tests/rtest_expintegral.mac

   1 /*******************************************************************************
   2
   3   Test file for the Exponential Integrals
   4
   5   by Dieter Kaiser (2008)
   6 *******************************************************************************/
   7
   8 kill(all);
   9 done;
  10
  11 /* Define a test function for single values */
  12
  13 closeto(e,tol):=block([numer:true,abse],abse:abs(e),if(abse<tol) then true else abse);
  14 closeto(e,tol):=block([numer:true,abse],abse:abs(e),if(abse<tol) then true else abse);
  15
  16 (test_value(actual, ref, digits) := closeto(realpart(actual)-realpart(ref), 10^(-digits)) and closeto(imagpart(actual)-imagpart(ref), 10^(-digits)), done);
  17 done;
  18
  19 /*******************************************************************************
  20  At first we check the special values of the Exponential Integrals
  21 *******************************************************************************/
  22
  23 errcatch(expintegral_e(0,0));
  24 [];
  25 errcatch(expintegral_e(0,0.0));
  26 [];
  27 errcatch(expintegral_e(0,0.0b0));
  28 [];
  29
  30 errcatch(expintegral_ei(0));
  31 [];
  32 errcatch(expintegral_ei(0.0));
  33 [];
  34 errcatch(expintegral_ei(0.0b0));
  35 [];
  36
  37 expintegral_li(0);
  38 0;
  39 expintegral_li(0.0);
  40 0.0;
  41 expintegral_li(0.0b0);
  42 0.0b0;
  43
  44 errcatch(expintegral_li(1));
  45 [];
  46 errcatch(expintegral_li(1.0));
  47 [];
  48 errcatch(expintegral_li(1.0b0));
  49 [];
  50
  51 expintegral_si(0);
  52 0;
  53 expintegral_si(0.0);
  54 0.0;
  55 expintegral_si(0.0b0);
  56 0.0b0;
  57
  58 expintegral_shi(0);
  59 0;
  60 expintegral_shi(0.0);
  61 0.0;
  62 expintegral_shi(0.0b0);
  63 0.0b0;
  64
  65 errcatch(expintegral_ci(0));
  66 [];
  67 errcatch(expintegral_ci(0.0));
  68 [];
  69 errcatch(expintegral_ci(0.0b0));
  70 [];
  71
  72 errcatch(expintegral_chi(0));
  73 [];
  74 errcatch(expintegral_chi(0.0));
  75 [];
  76 errcatch(expintegral_chi(0.0b0));
  77 [];
  78
  79 /*******************************************************************************
  80    Simplifications for the Exponential Integrals
  81 *******************************************************************************/
  82
  83 expintegral_e(-1,z);
  84 %e^(-z)*(z+1)/z^2;
  85
  86 expintegral_e(-1,10);
  87 11*%e^-10/100;
  88
  89 expintegral_e(-1,10+%i);
  90 (%i+11)*%e^(-%i-10)/(%i+10)^2;
  91
  92 expintegral_e(0,z);
  93 %e^(-z)/z;
  94
  95 expintegral_e(0,10);
  96 %e^-10/10;
  97
  98 expintegral_e(0,10+%i);
  99 %e^(-10-%i)/(10+%i);
 100
 101 /* The case Ev(0) simplifies to 1/(v-1) only for Re(v)>1  */
 102
 103 (assume(v>1), expintegral_e(v,0));
 104 1/(v-1);
 105
 106 expintegral_e(3/2+%i,0);
 107 1/(%i+1/2);
 108
 109 errcatch(expintegral_e(1/2+%i,0));
 110 [];
 111
 112 /* We don't simplify for infinities as an argument */
 113
 114 expintegral_e(-1,inf);
 115 expintegral_e(-1,inf);
 116
 117 expintegral_e(v,infinity);
 118 expintegral_e(v,infinity);
 119
 120 expintegral_e(minf,z);
 121 expintegral_e(minf,z);
 122
 123 /* Realpart and imagpart for expintegral_e */
 124
 125 (assume(x_pos>0),done);
 126 done;
 127
 128 /* a positive real value */
 129 realpart(expintegral_e(v,x_pos));
 130 expintegral_e(v,x_pos);
 131 imagpart(expintegral_e(v,x_pos));
 132 0;
 133
 134 /* a complex value */
 135 realpart(expintegral_e(v,1+%i));
 136 1/2*(expintegral_e(v,1-%i)+expintegral_e(v,1+%i));
 137 imagpart(expintegral_e(v,1+%i));
 138 1/2*%i*(expintegral_e(v,1-%i)-expintegral_e(v,1+%i));
 139
 140 /* Realpart and imagpart for expintegral_ei */
 141
 142 /* a real value */
 143 realpart(expintegral_ei(x));
 144 expintegral_ei(x);
 145 imagpart(expintegral_ei(x));
 146 0;
 147
 148 /* a complex value */
 149 realpart(expintegral_ei(1+%i));
 150 1/2*(expintegral_ei(1-%i)+expintegral_ei(1+%i));
 151 imagpart(expintegral_ei(1+%i));
 152 1/2*%i*(expintegral_ei(1-%i)-expintegral_ei(1+%i));
 153
 154 /* Realpart and imagpart for expintegral_si
 155    expintegral_si has in addition reflection symmetry and
 156    for a pure imaginary argument the result is pure imaginary */
 157
 158 /* a real value */
 159 realpart(expintegral_si(x));
 160 expintegral_si(x);
 161 imagpart(expintegral_si(x));
 162 0;
 163
 164 /* an imaginary value */
 165 realpart(expintegral_si(%i));
 166 0;
 167 imagpart(expintegral_si(%i));
 168 -%i*expintegral_si(%i);
 169
 170 /* a complex value */
 171 realpart(expintegral_si(1+%i));
 172 1/2*(expintegral_si(1-%i)+expintegral_si(1+%i));
 173 imagpart(expintegral_si(1+%i));
 174 1/2*%i*(expintegral_si(1-%i)-expintegral_si(1+%i));
 175
 176 /* Limits of expintegral_si */
 177
 178 limit(expintegral_si(x),x,inf);
 179 %pi/2;
 180
 181 limit(expintegral_si(x),x,minf);
 182 -%pi/2;
 183
 184 /* Limits of expintegral_shi */
 185
 186 limit(expintegral_shi(x),x,inf);
 187 inf;
 188
 189 limit(expintegral_shi(x),x,minf);
 190 minf;
 191
 192 /* Limits of expintegral_chi */
 193
 194 limit(expintegral_chi(x),x,inf);
 195 inf;
 196
 197 limit(expintegral_chi(x),x,minf);
 198 inf;
 199
 200 /*******************************************************************************
 201    For a negative integer parameter we expand in a series
 202 *******************************************************************************/
 203
 204 expintegral_e(-5,z);
 205 120*(z^5/120+z^4/24+z^3/6+z^2/2+z+1)*%e^-z/z^6;
 206
 207 /*******************************************************************************
 208    For a half integral parameter we can expand in terms of the
 209    Erfc or Erf function using the flag expintexpand.
 210 *******************************************************************************/
 211
 212 expintexpand:true;
 213 true;
 214
 215 expintegral_e(1/2,z);
 216 sqrt(%pi)/sqrt(z)*erfc(sqrt(z));
 217
 218 expintegral_e(-1/2,z);
 219 sqrt(%pi)/(2*z^(3/2))*erfc(sqrt(z))+%e^(-z)/z;
 220
 221 /* Expansion in terms of Erf for z = 1/2
 222    We test the Expansion against a pure numerical evaluation.
 223    The expansion works for complex argument too.
 224    Attention: We have no support for Complex arguments of the Erf function. */
 225
 226 expintexpand:erf;
 227 erf;
 228
 229 expintegral_e(5/2,1/2);
 230 2*sqrt(%pi)*(1-erf(1/sqrt(2)))/(3*sqrt(2));
 231
 232 /* We test the expansion against a pure numerical evaluation */
 233 test_value(%,expintegral_e(2.5,0.5),15),numer;
 234 true;
 235
 236 expintegral_e(-5/2,1/2);
 237 /*
 238 8*sqrt(2)*(15*sqrt(%pi)*(1-erf(1/sqrt(2)))/8+21/(4*sqrt(2)*sqrt(%e)));
 239 */
 240
 241 2^(7/2)*(15*sqrt(%pi)*(1-erf(1/sqrt(2)))/8+(5/sqrt(2)+1/2^(5/2))/sqrt(%e));
 242
 243 test_value(%,expintegral_e(-2.5,0.5),13),numer;
 244 true;
 245
 246 /*******************************************************************************
 247    For a parameter which is a positive integer we can expand
 248    the function in terms of the Exponential Integral Ei.
