-01 1.161107492592e-01 2.762836857714e-01 465 4.1 -1.995797831864e-01 9.152296666996e-02 2.665478075791e-01 466 4.2 -2.075180410509e-01 6.731970959282e-02 2.556035479029e-01 467 4.3 -2.130618457557e-01 4.365984333123e-02 2.435222015682e-01 468 4.4 -2.162731986113e-01 2.069537985617e-02 2.303836848768e-01 469 4.5 -2.172289150367e-01 -1.429581245757e-03 2.162758608729e-01 470 4.6 -2.160197833986e-01 -2.257983836164e-02 2.012938018584e-01 471 4.7 -2.127496292690e-01 -4.262999272469e-02 1.855389956149e-01 472 4.8 -2.075342935075e-01 -6.146526603061e-02 1.691185022383e-01 473 4.9 -2.005005331886e-01 -7.898222502270e-02 1.521440688890e-01 474 5.0 -1.917848549326e-01 -9.508940807917e-02 1.347312100851e-01 475 476 458 BESSEL FUNCTIONS OF FRACTIONAL ORDER 477 478 Table 10.1 SPHERICAL BESSEL FUNCTIONS--ORDER 0, 1 AND 2 479 480 x j0(x) j1(x) j2(x) 481 5.0 -1.917848549326e-01 -9.508940807917e-02 1.347312100851e-01 482 5.1 -1.815322906525e-01 -1.097078496795e-01 1.169982614293e-01 483 5.2 -1.698951261000e-01 -1.227714995001e-01 9.906541484998e-02 484 5.3 -1.570315928724e-01 -1.342275337833e-01 8.105374356112e-02 485 5.4 -1.431045347326e-01 -1.440365575324e-01 6.308422499234e-02 486 5.5 -1.282800591946e-01 -1.521726969974e-01 4.527676992329e-02 487 5.6 -1.127261853343e-01 -1.586235828294e-01 2.774926596146e-02 488 5.7 -9.661149870134e-02 -1.633902251825e-01 1.061664334211e-02 489 5.8 -8.010382403686e-02 -1.664867829273e-01 -6.010029201389e-03 490 5.9 -6.336892624241e-02 -1.679402299977e-01 -2.202441104455e-02 491 6.0 -4.656924969982e-02 -1.677899227250e-01 -3.732571166269e-02 492 6.1 -2.986270561838e-02 -1.660870727969e-01 -5.181946133091e-02 493 6.2 -1.340151658347e-02 -1.628941312269e-01 -6.541822433277e-02 494 6.3 2.668873092753e-03 -1.582840894112e-01 -7.804224900286e-02 495 6.4 1.821081325789e-02 -1.523397039844e-01 -8.962004950059e-02 496 6.5 3.309538278274e-02 -1.451526527608e-01 -1.000889148262e-01 497 6.6 4.720323689597e-02 -1.368226295549e-01 -1.093953412391e-01 498 6.7 6.042536128606e-02 -1.274563861118e-01 -1.174953849182e-01 499 6.8 7.266372810862e-02 -1.171667297414e-01 -1.243549324063e-01 500 6.9 8.383184991133e-02 -1.060714855383e-01 -1.299498871019e-01 501 7.0 9.385522838840e-02 -9.429243227927e-02 -1.342662707938e-01 502 7.1 1.026716957924e-01 -8.195422121837e-02 -1.373002399692e-01 503 7.2 1.102316477568e-01 -6.918328705214e-02 -1.390580173619e-01 504 7.3 1.164981672094e-01 -5.610676029750e-02 -1.395557399344e-01 505 7.4 1.214470399745e-01 -4.285139021620e-02 -1.388192251973e-01 506 7.5 1.250666635700e-01 -2.954248723534e-02 -1.368836584641e-01 507 7.6 1.273578515831e-01 -1.630289355252e-02 -1.337932043012e-01 508 7.7 1.283335368671e-01 -3.251990281913e-03 -1.296005460679e-01 509 7.8 1.280183776121e-01 9.495250903778e-03 -1.243663580338e-01 510 7.9 1.264482711190e-01 2.182916414664e-02 -1.181587151139e-01 511 8.0 1.236697808279e-01 3.364622682957e-02 -1.110524457668e-01 512 8.1 1.197394828204e-01 4.484983167360e-02 -1.031284340524e-01 513 8.2 1.147232386195e-01 5.535098775650e-02 -9.447287724516e-02 514 8.3 1.086954016574e-01 6.506894537687e-02 -8.517650573804e-02 515 8.4 1.017379652486e-01 7.393174040063e-02 -7.533377224838e-02 516 8.5 9.393966030865e-02 8.187665446982e-02 -6.504201755459e-02 517 8.6 8.539501138071e-02 8.885058822101e-02 -5.440062014082e-02 518 8.7 7.620335977956e-02 9.481034544563e-02 -4.351013721210e-02 519 8.8 6.646786282861e-02 9.972282691935e-02 -3.247144456065e-02 520 8.9 5.629447825370e-02 1.035651334264e-01 -2.138488271671e-02 521 9.0 4.579094280464e-02 1.063245782988e-01 -1.034941670504e-02 522 9.1 3.506575410433e-02 1.079986105753e-01 5.381834479782e-04 523 9.2 2.422716457612e-02 1.085946506500e-01 1.118413454889e-02 524 9.3 1.338219607603e-02 1.081298410246e-01 2.149839780287e-02 525 9.4 2.635683558870e-03 1.066307154887e-01 3.139539585243e-02 526 9.5 -7.910644259136e-03 1.041327907302e-01 4.079468343711e-02 527 9.6 -1.815903971073e-02 1.006800850087e-01 4.962156627594e-02 528 9.7 -2.801655942381e-02 9.632456911418e-02 5.780766327355e-02 529 9.8 -3.739582951550e-02 9.112555536625e-02 6.529140768884e-02 530 9.9 -4.621574684599e-02 8.514903088681e-02 7.201848347836e-02 531 10.0 -5.440211108894e-02 7.846694179875e-02 7.794219362856e-02 532 533 458bis 534 535 10.0 -5.440211108894e-02 7.846694179875e-02 7.794219362856e-02 536 10.1 -6.188818305870e-02 7.115643535747e-02 8.302375791736e-02 537 10.2 -6.861516545035e-02 6.329906731146e-02 8.723253818901e-02 538 10.3 -7.453260288967e-02 5.497998201023e-02 9.054618988294e-02 539 10.4 -7.959869895054e-02 4.628707293045e-02 9.295073921894e-02 540 10.5 -8.378054856873e-02 3.731013137403e-02 9.444058610417e-02 541 10.6 -8.705428505781e-02 2.813999110510e-02 9.501843348378e-02 542 10.7 -8.940514170749e-02 1.886767663681e-02 9.469514450286e-02 543 10.8 -9.082742870986e-02 9.583562754213e-03 9.348952947492e-02 544 10.9 -9.132442690884e-02 3.765526690027e-04 9.142806525811e-02 545 11.0 -9.090820059552e-02 -8.666718053050e-03 8.854455021741e-02 546 11.1 -8.959933227063e-02 -1.746264496723e-02 8.487969849570e-02 547 11.2 -8.742658295994e-02 -2.593137917667e-02 8.048067782333e-02 548 11.3 -8.442648229222e-02 -3.399750668754e-02 7.540059556102e-02 549 11.4 -8.064285312848e-02 -4.159067714705e-02 6.969793808978e-02 550 11.5 -7.612627605986e-02 -4.864617694025e-02 6.343596903197e-02 551 11.6 -7.093349956627e-02 -5.510544218398e-02 5.668209210489e-02 552 11.7 -6.512680204436e-02 -6.091650775198e-02 4.950718467206e-02 553 11.8 -5.877331226925e-02 -6.603438908984e-02 4.198490826336e-02 554 11.9 -5.194429514597e-02 -7.042139422580e-02 3.419100248400e-02 555 12.0 -4.471440983337e-02 -7.404736404715e-02 2.620256882158e-02 556 12.1 -3.716094748220e-02 -7.688983958608e-02 1.809735089061e-02 557 12.2 -2.936305592105e-02 -7.893415573735e-02 9.953017624984e-03 558 12.3 -2.140095864763e-02 -8.017346150685e-02 1.846455841080e-03 559 12.4 -1.335517543938e-02 -8.060866755792e-02 -6.146921550438e-03 560 12.5 -5.305751788098e-03 -8.024832247733e-02 -1.395384560646e-02 561 12.6 2.668495811199e-03 -7.910841981665e-02 -2.150383386278e-02 562 12.7 1.049071192283e-02 -7.721213857413e-02 -2.872979977498e-02 563 12.8 1.808670508606e-02 -7.458952036026e-02 -3.556862392049e-02 564 12.9 2.538561543703e-02 -7.127708703278e-02 -4.196168218884e-02 565 13.0 3.232054129436e-02 -6.731740308891e-02 -4.785532662257e-02 566 13.1 3.882911941774e-02 -6.275858756066e-02 -5.320131504231e-02 567 13.2 4.485405414448e-02 -5.765378056788e-02 -5.795718609173e-02 568 13.3 5.034359114260e-02 -5.206057004219e-02 -6.208657686640e-02 569 13.4 5.525193208600e-02 -4.604038443787e-02 -6.555948084075e-02 570 13.5 5.953958715197e-02 -3.965785749318e-02 -6.835244437268e-02 571 13.6 6.317366285709e-02 -3.298017129383e-02 -7.044870064250e-02 572 13.7 6.612808338018e-02 -2.607638401975e-02 -7.183824046479e-02 573 13.8 6.838374416262e-02 -1.901674882511e-02 -7.251781999416e-02 574 13.9 6.992859722266e-02 -1.187203031059e-02 -7.249090592279e-02 575 14.0 7.075766826392e-02 -4.712824995996e-03 -7.176755933449e-02 576 14.1 7.087300629418e-02 2.391107908037e-03 -7.036425993077e-02 577 14.2 7.028356709270e-02 9.371499296954e-03 -6.830367287504e-02 578 14.3 6.900503246676e-02 1.616220062702e-02 -6.561436100654e-02 579 14.4 6.705956781592e-02 2.269978639095e-02 -6.233044565114e-02 580 14.5 6.447552107067e-02 2.892412330064e-02 -5.849121969812e-02 581 14.6 6.128706658497e-02 3.477890123269e-02 -5.414071701661e-02 582 14.7 5.753379803694e-02 4.021212140313e-02 -4.932724264854e-02 583 14.8 5.326027482266e-02 4.517653768766e-02 -4.410286853462e-02 584 14.9 4.851552681289e-02 4.963004748537e-02 -3.852289980241e-02 585 15.0 4.335252267714e-02 5.353602903573e-02 -3.264531687000e-02 586 587 458ter 588 589 15.0 4.335252267714e-02 5.353602903573e-02 -3.264531687000e-02 590 15.1 3.782760726225e-02 5.686362263278e-02 -2.653019879217e-02 591 15.2 3.199991374038e-02 5.958795373365e-02 -2.023913339822e-02 592 15.3 2.593075641377e-02 6.169029652471e-02 -1.383461984030e-02 593 15.4 1.968301017829e-02 6.315817708217e-02 -7.379469188261e-03 594 15.5 1.332048270567e-02 6.398541584029e-02 -9.362086720615e-04 595 15.6 6.907285403813e-03 6.417210965305e-02 5.433504914082e-03 596 15.7 5.072091583423e-04 6.372455429985e-02 1.166945726838e-02 597 15.8 -5.816889254914e-03 6.265510883661e-02 1.771342890743e-02 598 15.9 -1.200368436315e-02 6.098200372599e-02 2.350972280202e-02 599 16.0 -1.799395729156e-02 5.872909518949e-02 2.900566263959e-02 600 16.1 -2.373114392447e-02 5.592556870489e-02 3.415205734774e-02 601 16.2 -2.916185101225e-02 5.260559502161e-02 3.890362786810e-02 602 16.3 -3.423633566176e-02 4.880794247907e-02 4.321939255975e-02 603 16.4 -3.890894404561e-02 4.457554978637e-02 4.706300803092e-02 604 16.5 -4.313850559813e-02 3.995506375179e-02 5.040306264391e-02 605 16.6 -4.688867942978e-02 3.499634673594e-02 5.321332040615e-02 606 16.7 -5.012825018082e-02 2.975195883913e-02 5.547291344534e-02 607 16.8 -5.283137104652e-02 2.427662002172e-02 5.716648176468e-02 608 16.9 -5.497775223280e-02 1.862665749292e-02 5.828425948007e-02 609 17.0 -5.655279363997e-02 1.285944378892e-02 5.882210724978e-02 610 17.1 -5.754766111589e-02 7.032830994576e-03 5.878149111493e-02 611 17.2 -5.795930616521e-02 1.204586544573e-03 5.816940846949e-02 612 17.3 -5.779042954179e-02 -4.568164029793e-03 5.699826236900e-02 613 17.4 -5.704938968222e-02 -1.022948216153e-02 5.528568586127e-02 614 17.5 -5.575005745532e-02 -1.572514403810e-02 5.305431847736e-02 615 17.6 -5.391161919989e-02 -2.100314625677e-02 5.033153745158e-02 616 17.7 -5.155833049668e-02 -2.601418473986e-02 4.714914664246e-02 617 17.8 -4.871922356661e-02 -3.071210260421e-02 4.354302649848e-02 618 17.9 -4.542777160120e-02 -3.505430301694e-02 3.955274874920e-02 619 18.0 -4.172151370954e-02 -3.900212344187e-02 3.522115980256e-02 620 18.1 -3.764164450583e-02 -4.252116808369e-02 3.059393708864e-02 621 18.2 -3.323257265969e-02 -4.558159576657e-02 2.571912280805e-02 622 18.3 -2.854145298507e-02 -4.815836093530e-02 2.064663971699e-02 623 18.4 -2.361769685173e-02 -5.023140593677e-02 1.542779370987e-02 624 18.5 -1.851246586322e-02 -5.178580322224e-02 1.011476804340e-02 625 18.6 -1.327815385681e-02 -5.281184660193e-02 4.760114082303e-03 626 18.7 -7.967862343005e-03 -5.330509117846e-02 -5.837565626296e-04 627 18.8 -2.634874514807e-03 -5.326634207989e-02 -5.865073689432e-03 628 18.9 2.667867079726e-03 -5.270159260126e-02 -1.103319923866e-02 629 19.0 7.888274192787e-03 -5.162191284178e-02 -1.603910253623e-02 630 19.1 1.297561298340e-02 -5.004329038868e-02 -2.083581566225e-02 631 19.2 1.788098587604e-02 -4.798642504275e-02 -2.537886478897e-02 632 19.3 2.255779069290e-02 -4.547648000282e-02 -2.962667359489e-02 633 19.4 2.696215284318e-02 -4.254279232074e-02 -3.354093516082e-02 634 19.5 3.105332665229e-02 -3.921854580316e-02 -3.708694908354e-02 635 19.6 3.479406224837e-02 -3.554040986720e-02 -4.023392090152e-02 636 19.7 3.815093478945e-02 -3.154814815147e-02 -4.295522130998e-02 637 19.8 4.109463320743e-02 -2.728420093975e-02 -4.522860304679e-02 638 19.9 4.360020606240e-02 -2.279324566878e-02 -4.703637375116e-02 639 20.0 4.564726253638e-02 -1.812173996385e-02 -4.836552353096e-02 640 */ 641 642 /* 643 complex Exponential Integral E1(z) = \f$\int_z^{\infty} t^{-1} exp(-t) dt\f$ 644 = \f$-\gamma - log(z) + \sum_{n>0} (-z)^n / n n!\f$ 645 */ 646 complex_t e1z(const complex_t&); //!< return E1(z) 647 648 complex_t eInz(const complex_t& z); //!< return \f$E1(z) + gamma + log(z) = \sum_{n>0} (-z)^n / n n!\f$ 649 // http://www.math.sfu.ca/~cbm/aands/page_228.htm (see footnote) 650 651 complex_t expzE1z(const complex_t&); //!< return exp(z)*E1(z) 652 653 complex_t zexpzE1z(const complex_t&); //!< return z*exp(z)*E1(z) 654 655 complex_t ascendingSeriesOfE1(const complex_t& z); //!