 249 *******************************************************************************/
 250
 251 expintegral_e(3,z);
 252 -z^2*(log(z)+(log(-1/z)-log(-z))/2+expintegral_ei(-z))/2-(z/2-1/2)*%e^-z;
 253
 254 /* We compare this with numerical values */
 255
 256 ratsimp(expintegral_e(3,1/2));
 257 (sqrt(%e)*(2*log(2)+log(-1/2)-2*expintegral_ei(-1/2)-log(-2))+4)/(16*sqrt(%e));
 258
 259 test_value(%,expintegral_e(3,0.5),15),numer;
 260 true;
 261
 262 (expand:expintegral_e(10,1/2),done);
 263 done;
 264
 265 test_value(expand,expintegral_e(10,0.5),15),numer;
 266 true;
 267
 268 (expand:expintegral_e(100,1/2),done);
 269 done;
 270
 271 test_value(expand,expintegral_e(100,0.5),15),numer;
 272 true;
 273
 274 expintexpand:false;
 275 false;
 276
 277 /*******************************************************************************
 278    Do tests for additional float evaluation
 279 *******************************************************************************/
 280
 281 test_value(
 282   expintegral_e(1,-1.700598-0.612828*%i),
 283   1.229787425764198*%i-3.675726471068782,15);
 284 true;
 285
 286 test_value(
 287   expintegral_e(1,-1.5-%i*18.0),
 288   0.181696882955049 + 0.16898654452488*%i,15);
 289 true;
 290
 291 test_value(
 292   expintegral_e(1,-1.5-%i*180.0),
 293   .01998793885396577-.01484463667769751*%i,15);
 294 true;
 295
 296 test_value(
 297   expintegral_e(1,-15.0-%i*50.0),
 298   62936.65453487506*%i-462.2396897671588,10);
 299 true;
 300
 301 test_value(
 302   expintegral_e(1,-15.0-%i*180.0),
 303   15302.16784585461-9678.322938932864*%i,10);
 304 true;
 305
 306 test_value(
 307   expintegral_e(1,-150.0-%i*540.0),
 308   2.479041623323267e+62*%i+2.105063890337474e+61,13-62);
 309 true;
 310
 311 test_value(
 312   expintegral_e(1,1.5+%i*18),
 313   .01021327940204757-.006754558125326895*%i,15);
 314 true;
 315
 316 test_value(
 317   expintegral_e(1,15.0+%i*180.0),
 318   1.129118870333903e-9*%i+1.261123517801268e-9,15);
 319 true;
 320
 321 test_value(
 322   expintegral_e(1,150.0+%i*540.0),
 323   7.503677061484707e-69-1.036533879419174e-68*%i,15-69);
 324 true;
 325
 326
 327
 328
 329
 330 /*******************************************************************************
 331    Do tests for Bigfloat evaluation (values from functions.wolfram.com)
 332 *******************************************************************************/
 333
 334 /* Remember actual fpprec */
 335 (oldfpprec:fpprec, fpprec:64, done);
 336 done;
 337
 338 test_value(
 339   expintegral_e(1,0.25b0),
 340   1.044282634443738194536438161232282251891528374744802718635140468b0,
 341   63);
 342 true;
 343
 344 test_value(
 345   expintegral_e(1,0.50b0),
 346   0.5597735947761608117467959393150852352268468903163535152482932191b0,
 347   63);
 348 true;
 349
 350 test_value(
 351   expintegral_e(1,1.50b0),
 352   0.1000195824066326519019093399116669782617300061403505850505670604b0,
 353   63);
 354 true;
 355
 356 test_value(
 357   expintegral_e(1,2.50b0),
 358   0.02491491787026973549562801227460963594584838471142737701193454450b0,
 359   64);
 360 true;
 361
 362 test_value(
 363   expintegral_e(1,0.5b0+%i),
 364   - 0.0713947110424527235558849799368449390033695834555289289265924760b0 +
 365   - 0.3574937736521626512548586934573247791553769710980144511429423771b0*%i,
 366   62);
 367 true;
 368
 369 test_value(
 370   expintegral_ei(-0.5b0),
 371   -0.5597735947761608117467959393150852352268468903163535152482932191b0,
 372   63);
 373 true;
 374
 375 test_value(
 376   expintegral_ei(0.5b0),
 377   0.4542199048631735799205238126628023652814055543526420451628177867b0,
 378   63);
 379 true;
 380
 381 test_value(
 382   expintegral_ei(1.5b0),
 383   3.301285449129797837957411316134742787656606985453422100762913970b0,
 384   63);
 385 true;
 386
 387 test_value(
 388   expintegral_ei(1.5b0+%i),
 389   2.799671509755731161198928033788660326920401659332834224954077897b0 +
 390   2.737079221508684508327603325281307039723914975958035767630503279b0*%i,
 391   63);
 392 true;
 393
 394 test_value(
 395   expintegral_li(-0.5b0),
 396   0.058454819131659372625658464865614145649609292000300491927658936b0 +
 397   3.265670174417890088883763125722547571780846521093778812746639381b0*%i,
 398   63);
 399 true;
 400
 401 test_value(
 402   expintegral_li(0.5b0),
 403   -0.3786710430610879767272071846365609805512340409782139969444209417b0,
 404   63);
 405 true;
 406
 407 test_value(
 408   expintegral_li(1.5b0),
 409   0.1250649863152963559943500047955129365420883239309922910956161087b0,
 410   63);
 411 true;
 412
 413 test_value(
 414   expintegral_li(1.5b0+%i),
 415   0.955549209862142878221358892328220253212569406502554250070344843b0 +
 416   1.567751569664112401647194996566519972267813732803503976674771406b0*%i,
 417   62);
 418 true;
 419
 420 test_value(
 421   expintegral_si(-0.5b0),
 422   -0.4931074180430666891616267075727646536413371384287211316602426140b0,
 423   63);
 424 true;
 425
 426 test_value(
 427   expintegral_si(0.5b0),
 428   0.4931074180430666891616267075727646536413371384287211316602426140b0,
 429   63);
 430 true;
 431
 432 test_value(
 433   expintegral_si(1.5b0),
 434   1.324683531172119680370472846875214042814140454625112248480722201b0,
 435   62.3b0);
 436 true;
 437
 438 test_value(
 439   expintegral_si(1.5b0+%i),
 440   1.5324237219775529809013400344036774093828247429181012177841253510b0 +
 441   0.6883092845662321107998443833405013522538814097236134325126725862b0*%i,
 442   63);
 443 true;
 444
 445 test_value(
 446   expintegral_ci(-0.5b0),
 447   -0.177784078806612901335810271070569078090519474812621968666825358b0 +
 448   3.141592653589793238462643383279502884197169399375105820974944592b0*%i,
 449   63);
 450 true;
 451
 452 test_value(
 453   expintegral_ci(0.5b0),
 454    -0.1777840788066129013358102710705690780905194748126219686668253576b0,
 455   63);
 456 true;
 457
 458 test_value(
 459   expintegral_ci(1.5b0),
 460   0.4703563171953998866750821522365605516152327005730752953941674282b0,
 461   62.2b0);
 462 true;
 463
 464 test_value(
 465   expintegral_ci(1.5b0+%i),
 466   0.7839176551092182755417770543442248219437012644680041434724902541b0 -
 467   0.0812194103150585766889706525256841552731980806778019973545925276b0*%i,
 468   63);
 469 true;
 470
 471 test_value(
 472   expintegral_shi(-0.5b0),
 473   -0.5069967498196671958336598759889438002541262223344977802055555029b0,
 474   63);
 475 true;
 476
 477 test_value(
 478   expintegral_shi(0.5b0),
 479   0.5069967498196671958336598759889438002541262223344977802055555029b0,
 480   63);
 481 true;
 482
 483 test_value(
 484   expintegral_shi(1.5b0),
 485   1.700652515768215244929660328023204882959168495796886342906740515b0,
 486   63);
 487 true;
 488
 489 test_value(
 490   expintegral_shi(1.5b0+%i),
 491   1.405456250564077987167827011345745730833834776665624179949753110b0 +
 492   1.324296049597592107364382165946694647067190542728881672301521998b0*%i,
 493   63);
 494 true;
 495
 496 test_value(
 497   expintegral_chi(-0.5b0),
 498   -0.052776844956493615913136063326141434972720667981855735042737716b0 +
 499   3.141592653589793238462643383279502884197169399375105820974944592b0*%i,
 500   63);
 501 true;
 502
 503 test_value(
 504   expintegral_chi(0.5b0),
 505   -0.05277684495649361591313606332614143497272066798185573504273771621b0,
 506   63);
 507 true;
 508
 509 test_value(
 510   expintegral_chi(1.5b0),
 511   1.600632933361582593027750988111537904697438489656535757856173455b0,
 512   63);
 513 true;
 514
 515 test_value(
 516   expintegral_chi(1.5b0+%i),
 517   1.394215259191653174031101022442914596086566882667210045004324787b0 +
 518   1.412783171911092400963221159334612392656724433229154095328981281b0*%i,
 519   63);
 520 true;
 521
 522
 523 /* restore fpprec */
 524 (fpprec:oldfpprec, done);
 525 done;
 526
 527 /*******************************************************************************
 528    Do tests for additional big float evaluation
 529 *******************************************************************************/
 530
 531 /* Remember actual fpprec */
 532 (oldfpprec:fpprec, fpprec:64, done);
 533 done;
 534
 535 test_value(
 536   expintegral_e(1,-1.5B0-%i*18.0B0),
 537   1.689865445248795461022355866185975300284302677290211138169639712b-1*%i
 538  +1.816968829550486868009237590887639256845378383882275967782801257b-1,63);
 539 true;
 540
 541 test_value(
 542   expintegral_e(1,-15.0B0-%i*50.0B0),
 543   6.293665453487508854049428479450922789578087068032898765875144175b4*%i
 544  -4.62239689767167158016168999268727094583699040537958778438547305b2,59);
 545 true;
 546
 547 test_value(
 548   expintegral_e(1,-1.5B0-%i*180.0B0),
 549   1.998793885396577954161433015440646790143523658773070463290940391b-2
 550  -1.48446366776975235341069111717359420933661819609933627048078479b-2*%i,64);
 551 true;
 552
 553 test_value(
 554   expintegral_e(1,-15.0B0-%i*180.0B0),
 555   1.530216784585460659290957316900806622008438558612666623998533076b4
 556  -9.678322938932862392009065659301165442754946683378713459319981218b3*%i,59);
 557 true;
 558
 559 test_value(
 560   expintegral_e(1,-150.0B0-%i*540.0B0),
 561   2.47904162332326778788226522449358779416653162097669293288256403b62*%i
 562  +2.105063890337477483794346786409993155624883726578689067616685218b61,1);
 563 true;
 564
 565 test_value(
 566   expintegral_e(1,1.5B0+%i*18.0B0),
 567   1.021327940204755864240745590773899499558121729025330486880547885b-2
 568  -6.754558125326884112691473105665013998699325818242672495399974279b-3*%i,65);
 569 true;
 570
 571 test_value(
 572   expintegral_e(1,15.0B0+%i*180.0B0),
 573   1.129118870333903101943908390287222351537018836722867882019975027b-9*%i
 574  +1.261123517801267593713906360450736314138456246082714062225342743b-9,72);
 575 true;
 576
 577 test_value(
 578   expintegral_e(1,150.0B0+%i*540.0B0),
 579   7.50367706148470796758191535417179842322266910293477644515984948b-69
 580  -1.03653387941917390868865781356557208741441722016128718318801276b-68*%i,-64-69);
 581 true;
 582
 583
 584 /* restore fpprec */
 585 (fpprec:oldfpprec, done);
 586 done;
 587
 588 /*******************************************************************************
 589    Do tests for a parameter not an positive integer (double float arithmetic)
 590 *******************************************************************************/
 591
 592 test_value(
 593   expintegral_e(0.5,0.5),
 594   0.7953794908467029,
 595   15);
 596 true;
 597
 598 /* functions.wolfram.com get a small imaginary part.