< ascending series in E1 formula (used for 'small' z) 656 // http://www.math.sfu.ca/~cbm/aands/page_229.htm (Formula 5.1.11) 657 658 complex_t continuedFractionOfE1(const complex_t& z); //!< continued fraction in E1 formula (used for 'large' z) 659 // http://www.math.sfu.ca/~cbm/aands/page_229.htm (Formula 5.1.22) 660 661 //! For tests 662 void e1Test(std::ostream&); 663 /* 664 . EXPONENTIAL INTEGRAl AND RELATED FUNCTIONS 249 665 . 666 . EXPONENTIAL INTEGRAL FOR COMPLEX ARGUMENTS Table 5.6 667 . Z*EXP(Z)*E1(Z) 668 669 y / x -19 -18 -17 -16 -15 670 0 1.059305e+00+i*3.344324e-07 1.063087e+00+i*8.612351e-07 1.067394e+00+i*2.211020e-06 1.072345e+00+i*5.656635e-06 1.078103e+00+i*1.441531e-05 671 1 1.059090e+00+i*3.539294e-03 1.062827e+00+i*4.010195e-03 1.067073e+00+i*4.584363e-03 1.071942e+00+i*5.295878e-03 1.077584e+00+i*6.195111e-03 672 2 1.058456e+00+i*6.999918e-03 1.062061e+00+i*7.917903e-03 1.066135e+00+i*9.031762e-03 1.070775e+00+i*1.040265e-02 1.076102e+00+i*1.211804e-02 673 3 1.057431e+00+i*1.031039e-02 1.060829e+00+i*1.163254e-02 1.064636e+00+i*1.322636e-02 1.068925e+00+i*1.517151e-02 1.073783e+00+i*1.757854e-02 674 4 1.056058e+00+i*1.341001e-02 1.059190e+00+i*1.507895e-02 1.062657e+00+i*1.707464e-02 1.066508e+00+i*1.948587e-02 1.070793e+00+i*2.243201e-02 675 5 1.054391e+00+i*1.625223e-02 1.057215e+00+i*1.820220e-02 1.060297e+00+i*2.051241e-02 1.063659e+00+i*2.327231e-02 1.067318e+00+i*2.659819e-02 676 6 1.052490e+00+i*1.880591e-02 1.054981e+00+i*2.096854e-02 1.057655e+00+i*2.350489e-02 1.060510e+00+i*2.649860e-02 1.063538e+00+i*3.005443e-02 677 7 1.050413e+00+i*2.105479e-02 1.052565e+00+i*2.336398e-02 1.054829e+00+i*2.604354e-02 1.057187e+00+i*2.916706e-02 1.059610e+00+i*3.282308e-02 678 8 1.048217e+00+i*2.299577e-02 1.050037e+00+i*2.539119e-02 1.051905e+00+i*2.814072e-02 1.053795e+00+i*3.130579e-02 1.055664e+00+i*3.495705e-02 679 9 1.045956e+00+i*2.463653e-02 1.047458e+00+i*2.706578e-02 1.048958e+00+i*2.982395e-02 1.050421e+00+i*3.295981e-02 1.051797e+00+i*3.652667e-02 680 10 1.043672e+00+i*2.599276e-02 1.044880e+00+i*2.841236e-02 1.046045e+00+i*3.113026e-02 1.047129e+00+i*3.418330e-02 1.048081e+00+i*3.760927e-02 681 11 1.041402e+00+i*2.708569e-02 1.042345e+00+i*2.946107e-02 1.043212e+00+i*3.210162e-02 1.043967e+00+i*3.503363e-02 1.044559e+00+i*3.828184e-02 682 12 1.039177e+00+i*2.793973e-02 1.039882e+00+i*3.024464e-02 1.040490e+00+i*3.278131e-02 1.040965e+00+i*3.556711e-02 1.041259e+00+i*3.861632e-02 683 13 1.037018e+00+i*2.858068e-02 1.037515e+00+i*3.079624e-02 1.037901e+00+i*3.321138e-02 1.038140e+00+i*3.583627e-02 1.038192e+00+i*3.867717e-02 684 14 1.034942e+00+i*2.903433e-02 1.035259e+00+i*3.114789e-02 1.035456e+00+i*3.343107e-02 1.035501e+00+i*3.588844e-02 1.035359e+00+i*3.852032e-02 685 15 1.032959e+00+i*2.932546e-02 1.033123e+00+i*3.132946e-02 1.033162e+00+i*3.347589e-02 1.033049e+00+i*3.576510e-02 1.032754e+00+i*3.819324e-02 686 16 1.031076e+00+i*2.947720e-02 1.031110e+00+i*3.136810e-02 1.031019e+00+i*3.337723e-02 1.030780e+00+i*3.550190e-02 1.030365e+00+i*3.773545e-02 687 17 1.029296e+00+i*2.951065e-02 1.029222e+00+i*3.128800e-02 1.029025e+00+i*3.316238e-02 1.028685e+00+i*3.512899e-02 1.028180e+00+i*3.717942e-02 688 18 1.027620e+00+i*2.944477e-02 1.027456e+00+i*3.111039e-02 1.027174e+00+i*3.285471e-02 1.026756e+00+i*3.467154e-02 1.026183e+00+i*3.655150e-02 689 19 1.026046e+00+i*2.929630e-02 1.025809e+00+i*3.085365e-02 1.025459e+00+i*3.247400e-02 1.024981e+00+i*3.415034e-02 1.024360e+00+i*3.587287e-02 690 20 1.024570e+00+i*2.907988e-02 1.024275e+00+i*3.053353e-02 1.023872e+00+i*3.203684e-02 1.023349e+00+i*3.358240e-02 1.022695e+00+i*3.516042e-02 691 y / x -14 -13 -12 -11 -10 692 0 1.084892e+00+i*3.657254e-05 1.093027e+00+i*9.231345e-05 1.102975e+00+i*2.316313e-04 1.115431e+00+i*5.771693e-04 1.131470e+00+i*1.426281e-03 693 1 1.084200e+00+i*7.359438e-03 1.092067e+00+i*8.912820e-03 1.101566e+00+i*1.106249e-02 1.113230e+00+i*1.416878e-02 1.127796e+00+i*1.887846e-02 694 2 1.082276e+00+i*1.430627e-02 1.089498e+00+i*1.716091e-02 1.098025e+00+i*2.098137e-02 1.108171e+00+i*2.624109e-02 1.120286e+00+i*3.369962e-02 695 3 1.079313e+00+i*2.060362e-02 1.085635e+00+i*2.447099e-02 1.092873e+00+i*2.950686e-02 1.101137e+00+i*3.618934e-02 1.110462e+00+i*4.521747e-02 696 4 1.075560e+00+i*2.607524e-02 1.080853e+00+i*3.063731e-02 1.086686e+00+i*3.642205e-02 1.093013e+00+i*4.384283e-02 1.099666e+00+i*5.345062e-02 697 5 1.071279e+00+i*3.064152e-02 1.075522e+00+i*3.559918e-02 1.079985e+00+i*4.172443e-02 1.084526e+00+i*4.933578e-02 1.088877e+00+i*5.881666e-02 698 6 1.066708e+00+i*3.430290e-02 1.069960e+00+i*3.940455e-02 1.073184e+00+i*4.555214e-02 1.076197e+00+i*5.296734e-02 1.078701e+00+i*6.188566e-02 699 7 1.062046e+00+i*3.711662e-02 1.064412e+00+i*4.216911e-02 1.066578e+00+i*4.811513e-02 1.068350e+00+i*5.509292e-02 1.069450e+00+i*6.322454e-02 700 8 1.057448e+00+i*3.917375e-02 1.059054e+00+i*4.404105e-02 1.060352e+00+i*4.964407e-02 1.061159e+00+i*5.605661e-02 1.061235e+00+i*6.332208e-02 701 9 1.053020e+00+i*4.058040e-02 1.053997e+00+i*4.517557e-02 1.054606e+00+i*5.035874e-02 1.054687e+00+i*5.615779e-02 1.054046e+00+i*6.256630e-02 702 10 1.048834e+00+i*4.144439e-02 1.049303e+00+i*4.571913e-02 1.049380e+00+i*5.045196e-02 1.048933e+00+i*5.564043e-02 1.047807e+00+i*6.124942e-02 703 11 1.044928e+00+i*4.186683e-02 1.044997e+00+i*4.580119e-02 1.044674e+00+i*5.008416e-02 1.043853e+00+i*5.469478e-02 1.042416e+00+i*5.958359e-02 704 12 1.041320e+00+i*4.193773e-02 1.041080e+00+i*4.553131e-02 1.040464e+00+i*4.938399e-02 1.039389e+00+i*5.346469e-02 1.037766e+00+i*5.771874e-02 705 13 1.038010e+00+i*4.173433e-02 1.037537e+00+i*4.499932e-02 1.036713e+00+i*4.845189e-02 1.035473e+00+i*5.205641e-02 1.033752e+00+i*5.575846e-02 706 14 1.034989e+00+i*4.132112e-02 1.034344e+00+i*4.427721e-02 1.033378e+00+i*4.736467e-02 1.032040e+00+i*5.054696e-02 1.030282e+00+i*5.377284e-02 707 15 1.032241e+00+i*4.075091e-02 1.031474e+00+i*4.342161e-02 1.030414e+00+i*4.618016e-02 1.029026e+00+i*4.899121e-02 1.027274e+00+i*5.180814e-02 708 16 1.029747e+00+i*4.006622e-02 1.028895e+00+i*4.247646e-02 1.027781e+00+i*4.494124e-02 1.026377e+00+i*4.742760e-02 1.024658e+00+i*4.989400e-02 709 17 1.027486e+00+i*3.930087e-02 1.026579e+00+i*4.147542e-02 1.025438e+00+i*4.367931e-02 1.024043e+00+i*4.588251e-02 1.022375e+00+i*4.804852e-02 710 18 1.025437e+00+i*3.848144e-02 1.024499e+00+i*4.044397e-02 1.023352e+00+i*4.241699e-02 1.021981e+00+i*4.437356e-02 1.020375e+00+i*4.628190e-02 711 19 1.023580e+00+i*3.762865e-02 1.022627e+00+i*3.940119e-02 1.021489e+00+i*4.117033e-02 1.020155e+00+i*4.291210e-02 1.018617e+00+i*4.459902e-02 712 20 1.021896e+00+i*3.675846e-02 1.020942e+00+i*3.836120e-02 1.019824e+00+i*3.995041e-02 1.018533e+00+i*4.150499e-02 1.017066e+00+i*4.300118e-02 713 y / x -9 -8 -7 -6 -5 714 0 1.152759e+00+i*3.489330e-03 1.181848e+00+i*8.431095e-03 1.222408e+00+i*2.005333e-02 1.278884e+00+i*4.672338e-02 1.353831e+00+i*1.058394e-01 715 1 1.146232e+00+i*2.637602e-02 1.169677e+00+i*3.884079e-02 1.199049e+00+i*6.021873e-02 1.233798e+00+i*9.733053e-02 1.268723e+00+i*1.608255e-01 716 2 1.134679e+00+i*4.457874e-02 1.151385e+00+i*6.081350e-02 1.169639e+00+i*8.533460e-02 1.186778e+00+i*1.221620e-01 1.196351e+00+i*1.756458e-01 717 3 1.120694e+00+i*5.759502e-02 1.131255e+00+i*7.470077e-02 1.140733e+00+i*9.825884e-02 1.146266e+00+i*1.300051e-01 1.142853e+00+i*1.706715e-01 718 4 1.106249e+00+i*6.594813e-02 1.111968e+00+i*8.215578e-02 1.115404e+00+i*1.028610e-01 1.114273e+00+i*1.284402e-01 1.105376e+00+i*1.581339e-01 719 5 1.092564e+00+i*7.059178e-02 1.094817e+00+i*8.505484e-02 1.094475e+00+i*1.024110e-01 1.089952e+00+i*1.223967e-01 1.079407e+00+i*1.438790e-01 720 6 1.080246e+00+i*7.251962e-02 1.080188e+00+i*8.498655e-02 1.077672e+00+i*9.918770e-02 1.071684e+00+i*1.146377e-01 1.061236e+00+i*1.302800e-01 721 7 1.069494e+00+i*7.258002e-02 1.067987e+00+i*8.312034e-02 1.064339e+00+i*9.461831e-02 1.057935e+00+i*1.065680e-01 1.048279e+00+i*1.181160e-01 722 8 1.060276e+00+i*7.142463e-02 1.057920e+00+i*8.024960e-02 1.053778e+00+i*8.953705e-02 1.047493e+00+i*9.883986e-02 1.038838e+00+i*1.075084e-01 723 9 1.052450e+00+i*6.952255e-02 1.049645e+00+i*7.688449e-02 1.045382e+00+i*8.440509e-02 1.039464e+00+i*9.171665e-02 1.031806e+00+i*9.833700e-02 724 10 1.045832e+00+i*6.719704e-02 1.042834e+00+i*7.334001e-02 1.038659e+00+i*7.946209e-02 1.033205e+00+i*8.527126e-02 1.026459e+00+i*9.041253e-02 725 11 1.040241e+00+i*6.466378e-02 1.037209e+00+i*6.980337e-02 1.033231e+00+i*7.482090e-02 1.028260e+00+i*7.948827e-02 1.022317e+00+i*8.354394e-02 726 12 1.035508e+00+i*6.206271e-02 1.032539e+00+i*6.638071e-02 1.028808e+00+i*7.052392e-02 1.024300e+00+i*7.431533e-02 1.019052e+00+i*7.756114e-02 727 13 1.031490e+00+i*5.948197e-02 1.028638e+00+i*6.312794e-02 1.025171e+00+i*6.657575e-02 1.021090e+00+i*6.968811e-02 1.016439e+00+i*7.232025e-02 728 14 1.028065e+00+i*5.697507e-02 1.025359e+00+i*6.007042e-02 1.022152e+00+i*6.296166e-02 1.018458e+00+i*6.554199e-02 1.014319e+00+i*6.770224e-02 729 15 1.025132e+00+i*5.457264e-02 1.022583e+00+i*5.721535e-02 1.019626e+00+i*5.965799e-02 1.016277e+00+i*6.181710e-02 1.012577e+00+i*6.360948e-02 730 16 1.022608e+00+i*5.229047e-02 1.020218e+00+i*5.455946e-02 1.017494e+00+i*5.663786e-02 1.014452e+00+i*5.846006e-02 1.011130e+00+i*5.996204e-02 731 17 1.020426e+00+i*5.013480e-02 1.018192e+00+i*5.209370e-02 1.015681e+00+i*5.387418e-02 1.012912e+00+i*5.542432e-02 1.009915e+00+i*5.669434e-02 732 18 1.018530e+00+i*4.810593e-02 1.016444e+00+i*4.980617e-02 1.014129e+00+i*5.134120e-02 1.011600e+00+i*5.266963e-02 1.008887e+00+i*5.375239e-02 733 19 1.016874e+00+i*4.620052e-02 1.014929e+00+i*4.768385e-02 1.012790e+00+i*4.901522e-02 1.010476e+00+i*5.016141e-02 1.008009e+00+i*5.109149e-02 734 20 1.015422e+00+i*4.441312e-02 1.013607e+00+i*4.571355e-02 1.011629e+00+i*4.687480e-02 1.009505e+00+i*4.787001e-02 1.007254e+00+i*4.867446e-02 735 736 737 250 EXPONENTIAL INTEGRAl AND RELATED FUNCTIONS 738 . 739 . EXPONENTIAL INTEGRAL FOR COMPLEX ARGUMENTS 740 . Z*EXP(Z)*E1(Z) 741 742 y / x -4 -3 -2 -1 0 743 0 1.438208e+00+i*2.301611e-01 1.483729e+00+i*4.692321e-01 1.340965e+00+i*8.503367e-01 6.971749e-01+i*1.155727e+00 0.000000e+00+i*0.000000e+00 744 1 1.287244e+00+i*2.637045e-01 1.251069e+00+i*4.104125e-01 1.098808e+00+i*5.619160e-01 8.134864e-01+i*5.786973e-01 6.214496e-01+i*3.433780e-01 745 2 1.185758e+00+i*2.473563e-01 1.136171e+00+i*3.284389e-01 1.032990e+00+i*3.884283e-01 8.964185e-01+i*3.788375e-01 7.980420e-01+i*2.890906e-01 746 3 1.123282e+00+i*2.178354e-01 1.080316e+00+i*2.628143e-01 1.013205e+00+i*2.893658e-01 9.362826e-01+i*2.809059e-01 8.758731e-01+i*2.376646e-01 747 4 1.085153e+00+i*1.890034e-01 1.051401e+00+i*2.151178e-01 1.006122e+00+i*2.283987e-01 9.574456e-01+i*2.226117e-01 9.167703e-01+i*1.987126e-01 748 5 1.061263e+00+i*1.644661e-01 1.035185e+00+i*1.804866e-01 1.003172e+00+i*1.878565e-01 9.698086e-01+i*1.839629e-01 9.407139e-01+i*1.694811e-01 749 6 1.045719e+00+i*1.443912e-01 1.025396e+00+i*1.547459e-01 1.001788e+00+i*1.591888e-01 9.775822e-01+i*1.565106e-01 9.558333e-01+i*1.471289e-01 750 7 1.035205e+00+i*1.280734e-01 1.019109e+00+i*1.350792e-01 1.001077e+00+i*1.379386e-01 9.827555e-01+i*1.360424e-01 9.659370e-01+i*1.296464e-01 751 8 1.027834e+00+i*1.147320e-01 1.014861e+00+i*1.196603e-01 1.000684e+00+i*1.215989e-01 9.863563e-01+i*1.202182e-01 9.729943e-01+i*1.156778e-01 752 9 1.022501e+00+i*1.037111e-01 1.011869e+00+i*1.072943e-01 1.000453e+00+i*1.086649e-01 9.889550e-01+i*1.076338e-01 9.781027e-01+i*1.043030e-01 753 10 1.018534e+00+i*9.450185e-02 1.009688e+00+i*9.718095e-02 1.000312e+00+i*9.818387e-02 9.908873e-01+i*9.739625e-02 9.819104e-01+i*9.488539e-02 754 11 1.015513e+00+i*8.671835e-02 1.008052e+00+i*8.876989e-02 1.000221e+00+i*8.952494e-02 9.923606e-01+i*8.891135e-02 9.848192e-01+i*8.697465e-02 755 12 1.013163e+00+i*8.006921e-02 1.006795e+00+i*8.167276e-02 1.000160e+00+i*8.225494e-02 9.935080e-01+i*8.176857e-02 9.870883e-01+i*8.024517e-02 756 13 1.011303e+00+i*7.433307e-02 1.005809e+00+i*7.560891e-02 