 599    Our algorithm gives a pure real result. That's correct. */
 600
 601 test_value(
 602   expintegral_e(1.5,0.5),
 603   0.4176818285785640 + 0.10d-16*%i,
 604   15);
 605 true;
 606
 607 test_value(
 608   expintegral_e(1.5,1.5),
 609   0.08475846798926254 + 0.10d-17*%i,
 610   15);
 611 true;
 612
 613 /* The algorithm works for an Complex parameter (not Bigfloat) too */
 614
 615 test_value(
 616   expintegral_e(-0.25+%i,0.5),
 617   0.7029675553348383 - 1.0854583859408265*%i,
 618   15);
 619 true;
 620
 621 test_value(
 622   expintegral_e(0.5+%i,0.5),
 623   0.5037890474837921 - 0.4713445822591324*%i,
 624   15);
 625 true;
 626
 627 test_value(
 628   expintegral_e(0.5+%i,1.5),
 629   0.10617497621984483 - 0.04370294969886679*%i,
 630   15);
 631 true;
 632
 633 /* For a negative integer as parameter the function will be expanded in a
 634    finite series. We compare this result with the direct numerically
 635    evaluation of the function. */
 636
 637 test_value(expintegral_e(-3,0.5)-expintegral_e(-3.0,0.5),0.0,13);
 638 true;
 639
 640 /*******************************************************************************
 641    Do tests for a parameter not an integer (Bigfloat arithmetic)
 642 *******************************************************************************/
 643
 644 /* Remember actual fpprec */
 645 (oldfpprec:fpprec, fpprec:64, done);
 646 done;
 647
 648 test_value(
 649   expintegral_e(0.5,0.5b0),
 650   0.7953794908467028960691560442509551305541581704269437959894550131b0,
 651   61);
 652 true;
 653
 654 /* functions.wolfram.com get a small imaginary part.
 655    Our algorithm gives a pure real result. That's correct. */
 656
 657 test_value(
 658   expintegral_e(1.5,0.5b0),
 659   0.4176818285785639511384430257314057763296781005474301153763293043b0 +
 660   0.10b-64*%i,
 661   62);
 662 true;
 663
 664 test_value(
 665   expintegral_e(1.5,1.5b0),
 666   0.08475846798926253566550159750486026149596073485977905694436004042b0 +
 667   0.10b-65*%i,
 668   63);
 669 true;
 670
 671 /* restore fpprec */
 672 (fpprec:oldfpprec, done);
 673 done;
 674
 675 /*******************************************************************************
 676    We check the transformation to another representation
 677 *******************************************************************************/
 678
 679 /* to prevent the simplification of 1/2*(log(z)-log(1/z))=0 which is wrong
 680    for z a negative and real value we switch of logexpand */
 681
 682 (oldlogexpand:logexpand, logexpand:false);
 683 false;
 684
 685 expintrep:gamma_incomplete;
 686 gamma_incomplete;
 687
 688 expintegral_e(n,z);
 689 gamma_incomplete(1-n,z)*z^(n-1);
 690
 691 expintegral_e(n,z);
 692 z^(n-1)*gamma_incomplete(1-n,z);
 693
 694 expintegral_e1(z);
 695 gamma_incomplete(0,z);
 696
 697 expintegral_ei(z);
 698 (log(z)-log(1/z))/2-log(-z)-gamma_incomplete(0,-z);
 699
 700 expintegral_li(z);
 701 (log(log(z))-log(1/log(z)))/2-log(-log(z))-gamma_incomplete(0,-log(z));
 702
 703 expintegral_si(z);
 704 %i*(-log(%i*z)+log(-%i*z)-gamma_incomplete(0,%i*z)+gamma_incomplete(0,-%i*z))/2;
 705
 706 expintegral_ci(z);
 707 log(z)-(log(%i*z)+log(-%i*z)+gamma_incomplete(0,%i*z)+gamma_incomplete(0,-%i*z))/2;
 708
 709 expintegral_shi(z);
 710 (log(z)-log(-z)+gamma_incomplete(0,z)-gamma_incomplete(0,-z))/2;
 711
 712 expintegral_chi(z);
 713 (-log(z)+log(-z)+gamma_incomplete(0,z)+gamma_incomplete(0,-z))/-2;
 714
 715 expintrep: expintegral_ei;
 716 expintegral_ei;
 717
 718 expintegral_li(x);
 719 expintegral_ei(log(x));
 720
 721 expintrep:false;
 722 false;
 723
 724 (logexpand:oldlogexpand,done);
 725 done;
 726
 727 /* Do the functions have the appropriate symmetrie?
 728
 729    With the exception of exponential_ci we get for all tests a double float
 730    zero or a Bigfloat zero. Because we use numercial evaluation this is not
 731    natural and perhaps an effect only true with the GCL Compiler.
 732    For the function expintegral_ci we get a small imaginary contribution which
 733    is nearby zero.
 734
 735    Can this be verified by other Compilers?
 736 */
 737
 738 z:0.5+%i;
 739 0.5+%i;
 740
 741 expintegral_e(1,z) - conjugate(expintegral_e(1,conjugate(z)));
 742 0.0;
 743
 744 expintegral_e(2,z) - conjugate(expintegral_e(2,conjugate(z)));
 745 0.0;
 746
 747 expintegral_e(5,z) - conjugate(expintegral_e(5,conjugate(z)));
 748 0.0;
 749
 750 expintegral_e(10,z) - conjugate(expintegral_e(10,conjugate(z)));
 751 0.0;
 752
 753 expintegral_e1(z) - conjugate(expintegral_e1(conjugate(z)));
 754 0.0;
 755
 756 expintegral_ei(z) - conjugate(expintegral_ei(conjugate(z)));
 757 0.0;
 758
 759 expintegral_li(z) - conjugate(expintegral_li(conjugate(z)));
 760 0.0;
 761
 762 expintegral_si(z) - conjugate(expintegral_si(conjugate(z)));
 763 0.0;
 764
 765 test_value(
 766   expintegral_ci(z) - conjugate(expintegral_ci(conjugate(z))),
 767   0.0,15);
 768 true;
 769
 770 expintegral_shi(z) - conjugate(expintegral_shi(conjugate(z)));
 771 0.0;
 772
 773 expintegral_chi(z) - conjugate(expintegral_chi(conjugate(z)));
 774 0.0;
 775
 776 /* The same for Bigfloats */
 777
 778 z:0.5b0+%i;
 779 0.5b0+%i;
 780
 781 expintegral_e(1,z) - conjugate(expintegral_e(1,conjugate(z)));
 782 0.0b0;
 783
 784 expintegral_e(2,z) - conjugate(expintegral_e(2,conjugate(z)));
 785 0.0b0;
 786
 787 expintegral_e(5,z) - conjugate(expintegral_e(5,conjugate(z)));
 788 0.0b0;
 789
 790 expintegral_e(10,z) - conjugate(expintegral_e(10,conjugate(z)));
 791 0.0b0;
 792
 793 expintegral_e1(z) - conjugate(expintegral_e1(conjugate(z)));
 794 0.0b0;
 795
 796 expintegral_ei(z) - conjugate(expintegral_ei(conjugate(z)));
 797 0.0b0;
 798
 799 expintegral_li(z) - conjugate(expintegral_li(conjugate(z)));
 800 0.0b0;
 801
 802 expintegral_si(z) - conjugate(expintegral_si(conjugate(z)));
 803 0.0b0;
 804
 805 expintegral_ci(z) - conjugate(expintegral_ci(conjugate(z)));
 806 0.0b0;
 807
 808 expintegral_shi(z) - conjugate(expintegral_shi(conjugate(z)));
 809 0.0b0;
 810
 811 expintegral_chi(z) - conjugate(expintegral_chi(conjugate(z)));
 812 0.0b0;
 813
 814 kill(z);
 815 done;
 816
 817 /******************************************************************************
 818
 819    Define test functions to do the tests with the numerical data
 820    of the tables of A&S.
 821
 822    This routine is based on the algorithm to test the Wronskians in the
 823    file rtest14.mac which was implemented by Raymond Toy.