1.000119e+00+i*7.606702e-02 9.944180e-01+i*7.567555e-02 9.888906e-01+i*7.445680e-02 757 14 1.009806e+00+i*6.934018e-02 1.005022e+00+i*7.037112e-02 1.000090e+00+i*7.073795e-02 9.951514e-01+i*7.041856e-02 9.903445e-01+i*6.942905e-02 758 15 1.008585e+00+i*6.495894e-02 1.004384e+00+i*6.580341e-02 1.000070e+00+i*6.610163e-02 9.957505e-01+i*6.583787e-02 9.915336e-01+i*6.502399e-02 759 16 1.007577e+00+i*6.108619e-02 1.003859e+00+i*6.178629e-02 1.000055e+00+i*6.203197e-02 9.962461e-01+i*6.181179e-02 9.925179e-01+i*6.113463e-02 760 17 1.006735e+00+i*5.764015e-02 1.003423e+00+i*5.822682e-02 1.000043e+00+i*5.843158e-02 9.966606e-01+i*5.824600e-02 9.933416e-01+i*5.767679e-02 761 18 1.006025e+00+i*5.455533e-02 1.003057e+00+i*5.505167e-02 1.000035e+00+i*5.522412e-02 9.970105e-01+i*5.506631e-02 9.940375e-01+i*5.458342e-02 762 19 1.005420e+00+i*5.177870e-02 1.002747e+00+i*5.220227e-02 1.000028e+00+i*5.234885e-02 9.973086e-01+i*5.221359e-02 9.946306e-01+i*5.180052e-02 763 20 1.004902e+00+i*4.926699e-02 1.002481e+00+i*4.963127e-02 1.000023e+00+i*4.975691e-02 9.975645e-01+i*4.964014e-02 9.951401e-01+i*4.928413e-02 764 y / x 1 2 3 4 5 765 0 5.963474e-01+i*0.000000e+00 7.226572e-01+i*0.000000e+00 7.862512e-01+i*0.000000e+00 8.253826e-01+i*0.000000e+00 8.521109e-01+i*0.000000e+00 766 1 6.733212e-01+i*1.478639e-01 7.470123e-01+i*7.566082e-02 7.970360e-01+i*4.568600e-02 8.311262e-01+i*3.061932e-02 8.555441e-01+i*2.198492e-02 767 2 7.775143e-01+i*1.865704e-01 7.969645e-01+i*1.182282e-01 8.230547e-01+i*7.875296e-02 8.460965e-01+i*5.549408e-02 8.648796e-01+i*4.099909e-02 768 3 8.474683e-01+i*1.812256e-01 8.443610e-01+i*1.322519e-01 8.531756e-01+i*9.665867e-02 8.655213e-01+i*7.217976e-02 8.778601e-01+i*5.534102e-02 769 4 8.914598e-01+i*1.652071e-01 8.810360e-01+i*1.316856e-01 8.805838e-01+i*1.034034e-01 8.853082e-01+i*8.140837e-02 8.921432e-01+i*6.482487e-02 770 5 9.198259e-01+i*1.482714e-01 9.078731e-01+i*1.251357e-01 9.031515e-01+i*1.035771e-01 9.032310e-01+i*8.518728e-02 9.060575e-01+i*7.020916e-02 771 6 9.388274e-01+i*1.329855e-01 9.273841e-01+i*1.166564e-01 9.210060e-01+i*1.003572e-01 9.185274e-01+i*8.545965e-02 9.187081e-01+i*7.254384e-02 772 7 9.520321e-01+i*1.198069e-01 9.417224e-01+i*1.079899e-01 9.349580e-01+i*9.559814e-02 9.312090e-01+i*8.366602e-02 9.297649e-01+i*7.279171e-02 773 8 9.615119e-01+i*1.085892e-01 9.524347e-01+i*9.982963e-02 9.458679e-01+i*9.030246e-02 9.415938e-01+i*8.075486e-02 9.392208e-01+i*7.169964e-02 774 9 9.685124e-01+i*9.904509e-02 9.605815e-01+i*9.240757e-02 9.544569e-01+i*8.498561e-02 9.500717e-01+i*7.731270e-02 9.472192e-01+i*6.979846e-02 775 10 9.738096e-01+i*9.088790e-02 9.668852e-01+i*8.575822e-02 9.612831e-01+i*7.989763e-02 9.570069e-01+i*7.368796e-02 9.539552e-01+i*6.744669e-02 776 11 9.779038e-01+i*8.387054e-02 9.718424e-01+i*7.983585e-02 9.667655e-01+i*7.514726e-02 9.627077e-01+i*7.008034e-02 9.596264e-01+i*6.487834e-02 777 12 9.811269e-01+i*7.779040e-02 9.757989e-01+i*7.456687e-02 9.712156e-01+i*7.076943e-02 9.674232e-01+i*6.659917e-02 9.644116e-01+i*6.224206e-02 778 13 9.837057e-01+i*7.248439e-02 9.789995e-01+i*6.987272e-02 9.748652e-01+i*6.676167e-02 9.713510e-01+i*6.329980e-02 9.684639e-01+i*5.962967e-02 779 14 9.857986e-01+i*6.782190e-02 9.816205e-01+i*6.567935e-02 9.778877e-01+i*6.310360e-02 9.746461e-01+i*6.020579e-02 9.719112e-01+i*5.709578e-02 780 15 9.875187e-01+i*6.369816e-02 9.837908e-01+i*6.192064e-02 9.804140e-01+i*5.976732e-02 9.774301e-01+i*5.732235e-02 9.748584e-01+i*5.467101e-02 781 16 9.889486e-01+i*6.002876e-02 9.856060e-01+i*5.853909e-02 9.825436e-01+i*5.672283e-02 9.797985e-01+i*5.464444e-02 9.773910e-01+i*5.237061e-02 782 17 9.901491e-01+i*5.674524e-02 9.871381e-01+i*5.548534e-02 9.843531e-01+i*5.394080e-02 9.818266e-01+i*5.216158e-02 9.795786e-01+i*5.020021e-02 783 18 9.911665e-01+i*5.379171e-02 9.884422e-01+i*5.271726e-02 9.859021e-01+i*5.139390e-02 9.835744e-01+i*4.986072e-02 9.814779e-01+i*4.815948e-02 784 19 9.920356e-01+i*5.112223e-02 9.895607e-01+i*5.019898e-02 9.872371e-01+i*4.905726e-02 9.850894e-01+i*4.772796e-02 9.831349e-01+i*4.624451e-02 785 20 9.927838e-01+i*4.869875e-02 9.905266e-01+i*4.789993e-02 9.883949e-01+i*4.690860e-02 9.864099e-01+i*4.574944e-02 9.845874e-01+i*4.444941e-02 786 y / x 6 7 8 9 10 787 0 8.716058e-01+i*0.000000e+00 8.864877e-01+i*0.000000e+00 8.982371e-01+i*0.000000e+00 9.077576e-01+i*0.000000e+00 9.156333e-01+i*0.000000e+00 788 1 8.738266e-01+i*1.657045e-02 8.880094e-01+i*1.294731e-02 8.993265e-01+i*1.040125e-02 9.085650e-01+i*8.542458e-03 9.162487e-01+i*7.143197e-03 789 2 8.800226e-01+i*3.145396e-02 8.923265e-01+i*2.486638e-02 9.024530e-01+i*2.014019e-02 9.109012e-01+i*1.663901e-02 9.180401e-01+i*1.397515e-02 790 3 8.890290e-01+i*4.351690e-02 8.987929e-01+i*3.499514e-02 9.072361e-01+i*2.869295e-02 9.145310e-01+i*2.392055e-02 9.208560e-01+i*2.022951e-02 791 4 8.994841e-01+i*5.237988e-02 9.065911e-01+i*4.296734e-02 9.131673e-01+i*3.575449e-02 9.191270e-01+i*3.014473e-02 9.244789e-01+i*2.571647e-02 792 5 9.102418e-01+i*5.825896e-02 9.149522e-01+i*4.877954e-02 9.197292e-01+i*4.124174e-02 9.243364e-01+i*3.520748e-02 9.286641e-01+i*3.033439e-02 793 6 9.205338e-01+i*6.167553e-02 9.232825e-01+i*5.266669e-02 9.264810e-01+i*4.524182e-02 9.298362e-01+i*3.912266e-02 9.331750e-01+i*3.406280e-02 794 7 9.299448e-01+i*6.322038e-02 9.311926e-01+i*5.497073e-02 9.330955e-01+i*4.794215e-02 9.353646e-01+i*4.198566e-02 9.378069e-01+i*3.694383e-02 795 8 9.383126e-01+i*6.342479e-02 9.384687e-01+i*5.604653e-02 9.393594e-01+i*4.956954e-02 9.407312e-01+i*4.393612e-02 9.423984e-01+i*3.905997e-02 796 9 9.456287e-01+i*6.271438e-02 9.450233e-01+i*5.621112e-02 9.451535e-01+i*5.034896e-02 9.458115e-01+i*4.512821e-02 9.468331e-01+i*4.051390e-02 797 10 9.519648e-01+i*6.140810e-02 9.508496e-01+i*5.572482e-02 9.504272e-01+i*5.048129e-02 9.505347e-01+i*4.571108e-02 9.510346e-01+i*4.141323e-02 798 11 9.574268e-01+i*5.973504e-02 9.559865e-01+i*5.479032e-02 9.551756e-01+i*5.013466e-02 9.548699e-01+i*4.581814e-02 9.549587e-01+i*4.186055e-02 799 12 9.621284e-01+i*5.785504e-02 9.604951e-01+i*5.355999e-02 9.594210e-01+i*4.944388e-02 9.588136e-01+i*4.556283e-02 9.585857e-01+i*4.194796e-02 800 13 9.661780e-01+i*5.587714e-02 9.644445e-01+i*5.214554e-02 9.632008e-01+i*4.851388e-02 9.623793e-01+i*4.503822e-02 9.619129e-01+i*4.175474e-02 801 14 9.696733e-01+i*5.387419e-02 9.679030e-01+i*5.062727e-02 9.665585e-01+i*4.742454e-02 9.655908e-01+i*4.431875e-02 9.649490e-01+i*4.134718e-02 802 15 9.726987e-01+i*5.189373e-02 9.709347e-01+i*4.906194e-02 9.695390e-01+i*4.623569e-02 9.684768e-01+i*4.346284e-02 9.677098e-01+i*4.077949e-02 803 16 9.753264e-01+i*4.996582e-02 9.735970e-01+i*4.748902e-02 9.721854e-01+i*4.499147e-02 9.710674e-01+i*4.251567e-02 9.702145e-01+i*4.009527e-02 804 17 9.776172e-01+i*4.810861e-02 9.759404e-01+i*4.593538e-02 9.745377e-01+i*4.372399e-02 9.733925e-01+i*4.151169e-02 9.724841e-01+i*3.932908e-02 805 18 9.796222e-01+i*4.633214e-02 9.780086e-01+i*4.441884e-02 9.766318e-01+i*4.245632e-02 9.754805e-01+i*4.047692e-02 9.745398e-01+i*3.850806e-02 806 19 9.813837e-01+i*4.464105e-02 9.798394e-01+i*4.295077e-02 9.784996e-01+i*4.120470e-02 9.773573e-01+i*3.943076e-02 9.764021e-01+i*3.765328e-02 807 20 9.829375e-01+i*4.303640e-02 9.814648e-01+i*4.153798e-02 9.801692e-01+i*3.998033e-02 9.790468e-01+i*3.838752e-02 9.780902e-01+i*3.678094e-02 808 809 . EXPONENTIAL INTEGRAl AND RELATED FUNCTIONS 251 810 . 811 . EXPONENTIAL INTEGRAL FOR COMPLEX ARGUMENTS Table 5.6 812 . Z*EXP(Z)*E1(Z) 813 814 y / x 11 12 13 14 15 815 0 9.222596e-01+i*0.000000e+00 9.279136e-01+i*0.000000e+00 9.327959e-01+i*0.000000e+00 9.370553e-01+i*0.000000e+00 9.408042e-01+i*0.000000e+00 816 1 9.227395e-01+i*6.063064e-03 9.282952e-01+i*5.211591e-03 9.331045e-01+i*4.528313e-03 9.373084e-01+i*3.971556e-03 9.410144e-01+i*3.511823e-03 817 2 9.241432e-01+i*1.190225e-02 9.294157e-01+i*1.025800e-02 9.340131e-01+i*8.932070e-03 9.380553e-01+i*7.847404e-03 9.416359e-01+i*6.948885e-03 818 3 9.263697e-01+i*1.732110e-02 9.312053e-01+i*1.499143e-02 9.354725e-01+i*1.309811e-02 9.392607e-01+i*1.153952e-02 9.426427e-01+i*1.024173e-02 819 4 9.292701e-01+i*2.217111e-02 9.335600e-01+i*1.929513e-02 9.374082e-01+i*1.693392e-02 9.408699e-01+i*1.497395e-02 9.439942e-01+i*1.333071e-02 820 5 9.326718e-01+i*2.636133e-02 9.363555e-01+i*2.309076e-02 9.397293e-01+i*2.037321e-02 9.428156e-01+i*1.809503e-02 9.456395e-01+i*1.616919e-02 821 6 9.364004e-01+i*2.985697e-02 9.394622e-01+i*2.633925e-02 9.423383e-01+i*2.337750e-02 9.450235e-01+i*2.086702e-02 9.475216e-01+i*1.872496e-02 822 7 9.402971e-01+i*3.267012e-02 9.427571e-01+i*2.903635e-02 9.451399e-01+i*2.593412e-02 9.474193e-01+i*2.327343e-02 9.495821e-01+i*2.098029e-02 823 8 9.442291e-01+i*3.484671e-02 9.461324e-01+i*3.120500e-02 9.480470e-01+i*2.805186e-02 9.499329e-01+i*2.531451e-02 9.517646e-01+i*2.293048e-02 824 9 9.480934e-01+i*3.645322e-02 9.494999e-01+i*3.288683e-02 9.509854e-01+i*2.975554e-02 9.525023e-01+i*2.700388e-02 9.540175e-01+i*2.458175e-02 825 10 9.518159e-01+i*3.756554e-02 9.527915e-01+i*3.413423e-02 9.538946e-01+i*3.108052e-02 9.550752e-01+i*2.836471e-02 9.562961e-01+i*2.594869e-02 826 11 9.553474e-01+i*3.826070e-02 9.559582e-01+i*3.500395e-02 9.567287e-01+i*3.206787e-02 9.576095e-01+i*2.942626e-02 9.585627e-01+i*2.705173e-02 827 12 9.586593e-01+i*3.861152e-02 9.589675e-01+i*3.555239e-02 9.594541e-01+i*3.276061e-02 9.600731e-01+i*3.022082e-02 9.607874e-01+i*2.791481e-02 828 13 9.617386e-01+i*3.868357e-02 9.618002e-01+i*3.583260e-02 9.620487e-01+i*3.320089e-02 9.624426e-01+i*3.078145e-02 9.629472e-01+i*2.856350e-02 829 14 9.645833e-01+i*3.853397e-02 9.644472e-01+i*3.589257e-02 9.644990e-01+i*3.342826e-02 9.647023e-01+i*3.114025e-02 9.650256e-01+i*2.902355e-02 830 15 9.671991e-01+i*3.821118e-02 9.669071e-01+i*3.577447e-02 9.667988e-01+i*3.347864e-02 9.668428e-01+i*3.132729e-02 9.670112e-01+i*2.931980e-02 831 16 9.695966e-01+i*3.775559e-02 9.691837e-01+i*3.551456e-02 9.689470e-01+i*3.338384e-02 9.688595e-01+i*3.136997e-02 9.688972e-01+i*2.947551e-02 832 17 9.717894e-01+i*3.720030e-02 9.712847e-01+i*3.514349e-02 9.709463e-01+i*3.317151e-02 9.707517e-01+i*3.129276e-02 9.706801e-01+i*2.951196e-02 833 18 9.737924e-01+i*3.657216e-02 9.732195e-01+i*3.468684e-02 9.728020e-01+i*3.286534e-02 9.725212e-01+i*3.111710e-02 9.723593e-01+i*2.944826e-02 834 19 9.756209e-01+i*3.589271e-02 9.749990e-01+i*3.416572e-02 9.745211e-01+i*3.248539e-02 9.741719e-01+i*3.086159e-02 9.739364e-01+i*2.930131e-02 835 20 9.772898e-01+i*3.517907e-02 9.766344e-01+i*3.359739e-02 9.761116e-01+i*3.204846e-02 9.757090e-01+i*3.054215e-02 9.754141e-01+i*2.908589e-02 836 y / x 16 17 18 19 20 837 0 9.441297e-01+i*0.000000e+00 9.470998e-01+i*0.000000e+00 9.497690e-01+i*0.000000e+00 9.521808e-01+i*0.000000e+00 9.543709e-01+i*0.000000e+00 838 1 9.443062e-01+i*3.127757e-03 9.472495e-01+i*2.803583e-03 9.498970e-01+i*2.527437e-03 9.522912e-01+i*2.290262e-03 9.544667e-01+i*2.085040e-03 839 2 9.448289e-01+i*6.196262e-03 9.476933e-01+i*5.559594e-03 9.502771e-01+i*5.016238e-03 9.526191e-01+i*4.548817e-03 9.547517e-01+i*4.143816e-03 840 3 9.456783e-01+i*9.150002e-03 9.484165e-01+i*8.223154e-03 9.508977e-01+i*7.429737e-03 9.531557e-01+i*6.745425e-03 9.552187e-01+i*6.151167e-03 841 4 9.468238e-01+i*1.194046e-02 9.493954e-01+i*1.075445e-02 9.517406e-01+i*9.734998e-03 9.538865e-01+i*8.852629e-03 9.558563e-01+i*8.084041e-03 842 5 9.482263e-01+i*1.452847e-02 9.505999e-01+i*1.312053e-02 9.527822e-01+i*1.190419e-02 9.547929e-01+i*1.084683e-02 9.566496e-01+i*9.922327e-03 843 6 9.498418e-01+i*1.688556e-02 9.519954e-01+i*1.529640e-02 9.539950e-01+i*1.391549e-02 9.558529e-01+i*1.270899e-02 9.575809e-01+i*1.164944e-02 844 7 9.516239e-01+i*1.899412e-02 9.535450e-01+i*1.726537e-02 9.553494e-01+i*1.575344e-02 9.570425e-01+i*1.442503e-02 9.586307e-01+i*1.325266e-02 845 8 9.535270e-01+i*2.084675e-02 9.552117e-01+i*1.901870e-02 9.568153e-01+i*1.740886e-02 9.583371e-01+i*1.598581e-02 