 824
 825 ******************************************************************************/
 826
 827 (test_table(func,table,rows,eps) :=
 828 block([badpoints : [],
 829        abserr    : 0,
 830        maxerr    : -1,
 831        numer     : true],
 832   for i:1 thru rows step 1 do
 833   (
 834     z      : table[i,0],
 835     result : rectform(func(z)),
 836     answer : table[i,1],
 837     abserr : abs(result-answer),
 838     maxerr : max(maxerr,abserr),
 839     if abserr > eps then
 840     (
 841       badpoints : cons ([z,result,answer,abserr],badpoints)
 842     )
 843   ),
 844   if badpoints # [] then
 845     cons(maxerr,badpoints)
 846   else
 847     badpoints
 848 ),done);
 849 done;
 850
 851 /* Test function for a table with Complex values */
 852
 853 (test_complex_table(func,table,rows,eps) :=
 854 block([badpoints : [],
 855        abserr    : 0,
 856        maxerr    : -1,
 857        numer     : true],
 858   for i:1 thru rows step 1 do
 859   (
 860     z      : table[i,0]+%i*table[i,1],
 861     result : rectform(func(z)),
 862     answer : (table[i,2]+%i*table[i,3]),
 863     abserr : abs(result-answer),
 864     maxerr : max(maxerr,abserr),
 865     if abserr > eps then
 866     (
 867       badpoints : cons ([z,result,answer,abserr],badpoints)
 868     )
 869   ),
 870   if badpoints # [] then
 871     cons(maxerr,badpoints)
 872   else
 873     badpoints
 874 ),done);
 875 done;
 876
 877 /*****************************************************************************/
 878
 879 /* Values for E[-2](z), E[-3](z) and E[-4](z) */
 880
 881 block(
 882 em2 : make_array(flonum,5,2),
 883 fillarray(em2,[
 884 0.0,0.0,               /* the first entry isn't tested */
 885 0.5,15.7697971525285,
 886 1.0,1.83939720585721
 887 ]),done);
 888 done;
 889
 890 block(
 891 em3 : make_array(flonum,5,2),
 892 fillarray(em3,[
 893 0.0,   0.0,               /* the first entry isn't tested */
 894 0.5,  95.8318442345961,
 895 1.0,  5.88607105874308
 896 ]),done);
 897 done;
 898
 899 block(
 900 em4 : make_array(flonum,5,2),
 901 fillarray(em4,[
 902 0.0,   0.0,               /* the first entry isn't tested */
 903 0.5, 767.8678151961939,
 904 1.0, 23.91216367614375
 905 ]),done);
 906 done;
 907
 908 /* Values for E2(z) from wolfram.functions.com */
 909
 910 block(
 911 e2 : make_array(flonum,20,4),
 912 fillarray(e2,[
 913 0.0, 0.0,   0.0,                0.0,
 914 0.5, 0.0,   0.326643862324553,  0.0,
 915 1.0, 0.0,   0.148495506775922,  0.0,
 916 1.5, 0.0,   0.0731007865384809, 0.0,
 917
 918 -1.5, 0.0, -0.47023910335663,   -4.71238898038469,
 919 -1.0, 0.0,  0.82316401210311,   -3.14159265358979,
 920 -0.5, 0.0,  1.42161131826854,   -1.57079632679490,
 921
 922 0.5, 1.0,   0.005913495891524,  -0.260236353676039,
 923 1.0, 1.0,   0.0191599508550726, -0.1305169650657347,
 924 1.5, 1.0,   0.0152091306647864, -0.0662678635026173,
 925
 926 -1.5, 1.0,  -2.18255375782701,  -1.57830995398609,
 927 -1.0, 1.0,  -1.04975484772724,  -1.27655210369426,
 928 -0.5, 1.0,  -0.386354424417090, -0.872308287488887
 929 ]),done);
 930 done;
 931
 932 /* Do the tests with the above test data */
 933
 934 test_table(lambda([z],expintegral_e(-2,z)),'em2,2,3.5e-14);
 935 [];
 936
 937 test_table(lambda([z],expintegral_e(-3,z)),'em3,2,2.9e-14);
 938 [];
 939
 940 test_table(lambda([z],expintegral_e(-4,z)),'em4,2,1.0e-15);
 941 [];
 942
 943 test_complex_table(lambda([z],expintegral_e(2,z)),'e2,12,3.9e-15);
 944 [];
 945
 946 /*****************************************************************************
 947    A&S Table 5.1 p. 238, values for Si(x)/x from 0.00 through 0.50
 948 ******************************************************************************/
 949
 950 block(
 951 si_1 : make_array(flonum,51,2),
 952 fillarray(si_1,[   /* We start the loop for the test with i=1. */
 953 0.00,1.0000000000, /* Therefore we don't do the the test for the first value */
 954 0.01,0.9999944444,
 955 0.02,0.9999777781,
 956 0.03,0.9999500014,
 957 0.04,0.9999111154,
 958 0.05,0.9998611215,
 959 0.06,0.9998000216,
 960 0.07,0.9997278178,
 961 0.08,0.9996445127,
 962 0.09,0.9995501094,
 963 0.10,0.9994446111,
 964 0.11,0.9993280218,
 965 0.12,0.9992003455,
 966 0.13,0.9990615870,
 967 0.14,0.9989117512,
 968 0.15,0.9987508435,
 969 0.16,0.9985788696,
 970 0.17,0.9983958357,
 971 0.18,0.9982017486,
 972 0.19,0.9979966151,
 973 0.20,0.9977804427,
 974 0.21,0.9975532390,
 975 0.22,0.9973150122,
 976 0.23,0.9970657709,
 977 0.24,0.9968055242,
 978 0.25,0.9965342813,
 979 0.26,0.9962520519,
 980 0.27,0.9959588464,
 981 0.28,0.9956546750,
 982 0.29,0.9953395489,
 983 0.30,0.9950134793,
 984 0.31,0.9946764779,
 985 0.32,0.9943285570,
 986 0.33,0.9939697288,
 987 0.34,0.9936000064,
 988 0.35,0.9932194028,
 989
 990 0.36,0.9928279320,
 991 0.37,0.9924256078,
 992 0.38,0.9920124449,
 993 0.39,0.9915884579,
 994 0.40,0.9911536619,
 995 0.41,0.9907080728,
 996 0.42,0.9902517063,
 997 0.43,0.9897845790,
 998 0.44,0.9893067074,
 999 0.45,0.9888181089,
1000 0.46,0.9883188008,
1001 0.47,0.9878088010,
1002 0.48,0.9872881278,
1003 0.49,0.9867567998,
1004 0.50,0.9862148361
1005 ]),done);
1006 done;
1007
1008 test_table(lambda([z],expintegral_si(z)/z),'si_1,50,1.25e-10);
1009 [];
1010
1011 /******************************************************************************
1012    A&S Table 5.1 p. 239-243, values for Si(x) from 0.50 through 10.0
1013 ******************************************************************************/
1014
1015 block(
1016 si_2 : make_array(flonum,231,2),
1017 fillarray(si_2,[
1018
1019 0.50,0.4931074180,
1020 0.51,0.5026877506,
1021 0.52,0.5122515212,
1022 0.53,0.5217984228,
1023 0.54,0.5313281492,
1024 0.55,0.5408403951,
1025 0.56,0.5503348563,
1026 0.57,0.5598112298,
1027 0.58,0.5692692137,
1028 0.59,0.5787085069,
1029 0.60,0.5881288096,
1030 0.61,0.5975298233,
1031 0.62,0.6069112503,
1032 0.63,0.6162727944,
1033 0.64,0.6256141603,
1034 0.65,0.6349350541,
1035 0.66,0.6442351831,
1036 0.67,0.6535142557,
1037 0.68,0.6627719817,
1038 0.69,0.6720080721,
1039 0.70,0.6812222391,
1040 0.71,0.6904141965,
1041 0.72,0.6995836590,
1042 0.73,0.7087303430,
1043 0.74,0.7178539660,
1044 0.75,0.7269542472,
1045 0.76,0.7360309067,
1046 0.77,0.7450836664,
1047 0.78,0.7541122494,
1048 0.79,0.7631163804,
1049 0.80,0.7720957855,
1050 0.81,0.7810501921,
1051 0.82,0.7899793293,
1052 0.83,0.7988829277,
1053 0.84,0.8077607191,
1054 0.85,0.8166124372,
1055 0.86,0.8254378170,
1056 0.87,0.8342365953,
1057 0.88,0.8430085102,
1058 0.89,0.8517533016,
1059 0.90,0.8604707107,
1060 0.91,0.8691604808,
1061 0.92,0.8778223564,
1062 0.93,0.8864560839,
1063 0.94,0.8950614112,
1064 0.95,0.9036380880,
1065 0.96,0.9121858656,
1066 0.97,0.9207044970,
1067 0.98,0.9291937370,
1068 0.99,0.9376533420,
1069
1070 1.00,0.9460830704,
1071 1.01,0.9544826820,
1072 1.02,0.9628519387,
1073 1.03,0.9711906039,
1074 1.04,0.9794984431,
1075 1.05,0.9877752233,
1076 1.06,0.9960207135,
1077 1.07,1.0042346846,
1078 1.08,1.0124169091,
1079 1.09,1.0205671617,
1080 1.10,1.0286852187,
1081 1.11,1.0367708583,
1082 1.12,1.0448238608,
1083 1.13,1.0528440082,
1084 1.14,1.0608310845,
1085 1.15,1.0687848757,
1086 1.16,1.0767051696,
1087 1.17,1.0845917561,
1088 1.18,1.0924444270,
1089 1.19,1.1002629760,
1090 1.20,1.1080471990,
1091 1.21,1.1157968937,
1092 1.22,1.1235118599,
1093 1.23,1.1311918994,
1094 1.24,1.1388368160,
1095 1.25,1.1464464157,