9.597786e-01+i*1.472319e-02 846 9 9.555085e-01+i*2.244490e-02 9.569601e-01+i*2.055483e-02 9.583632e-01+i*1.887832e-02 9.597121e-01+i*1.738685e-02 9.610042e-01+i*1.605601e-02 847 10 9.575301e-01+i*2.379715e-02 9.587578e-01+i*2.187820e-02 9.599657e-01+i*2.016343e-02 9.611444e-01+i*1.862785e-02 9.622877e-01+i*1.724957e-02 848 11 9.595589e-01+i*2.491742e-02 9.605761e-01+i*2.299793e-02 9.615980e-01+i*2.126988e-02 9.626124e-01+i*1.971201e-02 9.636107e-01+i*1.830531e-02 849 12 9.615675e-01+i*2.582319e-02 9.623905e-01+i*2.392662e-02 9.632382e-01+i*2.220650e-02 9.640970e-01+i*2.064537e-02 9.649563e-01+i*1.922715e-02 850 13 9.635343e-01+i*2.653403e-02 9.641807e-01+i*2.467905e-02 9.648680e-01+i*2.298435e-02 9.655815e-01+i*2.143606e-02 9.663095e-01+i*2.002098e-02 851 14 9.654425e-01+i*2.707037e-02 9.659307e-01+i*2.527127e-02 9.664722e-01+i*2.361591e-02 9.670518e-01+i*2.209368e-02 9.676576e-01+i*2.069410e-02 852 15 9.672799e-01+i*2.745250e-02 9.676281e-01+i*2.571970e-02 9.680385e-01+i*2.411435e-02 9.684964e-01+i*2.262874e-02 9.689896e-01+i*2.125479e-02 853 16 9.690382e-01+i*2.769994e-02 9.692639e-01+i*2.604054e-02 9.695578e-01+i*2.449303e-02 9.699060e-01+i*2.305212e-02 9.702966e-01+i*2.171191e-02 854 17 9.707124e-01+i*2.783095e-02 9.708318e-01+i*2.624932e-02 9.710231e-01+i*2.476501e-02 9.712734e-01+i*2.337475e-02 9.715713e-01+i*2.207451e-02 855 18 9.723000e-01+i*2.786231e-02 9.723280e-01+i*2.636060e-02 9.724299e-01+i*2.494279e-02 9.725936e-01+i*2.360730e-02 9.728084e-01+i*2.235162e-02 856 19 9.738003e-01+i*2.780916e-02 9.737506e-01+i*2.638776e-02 9.737750e-01+i*2.503810e-02 9.738627e-01+i*2.375992e-02 9.740038e-01+i*2.255199e-02 857 20 9.752147e-01+i*2.768500e-02 9.750992e-01+i*2.634296e-02 9.750572e-01+i*2.506177e-02 9.750786e-01+i*2.384219e-02 9.751545e-01+i*2.268399e-02 858 859 . EXPONENTIAL INTEGRAL FOR SMALL COMPLEX ARGUMENTS 860 . EXP(Z)*E1(Z) 861 862 y / x -4.0 -3.5 -3.0 -2.5 -2.0 863 0.0 -3.595520e-01-i*5.754028e-02 -4.205093e-01-i*9.486788e-02 -4.945764e-01-i*1.564107e-01 -5.806501e-01-i*2.578776e-01 -6.704827e-01-i*4.251683e-01 864 0.2 -3.471790e-01-i*7.828255e-02 -4.005965e-01-i*1.199269e-01 -4.624933e-01-i*1.855729e-01 -5.289874e-01-i*2.890090e-01 -5.875585e-01-i*4.512253e-01 865 0.4 -3.333728e-01-i*9.664773e-02 -3.792781e-01-i*1.412209e-01 -4.295541e-01-i*2.087998e-01 -4.783030e-01-i*3.108836e-01 -5.105427e-01-i*4.631926e-01 866 0.6 -3.185556e-01-i*1.126328e-01 -3.572020e-01-i*1.588904e-01 -3.967301e-01-i*2.265753e-01 -4.299777e-01-i*3.247735e-01 -4.411279e-01-i*4.641632e-01 867 0.8 -3.031094e-01-i*1.263012e-01 -3.349227e-01-i*1.731693e-01 -3.647850e-01-i*2.395000e-01 -3.849405e-01-i*3.320471e-01 -3.800126e-01-i*4.570879e-01 868 1.0 -2.873689e-01-i*1.377684e-01 -3.128943e-01-i*1.843550e-01 -3.342796e-01-i*2.482307e-01 -3.437194e-01-i*3.340435e-01 -3.271401e-01-i*4.445281e-01 869 870 . E1(Z)+LOG(Z) 871 872 y / x -2.0 -1.5 -1.0 0.5 0.0 873 0.0 -4.261087e+00+i*0.000000e+00 -2.895820e+00+i*0.000000e+00 -1.895118e+00+i*0.000000e+00 -1.147367e+00+i*0.000000e+00 -5.77216 e+00+i*0.000000e+00 874 0.2 -4.219228e+00+i*6.367788e-01 -2.867070e+00+i*4.628041e-01 -1.875155e+00+i*3.426999e-01 -1.133341e+00+i*2.588397e-01 -5.672323e-01+i*1.995561e-01 875 0.4 -4.094686e+00+i*1.260867e+00 -2.781497e+00+i*9.171266e-01 -1.815717e+00+i*6.796906e-01 -1.091560e+00+i*5.138060e-01 -5.374814e-01+i*3.964615e-01 876 0.6 -3.890531e+00+i*1.859922e+00 -2.641121e+00+i*1.354712e+00 -1.718135e+00+i*1.005410e+00 -1.022911e+00+i*7.611217e-01 -4.885549e-01+i*5.881288e-01 877 0.8 -3.611783e+00+i*2.422284e+00 -2.449241e+00+i*1.767749e+00 -1.584591e+00+i*1.314586e+00 -9.288420e-01+i*9.971998e-01 -4.214222e-01+i*7.720958e-01 878 1.0 -3.265262e+00+i*2.937296e+00 -2.210344e+00+i*2.149077e+00 -1.418052e+00+i*1.602372e+00 -8.113274e-01+i*1.218731e+00 -3.374039e-01+i*9.460831e-01 879 y / x 0.5 1.0 1.5 2.0 2.5 880 0.0 -1.333736e-01+i*0.000000e+00 2.193839e-01+i*0.000000e+00 5.054847e-01+i*0.000000e+00 7.420477e-01+i*0.000000e+00 9.412056e-01+i*0.000000e+00 881 0.2 -1.261685e-01+i*1.570812e-01 2.246612e-01+i*1.262102e-01 5.094099e-01+i*1.034318e-01 7.450141e-01+i*8.635880e-02 9.434838e-01+i*7.335542e-02 882 0.4 -1.046871e-01+i*3.123312e-01 2.404021e-01+i*2.511426e-01 5.211237e-01+i*2.059620e-01 7.538706e-01+i*1.720741e-01 9.502886e-01+i*1.462458e-01 883 0.6 -6.932772e-02+i*4.639611e-01 2.663367e-01+i*3.735472e-01 5.404416e-01+i*3.067078e-01 7.684901e-01+i*2.565150e-01 9.615317e-01+i*2.182147e-01 884 0.8 -2.074303e-02+i*6.102641e-01 3.020225e-01+i*4.922290e-01 5.670609e-01+i*4.048231e-01 7.886640e-01+i*3.390751e-01 9.770680e-01+i*2.888224e-01 885 1.0 4.017706e-02+i*7.496549e-01 3.468552e-01+i*6.060736e-01 6.005685e-01+i*4.995155e-01 8.141071e-01+i*4.191846e-01 9.966987e-01+i*3.576536e-01 886 */ 887 888 /*! 889 Function \f$Gamma(z) = \int_0^{\infty} t^{z-1} exp(-t) dt for Re(z) > 0\f$ 890 - Gamma(1-z) = Pi / ( sin(Pi z) Gamma(z) ) 891 - Gamma(z+1) = z Gamma(z) 892 - Gamma(n+1) = n! for integer n > 0 893 */ 894 real_t gammaFunction(const int_t n); //!< return Gamma(n) = (n-1)! 895 896 real_t gammaFunction(const real_t); //!< return \f$\int_0^t dt t^{x-1} \exp(-t)\f$ for x > 0 897 898 complex_t gammaFunction(const complex_t&); //!< return \f$\int_0^t dt t^{x-1} \exp(-t)\f$ for x > 0 899 900 real_t logGamma(const real_t); //!< return Log(Gamma(x)) 901 902 complex_t logGamma1(const complex_t&); //!< return Log(Gamma(z)) 903 complex_t logGamma(const complex_t&); //!