1096 1.26,1.1540205063,
1097 1.27,1.1615588978,
1098 1.28,1.1690614023,
1099 1.29,1.1765278340,
1100 1.30,1.1839580091,
1101 1.31,1.1913517459,
1102 1.32,1.1987088649,
1103 1.33,1.2060291886,
1104 1.34,1.2133125418,
1105 1.35,1.2205587513,
1106 1.36,1.2277676460,
1107 1.37,1.2349390571,
1108 1.38,1.2420728180,
1109 1.39,1.2491687640,
1110 1.40,1.2562267328,
1111 1.41,1.2632465642,
1112 1.42,1.2702281004,
1113 1.43,1.2771711854,
1114 1.44,1.2840756658,
1115 1.45,1.2909413902,
1116 1.46,1.2977682094,
1117 1.47,1.3045559767,
1118 1.48,1.3113045473,
1119 1.49,1.3180137788,
1120
1121 1.50,1.3246835312,
1122 1.51,1.3313136664,
1123 1.52,1.3379040489,
1124 1.53,1.3444545453,
1125 1.54,1.3509650245,
1126 1.55,1.3574353577,
1127 1.56,1.3638654183,
1128 1.57,1.3702550823,
1129 1.58,1.3766042276,
1130 1.59,1.3829127345,
1131 1.60,1.3891804859,
1132 1.61,1.3954073666,
1133 1.62,1.4015932640,
1134 1.63,1.4077380678,
1135 1.64,1.4138416698,
1136 1.65,1.4199039644,
1137 1.66,1.4259248482,
1138 1.67,1.4319042202,
1139 1.68,1.4378419816,
1140 1.69,1.4437380361,
1141 1.70,1.4495922897,
1142 1.71,1.4554046507,
1143 1.72,1.4611750299,
1144 1.73,1.4669033404,
1145 1.74,1.4725894974,
1146 1.75,1.4782334189,
1147 1.76,1.4838350249,
1148 1.77,1.4893942379,
1149 1.78,1.4949109830,
1150 1.79,1.5003851872,
1151 1.80,1.5058167803,
1152 1.81,1.5112056942,
1153 1.82,1.5165518633,
1154 1.83,1.5218552243,
1155 1.84,1.5271157165,
1156 1.85,1.5323332813,
1157 1.86,1.5375078626,
1158 1.87,1.5426394066,
1159 1.88,1.5477278621,
1160 1.89,1.5527731800,
1161 1.90,1.5577753137,
1162 1.91,1.5627342192,
1163 1.92,1.5676498545,
1164 1.93,1.5725221801,
1165 1.94,1.5773511591,
1166 1.95,1.5821367567,
1167 1.96,1.5868789407,
1168 1.97,1.5915776810,
1169 1.98,1.5962329502,
1170 1.99,1.6008447231,
1171
1172 2.0,1.6054129768,
1173 2.1,1.6486986362,
1174 2.2,1.6876248272,
1175 2.3,1.7222074818,
1176 2.4,1.7524855008,
1177 2.5,1.7785201734,
1178 2.6,1.8003944505,
1179 2.7,1.8182120765,
1180 2.8,1.8320965891,
1181 2.9,1.8421901946,
1182 3.0,1.8486525280,
1183 3.1,1.8516593077,
1184 3.2,1.8514008970,
1185 3.3,1.8480807828,
1186 3.4,1.8419139833,
1187 3.5,1.8331253987,
1188 3.6,1.8219481156,
1189 3.7,1.8086216809,
1190 3.8,1.7933903548,
1191 3.9,1.7765013604,
1192 4.0,1.7582031389,
1193 4.1,1.7387436265,
1194 4.2,1.7183685637,
1195 4.3,1.6973198507,
1196 4.4,1.6758339594,
1197 4.5,1.6541404144,
1198 4.6,1.6324603525,
1199 4.7,1.6110051718,
1200 4.8,1.5899752782,
1201 4.9,1.5695589381,
1202 5.0,1.5499312449,
1203 5.1,1.5312532047,
1204 5.2,1.5136709468,
1205 5.3,1.4973150636,
1206 5.4,1.4823000826,
1207 5.5,1.4687240727,
1208 5.6,1.4566683847,
1209 5.7,1.4461975285,
1210 5.8,1.4373591823,
1211 5.9,1.4301843341,
1212 6.0,1.4246875513,
1213 6.1,1.4208673734,
1214 6.2,1.4187068241,
1215 6.3,1.4181740348,
1216 6.4,1.4192229740,
1217 6.5,1.4217942744,
1218 6.6,1.4258161486,
1219 6.7,1.4312053853,
1220 6.8,1.4378684161,
1221 6.9,1.4457024427,
1222
1223 7.0,1.4545966142,
1224 7.1,1.4644332441,
1225 7.2,1.4750890554,
1226 7.3,1.4864364451,
1227 7.4,1.4983447533,
1228 7.5,1.5106815309,
1229 7.6,1.5233137914,
1230 7.7,1.5361092381,
1231 7.8,1.5489374581,
1232 7.9,1.5616710702,
1233 8.0,1.5741868217,
1234 8.1,1.5863666225,
1235 8.2,1.5980985106,
1236 8.3,1.6092775419,
1237 8.4,1.6198065968,
1238 8.5,1.6295970994,
1239 8.6,1.6385696454,
1240 8.7,1.6466545309,
1241 8.8,1.6537921861,
1242 8.9,1.6599335052,
1243 9.0,1.6650400758,
1244 9.1,1.6690843056,
1245 9.2,1.6720494480,
1246 9.3,1.6739295283,
1247 9.4,1.6747291725,
1248 9.5,1.6744633423,
1249 9.6,1.6731569803,
1250 9.7,1.6708445697,
1251 9.8,1.6675696169,
1252 9.9,1.6633840566,
1253 10.,1.6583475942
1254 ]),done);
1255 done;
1256
1257 test_table(lambda([z],expintegral_si(z)),'si_2,230,1.95e-10);
1258 [];
1259
1260 /******************************************************************************
1261    A&S Table 5.1 p. 238, values for (Ci(x)-log(x)-%gamma)/x
1262                          from 0.00 through 0.50
1263 ******************************************************************************/
1264
1265 block(
1266 ci_1 : make_array(flonum,55,2),
1267 fillarray(ci_1,[
1268 0.00,-0.2500000000,
1269 0.01,-0.2499989583,
1270 0.02,-0.2499958333,
1271 0.03,-0.2499906250,
1272 0.04,-0.2499833339,
1273 0.05,-0.2499739598,
1274 0.06,-0.2499625030,
1275 0.07,-0.2499489639,
1276 0.08,-0.2499333429,
1277 0.09,-0.2499156402,
1278 0.10,-0.2498958564,
1279 0.11,-0.2498739923,
1280 0.12,-0.2498500480,
1281 0.13,-0.2498240244,
1282 0.14,-0.2497959223,
1283 0.15,-0.2497657422,
1284 0.16,-0.2497334850,
1285 0.17,-0.2496991516,
1286 0.18,-0.2496627429,
1287 0.19,-0.2496242598,
1288 0.20,-0.2495837035,
1289 0.21,-0.2495410749,
1290 0.22,-0.2494963752,
1291 0.23,-0.2494496056,
1292 0.24,-0.2494007674,
1293 0.25,-0.2493498618,
1294 0.26,-0.2492968902,
1295 0.27,-0.2492418540,
1296 0.28,-0.2491847546,
1297 0.29,-0.2491255938,
1298 0.30,-0.2490643727,
1299 0.31,-0.2490010933,
1300 0.32,-0.2489357573,
1301 0.33,-0.2488683662,
1302 0.34,-0.2487989219,
1303 0.35,-0.2487274263,
1304 0.36,-0.2486538813,
1305 0.37,-0.2485782887,
1306 0.38,-0.2485006507,
1307 0.39,-0.2484209693,
1308 0.40,-0.2483392466,
1309 0.41,-0.2482554849,
1310 0.42,-0.2481696860,
1311 0.43,-0.2480818528,
1312 0.44,-0.2479919870,
1313 0.45,-0.2479000913,
1314 0.46,-0.2478061685,
1315 0.47,-0.2477102206,
1316 0.48,-0.2476122500,
1317 0.49,-0.2475122600,
1318 0.50,-0.2474102526]), done);
1319 done;
1320
1321 test_table(
1322   lambda([z],(expintegral_ci(z)-log(z)-%gamma)/z^2),'ci_1,50,1.95e-10);
1323 [];
1324
1325 /******************************************************************************
1326    A&S Table 5.1 p. 239-243, values for Ci(x) from 0.50 through 10.0
1327 ******************************************************************************/
1328
1329 block(
1330 ci_2 : make_array(flonum,235,2),
1331 fillarray(ci_2,[
1332
1333 0.50,-0.1777840788,
1334 0.51,-0.1604532390,
1335 0.52,-0.1435537358,
1336 0.53,-0.1270707938,
1337 0.54,-0.1109904567,
1338 0.55,-0.0952995274,
1339 0.56,-0.0799855129,
1340 0.57,-0.0650365744,
1341 0.58,-0.0504414815,
1342 0.59,-0.0361895707,
1343 0.60,-0.0222707070,
1344 0.61,-0.0086752486,
1345 0.62,+0.0046059849,
1346 0.63,+0.0175817424,
1347 0.64,+0.0302603686,
1348 0.65,+0.0426498293,
1349 0.66,+0.0547577343,
1350 0.67,+0.0665913594,
1351 0.68,+0.0781576659,
1352 0.69,+0.0894633195,
1353 0.70,+0.1005147070,
1354 0.71,+0.1113179525,
1355 0.72,+0.1218789322,
1356 0.73,+0.1322032879,
1357 0.74,+0.1422964404,
1358 0.75,+0.1521636010,
1359 0.76,+0.1618097827,
1360 0.77,+0.1712398110,
1361 0.78,+0.1804583335,
1362 0.79,+0.1894698290,
1363 0.80,+0.1982786160,
1364 0.81,+0.2068888610,
1365 0.82,+0.2153045859,
1366 0.83,+0.2235296752,
1367 0.84,+0.2315678824,
1368 0.85,+0.2394228368,
1369 0.86,+0.2470980486,
1370 0.87,+0.2545969153,
1371 0.88,+0.2619227264,
1372 0.89,+0.2690786687,
1373 0.90,+0.2760678305,
1374 0.91,+0.2828932065,
1375 0.92,+0.2895577018,
1376 0.93,+0.2960641358,
1377 0.94,+0.3024152458,
1378 0.95,+0.3086136908,
1379 0.96,+0.3146620547,
1380 0.97,+0.3205628495,
1381 0.98,+0.3263185183,
1382 0.99,+0.3319314382,
1383
1384 1.00,0.3374039229,
1385 1.01,0.3427382254,
1386 1.02,0.3479365405,
1387 1.03,0.3530010067,
1388 1.04,0.3579337091,
1389 1.05,0.3627366810,
1390 1.06,0.3674119060,
1391 1.07,0.3719613201,
1392 1.08,0.3763868132,
1393 1.09,0.3806902312,
1394 1.10,0.3848733774,
1395 1.11,0.3889380142,
1396 1.12,0.3928858645,
1397 1.13,0.3967186134,
1398 1.14,0.4004379090,
1399 1.15,0.4040453647,
1400 1.16,0.4075425593,
1401 1.17,0.4109310390,
1402 1.18,0.4142123185,
1403 1.19,0.4173878816,
1404 1.20,0.4204591829,
1405 1.21,0.4234276482,
1406 1.22,0.4262946760,
1407 1.23,0.4290616379,
1408 1.24,0.4317298802,
1409 1.25,0.4343007240,
1410 1.26,0.4367754665,
1411 1.27,0.4391553815,
1412 1.28,0.4414417205,
1413 1.29,0.4436357130,
1414 1.30,0.4457385675,
1415 1.31,0.4477514723,
1416 1.32,0.4496755955,
1417 1.33,0.4515120863,
1418 1.34,0.4532620753,
1419 1.35,0.4549266752,
1420 1.36,0.4565069811,
1421 1.37,0.4580040711,
1422 1.38,0.4594190071,
1423 1.39,0.4607528349,
1424 1.40,0.4620065851,
1425 1.41,0.4631812730,
1426 1.42,0.4642778995,
1427 1.43,0.4652974513,
1428 1.44,0.4662409014,
1429 1.45,0.4671092094,
1430 1.46,0.4679033219,
1431 1.47,0.4686241732,
1432 1.48,0.4692726848,
1433 1.49,0.4698497667,
1434
1435 1.50,0.4703563172,
1436 1.51,0.4707932232,
1437 1.52,0.4711613608,
1438 1.53,0.4714615952,
1439 1.54,0.4716947815,
1440 1.55,0.4718617642,
1441 1.56,0.4719633785,
1442 1.57,0.4720004495,
1443 1.58,0.4719737932,
1444 1.59,0.4718842164,
1445 1.60,0.4717325169,
1446 1.61,0.4715194840,
1447 1.62,0.4712458984,
1448 1.63,0.4709125325,
1449 1.64,0.4705201507,
1450 1.65,0.4700695096,
1451 1.66,0.4695613580,
1452 1.67,0.4689964372,
1453 1.68,0.4683754812,
1454 1.69,0.4676992169,
1455 1.70,0.4669683642,
1456 1.71,0.4661836359,
1457 1.72,0.4653457385,
1458 1.73,0.4644553716,
1459 1.74,0.4635132286,
1460 1.75,0.4625199967,
1461 1.76,0.4614763568,
1462 1.77,0.4603829839,
1463 1.78,0.4592405471,
1464 1.79,0.4580497097,
1465 1.80,0.4568111294,
1466 1.81,0.4555254585,
1467 1.82,0.4541933436,
1468 1.83,0.4528154262,
1469 1.84,0.4513923427,
1470 1.85,0.4499247241,
1471 1.86,0.4484131966,
1472 1.87,0.4468583813,
1473 1.88,0.4452608948,
1474 1.89,0.4436213486,
1475 1.90,0.4419403497,
1476 1.91,0.4402185005,
1477 1.92,0.4384563991,
1478 1.93,0.4366546388,
1479 1.94,0.4348138088,
1480 1.95,0.4329344941,
1481 1.96,0.4310172752,
1482 1.97,0.4290627288,
1483 1.98,0.4270714273,
1484 1.99,0.4250439391,
1485
1486 2.0,+0.4229808288,
1487 2.1,+0.4005119878,
1488 2.2,+0.3750745990,
1489 2.3,+0.3471756175,
1490 2.4,+0.3172916174,
1491 2.5,+0.2858711964,
1492 2.6,+0.2533366161,
1493 2.7,+0.2200848786,
1494 2.8,+0.1864883896,
1495 2.9,+0.1528953242,
1496 3.0,+0.1196297860,
1497 3.1,+0.0869918312,
1498 3.2,+0.0552574117,
1499 3.3,+0.0246782846,
1500 3.4,-0.0045180779,
1501 3.5,-0.0321285485,
1502 3.6,-0.0579743519,
1503 3.7,-0.0819010013,
1504 3.8,-0.1037781504,
1505 3.9,-0.1234993492,
1506 4.0,-0.1409816979,
1507 4.1,-0.1561653918,
1508 4.2,-0.1690131568,
1509 4.3,-0.1795095725,
1510 4.4,-0.1876602868,
1511 4.5,-0.1934911221,
1512 4.6,-0.1970470797,
1513 4.7,-0.1983912468,
1514 4.8,-0.1976036133,
1515 4.9,-0.1947798060,
1516 5.0,-0.1900297497,
1517 5.1,-0.1834762632,
1518 5.2,-0.1752536023,
1519 5.3,-0.1655059586,
1520 5.4,-0.1543859262,
1521 5.5,-0.1420529476,
1522 5.6,-0.1286717494,
1523 5.7,-0.1144107808,
1524 5.8,-0.0994406647,
1525 5.9,-0.0839326741,
1526 6.0,-0.0680572439,
1527 6.1,-0.0519825290,
1528 6.2,-0.0358730193,
1529 6.3,-0.0198882206,
1530 6.4,-0.0041814110,
1531 6.5,+0.0111015195,
1532 6.6,+0.0258231381,
1533 6.7,+0.0398554400,
1534 6.8,+0.0530807167,
1535 6.9,+0.0653923140,
1536
1537 7.0,+0.0766952785,
1538 7.1,+0.0869068881,
1539 7.2,+0.0959570643,
1540 7.3,+0.1037886664,
1541 7.4,+0.1103576658,
1542 7.5,+0.1156332032,
1543 7.6,+0.1195975293,
1544 7.7,+0.1222458319,
1545 7.8,+0.1235859542,
1546 7.9,+0.1236380071,
1547 8.0,+0.1224338825,
1548 8.1,+0.1200166733,
1549 8.2,+0.1164400055,
1550 8.3,+0.1117672931,
1551 8.4,+0.1060709196,
1552 8.5,+0.0994313586,
1553 8.6,+0.0919362396,
1554 8.7,+0.0836793696,
1555 8.8,+0.0747597196,
1556 8.9,+0.0652803850,
1557 9.0,+0.0553475313,
1558 9.1,+0.0450693325,
1559 9.2,+0.0345549134,
1560 9.3,+0.0239133045,
1561 9.4,+0.0132524187,
1562 9.5,+0.0026780588,
1563 9.6,-0.0077070361,
1564 9.7,-0.0178040977,
1565 9.8,-0.0275191811,
1566 9.9,-0.0367639563,
1567 10.,-0.0454564330
1568 ]),done);
1569 done;
1570
1571 test_table(lambda([z],expintegral_ci(z)),'ci_2,230,3.2e-8);
1572 [];
1573
1574 /******************************************************************************
1575    A&S Table 5.1 p. 238, values for (Ei(x)-log(x)-%gamma)/x
1576                          from 0.00 through 0.50
1577 ******************************************************************************/
1578
1579 block(
1580 ei_1 : make_array(flonum,200,2),
1581 fillarray(ei_1,[
1582 0.00,1.000000000,
1583 0.01,1.002505566,
1584 0.02,1.005022306,
1585 0.03,1.007550283,
1586 0.04,1.010089560,
1587 0.05,1.012640202,
1588 0.06,1.015202272,
1589 0.07,1.017775836,
1590 0.08,1.020360958,
1591 0.09,1.022957705,
1592 0.10,1.025566141,
1593 0.11,1.028186335,
1594 0.12,1.030818352,
1595 0.13,1.033462259,
1596 0.14,1.036118125,
1597 0.15,1.038786018,
1598 0.16,1.041466006,
1599 0.17,1.044158158,
1600 0.18,1.046862544,
1601 0.19,1.049579234,
1602 0.20,1.052308298,
1603 0.21,1.055049807,
1604 0.22,1.057803833,
1605 0.23,1.060570446,
1606 0.24,1.063349719,
1607 0.25,1.066141726,
1608 0.26,1.068946539,
1609 0.27,1.071764232,
1610 0.28,1.074594879,
1611 0.29,1.077438555,
1612 0.30,1.080295334,
1613 0.31,1.083165293,
1614 0.32,1.086048507,
1615 0.33,1.088945053,
1616 0.34,1.091855008,
1617 0.35,1.094778451,
1618 0.36,1.097715458,
1619 0.37,1.100666108,
1620 0.38,1.103630481,
1621 0.39,1.106608656,
1622 0.40,1.109600714,
1623 0.41,1.112606735,
1624 0.42,1.115626800,
1625 0.43,1.118660991,
1626 0.44,1.121709391,
1627 0.45,1.124772082,
1628 0.46,1.127849147,
1629 0.47,1.130940671,
1630 0.48,1.134046738,
1631 0.49,1.137167432,
1632 0.50,1.140302841]),done);
1633 done;
1634
1635 test_table(lambda([z],(expintegral_ei(z)-log(z)-%gamma)/z),'ei_1,50,5.90e-10);
1636 [];
1637
1638 /******************************************************************************
1639    A&S Table 5.1 p. 239-241, values for Ei(x) from 0.50 through 2.00
1640 ******************************************************************************/
1641
1642 block(
1643 ei_2 : make_array(flonum,235,2),
1644 fillarray(ei_2,[
1645
1646 0.50,0.454219905,
1647 0.51,0.487032167,
1648 0.52,0.519530633,
1649 0.53,0.551730445,
1650 0.54,0.583645931,
1651 0.55,0.615290657,
1652 0.56,0.646677490,
1653 0.57,0.677818642,
1654 0.58,0.708725720,
1655 0.59,0.739409764,
1656 0.60,0.769881290,
1657 0.61,0.800150320,
1658 0.62,0.830226417,
1659 0.63,0.860118716,
1660 0.64,0.889835949,
1661 0.65,0.919386468,
1662 0.66,0.948778277,
1663 0.67,0.978019042,
1664 0.68,1.007116121,
1665 0.69,1.036076576,
1666 0.70,1.064907195,
1667 0.71,1.093614501,
1668 0.72,1.122204777,
1669 0.73,1.150684069,
1670 0.74,1.179058208,
1671 0.75,1.207332816,
1672 0.76,1.235513319,
1673 0.77,1.263604960,
1674 0.78,1.291612805,
1675 0.79,1.319541753,
1676 0.80,1.347396548,
1677 0.81,1.375181783,
1678 0.82,1.402901910,
1679 0.83,1.430561245,
1680 0.84,1.458163978,
1681 0.85,1.485714176,
1682 0.86,1.513215791,
1683 0.87,1.540672664,
1684 0.88,1.568088534,
1685 0.89,1.595467036,
1686 0.90,1.622811714,
1687 0.91,1.650126019,
1688 0.92,1.677413317,
1689 0.93,1.704676891,
1690 0.94,1.731919946,
1691 0.95,1.759145612,
1692 0.96,1.786356947,
1693 0.97,1.813556941,
1694 0.98,1.840748519,
1695 0.99,1.867934543,
1696
1697 1.00,1.895117816,
1698 1.01,1.922301085,
1699 1.02,1.949487042,