< Paul Godfrey's Lanczos implementation of the Gamma function 904 905 /* 906 -------------------------------------------------------------------------------- 907 Function DiGamma(z) = d/dz(Log(Gamma(z))) 908 -------------------------------------------------------------------------------- 909 */ 910 //@{ 911 /*! return \f$-gamma + \sum_1^{n-1} 1/n\f$ 912 where gamma is Euler-Mascheroni constant 913 */ 914 real_t diGamma(const int_t n); 915 real_t diGamma(const real_t); 916 complex_t diGamma(const complex_t&); 917 //@} 918 919 /* 920 -------------------------------------------------------------------------------- 921 For tests 922 -------------------------------------------------------------------------------- 923 */ 924 void gammaTest(std::ostream&); 925 926 //-------------------------------------------------------------------------------- 927 // OrthogonalPolynomials : some orthogonal polynomials on [-1 , 1] 928 // 929 // - ChebyshevPolynomials (of the first kind) on [-1, 1] up to order n 930 // . Tn = cos( n acos(x)) 931 // . Recurrence relation : 932 // . T_0(x) = 1, T_1(x) = x 933 // . T_n(x) = 2 x T_{n-1}(x) - T_{n-1}(x) , n > 1 934 // . http://en.wikipedia.org/wiki/Chebyshev_polynomials 935 // 936 // 937 // - GegenbauerPolynomials Gegenbauer ultraspherical polynomials on [-1, 1], with parameter lambda up to order n 938 // . Recurrence relation : 939 // . P_0 = 1, P_1 = 2*lambda*x 940 // . n*P_n(x) = 2*(n+lambda-1)*x*P_{n-1}(x) - (n+2*lambda-2)*P_{n-2}(x) , n > 1 941 // . P_n is Jacobi polynomial with a = b = lambda - 1/2 942 // . http://en.wikipedia.org/wiki/Gegenbauer_polynomials 943 // 944 // - JacobiPolynomials on [-1, 1] up to order n (cf. Abramowitz & Stegun p.382) 945 // . Orthogonal polynomials with weight (1-x)^a (1+x)^b 946 // . Recurrence relation : 947 // . P_0 = 1, P_1 = (2*(a+1)+a+b+2)*(x-1) / 2 948 // . P_{n+1}*{ 2*(n+1)(n+a+b+1)(2*n+a+b) } = 949 // . { (2*n+1+a+b)(a^2-b^2) + (2*n+a+b)(2*n+1+a+b)(2*n+2+a+b)*x } * P_{n} 950 // . -{ 2*(n+a)*(n+b)(2*n+2+a+b) } * P_{n-1} , n > 0 951 // . http://en.wikipedia.org/wiki/Jacobi_polynomials 952 // 953 // - LegendrePolynomials on [-1, 1] up to order n 954 // . Recurrence relation : 955 // . P_0(x) = 1, P_1(x) = x 956 // . n*P_n(x) = (2*n-1) x P_{n-1}(x) - (n-1) P_{n-2}(x) , n > 1 957 // . P_n is Gegenbauer ultraspherical polynomial with lambda = 1/2 958 // . or Jacobi polynomial with a = b = 0 959 // . http://en.wikipedia.org/wiki/Legendre_polynomials 960 // 961 // - LegendrePolynomialsDerivative 962 // . Recurrence relation 963 // . P'n satisfies : 964 // . P'_0(x) = 0, P'_1(x) = 1 965 // . (1-x^2) P'_n(x) = - n x P_n(x) + n P_{n-1}(x) , n > 1 966 // . P'_n is Gegenbauer ultraspherical polynomial with lambda = 3/2 967 // 968 // - LegendreFunctions : Associated Legendre functions on [-1, 1] up to order n 969 // . P^m_l(x) = (-1)^{m} * (1-x^2)^{m/2} d^m/dx^m[ P_l(x) 970 // . where P_l is Legendre polynomial of degree l 971 // . Recurrence relation : 972 // . P^{k}_{k}(x) = (2k-1)!! * (-\sqrt{1-x^2})^{l} 973 // . where (2k-1)!! = \Prod_{i=1,..,k}{2i-1) = (2k)! / 2^{k}*k! 974 // . P^{k}_{k+1} = (2k+1) * x * P^{k}_{k}(x) 975 // . and for l = k+2, ..,n 976 // . (l-k+1) P^k_{l+1}(x) = (2*l+1)*x*P^k_{l}(x) - (l+k)*P^k_{l-1}(x) 977 // . http://en.wikipedia.org/wiki/Associated_Legendre_polynomials 978 // 979 // - LegendreFunctionsDerivative : Associated Legendre functions derivatives on [-1, 1] up to order n 980 // . Case (x^2 = 1) : 981 // . P'_{l}^{1}(x) = - x^l l(l+1) / 2 982 // . P'_{l}^{m}(x) = 0 if m != 1 983 // . Case (x^2 != 1) : 984 // . P'_{0}^{0}(x) = 0 985 // . for l > 0 : 986 // . P'_{l}^{0} = P_{l}^{1} 987 // . for m = 1,..., l 988 // . P'_{l}^{m} = - (l-m+1) (l+m) P_{l}^{m-1} - x/sqrt(1-x^2) * m * P_{l}^{m} 989 //-------------------------------------------------------------------------------- 990 /*! 991 Chebyshev polynomials (of the first kind) on [-1, 1] up to order n 992 T_0(x) = 1, T_1(x) = x, T_n(x) = 2 x T_{n-1}(x) - T_{n-1}(x) , n > 1 993 */ 994 void chebyshevPolynomials(const real_t, std::vector<real_t>&); 995 /*! 996 Gegenbauer ultraspherical polynomials on [-1, 1], with parameter lambda up to order n 997 P_0 = 1, P_1 = 2*lambda*x 998 n*P_n(x) = 2*(n+lambda-1)*x*P_{n-1}(x) - (n+2*lambda-2)*P_{n-2}(x) , n > 1 999 */ 1000 void gegenbauerPolynomials(const real_t lambda, const real_t, std::vector<real_t>&); 1001 1002 /*! 1003 Jacobi polynomials on [-1, 1] up to order n 1004 P_0 = 1, P_1 = (2*(a+1)+a+b+2)*(x-1) / 2 1005 P_{n+1}*{ 2*(n+1)(n+a+b+1)(2*n+a+b) } = { (2*n+1+a+b)(a^2-b^2) + (2*n+a+b)(2*n+1+a+b)(2*n+2+a+b)*x } * P_{n} 1006 -{ 2*(n+a)*(n+b)(2*n+2+a+b) } * P_{n-1} , n > 0 1007 */ 1008 void jacobiPolynomials(const real_t a, const real_t b, const real_t, std::vector<real_t>&); 1009 void jacobiPolynomials01(const real_t a, const real_t b, const real_t, std::vector<real_t>&); //!< Jacobi polynomials on [0, 1] up to order n 1010 1011 /*! 1012 Legendre polynomials on [-1, 1] up to order n 1013 P_0(x) = 1, P_1(x) = x, 1014 n*P_n(x) = (2*n-1) x P_{n-1}(x) - (n-1) P_{n-2}(x) , n > 1 1015 */ 1016 void legendrePolynomials(const real_t, std::vector<real_t>&); 1017 1018 /*! 1019 derivatives of Legendre polynomials on [-1, 1] up to order n 1020 P'_0(x) = 0, P'_1(x) = 1 1021 (1-x^2) P'_n(x) = - n x P_n(x) + n P_{n-1}(x) , n > 1 1022 */ 1023 void legendrePolynomialsDerivative(const real_t, std::vector<real_t>&); 1024 1025 /*! 1026 compute all Legendre functions up to order n for any real x: P^m_l(x) 1027 where n = Pml.size()-1 and l = 0,.., n and m = 0,..,n 1028 P^m_l(x) = (-1)^{m} * (1-x^2)^{m/2} d^m/dx^m[ P_l(x) ] 1029 where P_l is Legendre polynomial of degree l 1030 */ 1031 void legendreFunctions(const real_t, std::vector<std::vector<real_t> >& Pml); 1032 1033 /*! 1034 compute all Legendre functions derivatives up to order n for any real x: P'^m_l(x) 1035 from given Legendre functions up to order n for any real x: P^m_l(x) 1036 where n = Pml.size()-1 and l = 0,.., n and m = 0,..,n 1037 */ 1038 void legendreFunctionsDerivative(const real_t, const std::vector<std::vector<real_t> >&, std::vector<std::vector<real_t> >&); 1039 1040 void legendreFunctionsTest(const real_t x, const number_t n, std::ostream&); //!< output function for test of Legendre Functions 1041 void LegendreFunctionsDerivativeTest(const real_t, const number_t, std::ostream&); //!< output function for test of Legendre Functions derivatives 1042 1043 } // end of namespace xlifepp 1044 1045 #endif /* SPECIAL_FUNCTIONS_HPP */ 1046