1700 1.03,1.976678325,
1701 1.04,2.003877525,
1702 1.05,2.031087184,
1703 1.06,2.058309800,
1704 1.07,2.085547825,
1705 1.08,2.112803672,
1706 1.09,2.140079712,
1707 1.10,2.167378280,
1708 1.11,2.194701672,
1709 1.12,2.222052152,
1710 1.13,2.249431949,
1711 1.14,2.276843260,
1712 1.15,2.304288252,
1713 1.16,2.331769062,
1714 1.17,2.359287800,
1715 1.18,2.386846549,
1716 1.19,2.414447367,
1717 1.20,2.442092285,
1718 1.21,2.469783315,
1719 1.22,2.497522442,
1720 1.23,2.525311634,
1721 1.24,2.553152836,
1722 1.25,2.581047974,
1723 1.26,2.608998956,
1724 1.27,2.637007673,
1725 1.28,2.665075997,
1726 1.29,2.693205785,
1727 1.30,2.721398880,
1728 1.31,2.749657110,
1729 1.32,2.777982287,
1730 1.33,2.806376214,
1731 1.34,2.834840677,
1732 1.35,2.863377453,
1733 1.36,2.891988308,
1734 1.37,2.920674997,
1735 1.38,2.949439263,
1736 1.39,2.978282844,
1737 1.40,3.007207464,
1738 1.41,3.036214843,
1739 1.42,3.065306691,
1740 1.43,3.094484712,
1741 1.44,3.123750601,
1742 1.45,3.153106049,
1743 1.46,3.182552741,
1744 1.47,3.212092355,
1745 1.48,3.241726566,
1746 1.49,3.271457042,
1747
1748 1.50,3.301285449,
1749 1.51,3.331213449,
1750 1.52,3.361242701,
1751 1.53,3.391374858,
1752 1.54,3.421611576,
1753 1.55,3.451954503,
1754 1.56,3.482405289,
1755 1.57,3.512965580,
1756 1.58,3.543637024,
1757 1.59,3.574421266,
1758 1.60,3.605319949,
1759 1.61,3.636334719,
1760 1.62,3.667467221,
1761 1.63,3.698719099,
1762 1.64,3.730091999,
1763 1.65,3.761587569,
1764 1.66,3.793207456,
1765 1.67,3.824953310,
1766 1.68,3.856826782,
1767 1.69,3.888829528,
1768 1.70,3.920963201,
1769 1.71,3.953229462,
1770 1.72,3.985629972,
1771 1.73,4.018166395,
1772 1.74,4.050840400,
1773 1.75,4.083653659,
1774 1.76,4.116607847,
1775 1.77,4.149704645,
1776 1.78,4.182945736,
1777 1.79,4.216332809,
1778 1.80,4.249867557,
1779 1.81,4.283551681,
1780 1.82,4.317386883,
1781 1.83,4.351374872,
1782 1.84,4.385517364,
1783 1.85,4.419816080,
1784 1.86,4.454272746,
1785 1.87,4.488889097,
1786 1.88,4.523666872,
1787 1.89,4.558607817,
1788 1.90,4.593713687,
1789 1.91,4.628986242,
1790 1.92,4.664427249,
1791 1.93,4.700038485,
1792 1.94,4.735821734,
1793 1.95,4.771778785,
1794 1.96,4.807911438,
1795 1.97,4.844221501,
1796 1.98,4.880710791,
1797 1.99,4.917381131,
1798 2.00,4.954234356
1799 ]),done);
1800 done;
1801
1802 test_table(lambda([z],expintegral_ei(z)),'ei_2,150,5.70e-10);
1803 [];
1804
1805 /******************************************************************************
1806   A&S Table 5.1 p. 242-243, values for x*%e^(-x)*Ei(x) from 2.00 through 10.0
1807 ******************************************************************************/
1808
1809 block(
1810 ei_3 : make_array(flonum,235,2),
1811 fillarray(ei_3,[
1812
1813 2.0,1.340965420,
1814 2.1,1.371486802,
1815 2.2,1.397421992,
1816 2.3,1.419171534,
1817 2.4,1.437118315,
1818 2.5,1.451625159,
1819 2.6,1.463033397,
1820 2.7,1.471662153,
1821 2.8,1.477808187,
1822 2.9,1.481746162,
1823 3.0,1.483729204,
1824 3.1,1.483989691,
1825 3.2,1.482740191,
1826 3.3,1.480174491,
1827 3.4,1.476468706,
1828 3.5,1.471782389,
1829 3.6,1.466259659,
1830 3.7,1.460030313,
1831 3.8,1.453210902,
1832 3.9,1.445905765,
1833 4.0,1.438208032,
1834 4.1,1.430200557,
1835 4.2,1.421956813,
1836 4.3,1.413541719,
1837 4.4,1.405012424,
1838 4.5,1.396419030,
1839 4.6,1.387805263,
1840 4.7,1.379209093,
1841 4.8,1.370663313,
1842 4.9,1.362196054,
1843 5.0,1.353831278,
1844 5.1,1.345589212,
1845 5.2,1.337486755,
1846 5.3,1.329537845,
1847 5.4,1.321753788,
1848 5.5,1.314143566,
1849 5.6,1.306714107,
1850 5.7,1.299470536,
1851 5.8,1.292416395,
1852 5.9,1.285553849,
1853 6.0,1.278883860,
1854 6.1,1.272406357,
1855 6.2,1.266120373,
1856 6.3,1.260024184,
1857 6.4,1.254115417,
1858 6.5,1.248391155,
1859 6.6,1.242848032,
1860 6.7,1.237482309,
1861 6.8,1.232289952,
1862 6.9,1.227266684,
1863 7.0,1.222408053,
1864 7.1,1.217709472,
1865 7.2,1.213166264,
1866 7.3,1.208773699,
1867 7.4,1.204527026,
1868 7.5,1.200421500,
1869 7.6,1.196452401,
1870 7.7,1.192615063,
1871 7.8,1.188904881,
1872 7.9,1.185317334,
1873 8.0,1.181847987,
1874 8.1,1.178492509,
1875 8.2,1.175246676,
1876 8.3,1.172106376,
1877 8.4,1.169067617,
1878 8.5,1.166126526,
1879 8.6,1.163279354,
1880 8.7,1.160522476,
1881 8.8,1.157852390,
1882 8.9,1.155265719,
1883 9.0,1.152759209,
1884 9.1,1.150329724,
1885 9.2,1.147974251,
1886 9.3,1.145689889,
1887 9.4,1.143473855,
1888 9.5,1.141323476,
1889 9.6,1.139236185,
1890 9.7,1.137209523,
1891 9.8,1.135241130,
1892 9.9,1.133328746,
1893 10.,1.131470205
1894 ]),done);
1895 done;
1896
1897 test_table(lambda([z],z*%e^(-z)*expintegral_ei(z)),'ei_3,80,8.95e-10);
1898 [];
1899
1900 /******************************************************************************
1901    A&S Table 5.1 p. 238, values for (E1(x)-log(x)-%gamma)/x
1902                          from 0.00 through 0.50
1903 ******************************************************************************/
1904
1905 block(
1906 e1_1 : make_array(flonum,200,2),
1907 fillarray(e1_1,[
1908
1909 0.00,1.0000000000,
1910 0.01,0.9975055452,
1911 0.02,0.9950221392,
1912 0.03,0.9925497201,
1913 0.04,0.9900882265,
1914 0.05,0.9876375971,
1915 0.06,0.9851977714,
1916 0.07,0.9827686889,
1917 0.08,0.9803502898,
1918 0.09,0.9779425142,
1919 0.10,0.9755453033,
1920 0.11,0.9731585980,
1921 0.12,0.9707823399,
1922 0.13,0.9684164710,
1923 0.14,0.9660609336,
1924 0.15,0.9637156702,
1925 0.16,0.9613806240,
1926 0.17,0.9590557383,
1927 0.18,0.9567409569,
1928 0.19,0.9544362237,
1929 0.20,0.9521414833,
1930 0.21,0.9498566804,
1931 0.22,0.9475817603,
1932 0.23,0.9453166684,
1933 0.24,0.9430613506,
1934 0.25,0.9408157528,
1935 0.26,0.9385798221,
1936 0.27,0.9363535046,
1937 0.28,0.9341367481,
1938 0.29,0.9319294997,
1939 0.30,0.9297317075,
1940 0.31,0.9275433196,
1941 0.32,0.9253642845,
1942 0.33,0.9231945510,
1943 0.34,0.9210340684,
1944 0.35,0.9188827858,
1945 0.36,0.9167406533,
1946 0.37,0.9146076209,
1947 0.38,0.9124836388,
1948 0.39,0.9103686582,
1949 0.40,0.9082626297,
1950 0.41,0.9061655048,
1951 0.42,0.9040772350,
1952 0.43,0.9019977725,
1953 0.44,0.8999270693,
1954 0.45,0.8978650778,
1955 0.46,0.8958117511,
1956 0.47,0.8937670423,
1957 0.48,0.8917309048,
1958 0.49,0.8897032920,
1959 0.50,0.8876841584
1960 ]),done);
1961 done;
1962
1963 test_table(lambda([z],(expintegral_e1(z)+log(z)+%gamma)/z),'e1_1,50,1.65e-10);
1964 [];
1965
1966 /******************************************************************************
1967    A&S Table 5.1 p. 239-241, values for E1(x) from 0.50 through 2.00
1968 ******************************************************************************/
1969
1970 block(
1971 e1_2 : make_array(flonum,200,2),
1972 fillarray(e1_2,[
1973
1974 0.50,0.559773595,
1975 0.51,0.547822352,
1976 0.52,0.536219798,
1977 0.53,0.524951510,
1978 0.54,0.514003886,
1979 0.55,0.503364081,
1980 0.56,0.493019959,
1981 0.57,0.482960034,
1982 0.58,0.473173433,
1983 0.59,0.463649849,
1984 0.60,0.454379503,
1985 0.61,0.445353112,
1986 0.62,0.436561854,
1987 0.63,0.427997338,
1988 0.64,0.419651581,
1989 0.65,0.411516976,
1990 0.66,0.403586275,
1991 0.67,0.395852563,
1992 0.68,0.388309243,
1993 0.69,0.380950010,
1994 0.70,0.373768843,
1995 0.71,0.366759981,
1996 0.72,0.359917914,
1997 0.73,0.353237364,
1998 0.74,0.346713279,
1999 0.75,0.340340813,
2000 0.76,0.334115321,
2001 0.77,0.328032346,
2002 0.78,0.322087610,
2003 0.79,0.316277004,
2004 0.80,0.310596579,
2005 0.81,0.305042539,
2006 0.82,0.299611236,
2007 0.83,0.294299155,
2008 0.84,0.289102918,
2009 0.85,0.284019269,
2010 0.86,0.279045070,
2011 0.87,0.274177301,
2012 0.88,0.269413046,
2013 0.89,0.264749496,
2014 0.90,0.260183939,
2015 0.91,0.255713758,
2016 0.92,0.251336425,
2017 0.93,0.247049501,
2018 0.94,0.242850627,
2019 0.95,0.238737524,
2020 0.96,0.234707988,
2021 0.97,0.230759890,
2022 0.98,0.226891167,
2023 0.99,0.223099826,
2024
2025 1.00,0.219383934,
2026 1.01,0.215741624,
2027 1.02,0.212171083,
2028 1.03,0.208670559,
2029 1.04,0.205238352,
2030 1.05,0.201872813,
2031 1.06,0.198572347,
2032 1.07,0.195335403,
2033 1.08,0.192160479,
2034 1.09,0.189046118,
2035 1.10,0.185990905,
2036 1.11,0.182993465,
2037 1.12,0.180052467,
2038 1.13,0.177166615,
2039 1.14,0.174334651,
2040 1.15,0.171555354,
2041 1.16,0.168827535,
2042 1.17,0.166150040,
2043 1.18,0.163521748,
2044 1.19,0.160941567,
2045 1.20,0.158408437,
2046 1.21,0.155921324,
2047 1.22,0.153479226,
2048 1.23,0.151081164,
2049 1.24,0.148726188,
2050 1.25,0.146413373,
2051 1.26,0.144141815,
2052 1.27,0.141910639,
2053 1.28,0.139718989,
2054 1.29,0.137566032,
2055 1.30,0.135450958,
2056 1.31,0.133372975,
2057 1.32,0.131331314,
2058 1.33,0.129325224,
2059 1.34,0.127353972,
2060 1.35,0.125416844,
2061 1.36,0.123513146,
2062 1.37,0.121642198,
2063 1.38,0.119803337,
2064 1.39,0.117995919,
2065 1.40,0.116219313,
2066 1.41,0.114472903,
2067 1.42,0.112756090,
2068 1.43,0.111068287,
2069 1.44,0.109408923,
2070 1.45,0.107777440,
2071 1.46,0.106173291,
2072 1.47,0.104595946,
2073 1.48,0.103044882,
2074 1.49,0.101519593,
2075
2076 1.50,0.100019582,
2077 1.51,0.098544365,
2078 1.52,0.097093466,
2079 1.53,0.095666424,
2080 1.54,0.094262786,
2081 1.55,0.092882108,
2082 1.56,0.091523960,
2083 1.57,0.090187917,
2084 1.58,0.088873566,
2085 1.59,0.087580504,
2086 1.60,0.086308334,
2087 1.61,0.085056670,
2088 1.62,0.083825133,
2089 1.63,0.082613354,
2090 1.64,0.081420970,
2091 1.65,0.080247627,
2092 1.66,0.079092978,
2093 1.67,0.077956684,
2094 1.68,0.076838412,
2095 1.69,0.075737839,
2096 1.70,0.074654644,
2097 1.71,0.073588518,
2098 1.72,0.072539154,
2099 1.73,0.071506255,
2100 1.74,0.070489527,
2101 1.75,0.069488685,
2102 1.76,0.068503447,
2103 1.77,0.067533539,
2104 1.78,0.066578691,
2105 1.79,0.065638641,
2106 1.80,0.064713129,
2107 1.81,0.063801903,
2108 1.82,0.062904715,
2109 1.83,0.062021320,
2110 1.84,0.061151482,
2111 1.85,0.060294967,
2112 1.86,0.059451545,
2113 1.87,0.058620994,
2114 1.88,0.057803091,
2115 1.89,0.056997623,
2116 1.90,0.056204378,
2117 1.91,0.055423149,
2118 1.92,0.054653731,
2119 1.93,0.053895927,
2120 1.94,0.053149540,
2121 1.95,0.052414380,
2122 1.96,0.051690257,
2123 1.97,0.050976988,
2124 1.98,0.050274392,
2125 1.99,0.049582291,
2126 2.00,0.048900511
2127 ]),done);
2128 done;
2129
2130 test_table(lambda([z],expintegral_e1(z)),'e1_2,150,5.35e-10);
2131 [];
2132
2133 /******************************************************************************
2134    A&S Table 5.1 p. 242-243, values for x*%e^x*E1(x) from 2.00 through 10.0
2135 ******************************************************************************/
2136
2137 block(
2138 e1_3 : make_array(flonum,200,2),
2139 fillarray(e1_3,[
2140
2141 2.0,0.722657234,
2142 2.1,0.730791502,
2143 2.2,0.738431132,
2144 2.3,0.745622149,
2145 2.4,0.752404829,
2146 2.5,0.758814592,
2147 2.6,0.764882722,
2148 2.7,0.770636987,
2149 2.8,0.776102123,
2150 2.9,0.781300252,
2151 3.0,0.786251221,
2152 3.1,0.790972900,
2153 3.2,0.795481422,
2154 3.3,0.799791408,
2155 3.4,0.803916127,
2156 3.5,0.807867661,
2157 3.6,0.811657037,
2158 3.7,0.815294342,
2159 3.8,0.818788821,
2160 3.9,0.822148967,
2161 4.0,0.825382600,
2162 4.1,0.828496926,
2163 4.2,0.831498602,
2164 4.3,0.834393794,
2165 4.4,0.837188207,
2166 4.5,0.839887144,
2167 4.6,0.842495539,
2168 4.7,0.845017971,
2169 4.8,0.847458721,
2170 4.9,0.849821778,
2171 5.0,0.852110880,
2172 5.1,0.854329519,
2173 5.2,0.856480958,
2174 5.3,0.858568275,
2175 5.4,0.860594348,
2176 5.5,0.862561885,
2177 5.6,0.864473436,
2178 5.7,0.866331399,
2179 5.8,0.868138040,
2180 5.9,0.869895494,
2181 6.0,0.871605775,
2182 6.1,0.873270793,
2183 6.2,0.874892347,
2184 6.3,0.876472150,
2185 6.4,0.878011816,
2186 6.5,0.879512881,
2187 6.6,0.880976797,
2188 6.7,0.882404955,
2189 6.8,0.883798662,
2190 6.9,0.885159176,
2191 7.0,0.886487675,
2192 7.1,0.887785294,
2193 7.2,0.889053119,
2194 7.3,0.890292173,
2195 7.4,0.891503440,
2196 7.5,0.892687854,
2197 7.6,0.893846312,
2198 7.7,0.894979666,
2199 7.8,0.896088737,
2200 7.9,0.897174302,
2201 8.0,0.898237113,
2202 8.1,0.899277888,
2203 8.2,0.900297306,
2204 8.3,0.901296033,
2205 8.4,0.902274699,
2206 8.5,0.903233900,
2207 8.6,0.904174228,
2208 8.7,0.905096235,
2209 8.8,0.906000459,
2210 8.9,0.906887415,
2211 9.0,0.907757602,
2212 9.1,0.908611483,
2213 9.2,0.909449530,
2214 9.3,0.910272177,
2215 9.4,0.911079850,
2216 9.5,0.911872958,
2217 9.6,0.912651897,
2218 9.7,0.913417043,
2219 9.8,0.914168766,
2220 9.9,0.914907418,
2221 10.,0.915633339
2222 ]),done);
2223 done;
2224
2225 test_table(lambda([z],z*%e^z*expintegral_e1(z)),'e1_3,80,2.50e-8);
2226 [];
2227
2228 /******************************************************************************
2229    A&S Table 5.6 p. 249-251, values for z*%e^z*E1(z) for Complex values
2230                              from -19 through 20 for the Real part and
2231                              from 0 through 20 for the Complex part
2232 ******************************************************************************/
2233
2234 block(
2235 ec_1 : make_array(flonum,855,4),
2236 fillarray(ec_1,[
2237
2238 /* Table 5.6 p. 249 */
2239
2240 -19,00,1.059305,0.000000,
2241 -18,00,1.063087,0.000001,
2242 -17,00,1.067394,0.000002,
2243 -16,00,1.072345,0.000006,
2244 -15,00,1.078103,0.000014,
2245
2246 -19,01,1.059090,0.003539,
2247 -18,01,1.062827,0.004010,
2248 -17,01,1.067073,0.004584,
2249 -16,01,1.071942,0.005296,
2250 -15,01,1.077584,0.006195,
2251
2252 -19,02,1.058456,0.007000,
2253 -18,02,1.062061,0.007918,
2254 -17,02,1.066135,0.009032,
2255 -16,02,1.070774,0.010403,
2256 -15,02,1.076102,0.012118,
2257
2258 -19,03,1.057431,0.010310,
2259 -18,03,1.060829,0.011633,
2260 -17,03,1.064636,0.013226,
2261 -16,03,1.068925,0.015172,
2262 -15,03,1.073783,0.017579,
2263
2264 -19,04,1.056058,0.013410,
2265 -18,04,1.059190,0.015079,
2266 -17,04,1.062657,0.017075,
2267 -16,04,1.066508,0.019486,
2268 -15,04,1.070793,0.022432,
2269
2270 -19,05,1.054391,0.016252,
2271 -18,05,1.057215,0.018202,
2272 -17,05,1.060297,0.020512,
2273 -16,05,1.063659,0.023272,
2274 -15,05,1.067318,0.026598,
2275
2276 -19,06,1.052490,0.018806,
2277 -18,06,1.054981,0.020969,
2278 -17,06,1.057655,0.023505,
2279 -16,06,1.060510,0.026499,
2280 -15,06,1.063538,0.030055,
2281
2282 -19,07,1.050413,0.021055,
2283 -18,07,1.052565,0.023364,
2284 -17,07,1.054829,0.026044,
2285 -16,07,1.057187,0.029167,
2286 -15,07,1.059610,0.032823,
2287
2288 -19,08,1.048217,0.022996,
2289 -18,08,1.050037,0.025391,
2290 -17,08,1.051905,0.028141,
2291 -16,08,1.053795,0.031306,
2292 -15,08,1.055664,0.034957,
2293
2294 -19,09,1.045956,0.024637,
2295 -18,09,1.047458,0.027066,
2296 -17,09,1.048958,0.029824,
2297 -16,09,1.050421,0.032960,
2298 -15,09,1.051797,0.036527,
2299
2300 -19,10,1.043672,0.025993,
2301 -18,10,1.044880,0.028412,
2302 -17,10,1.046045,0.031130,
2303 -16,10,1.047129,0.034183,
2304 -15,10,1.048081,0.037609,
2305
2306 -19,11,1.041402,0.027086,
2307 -18,11,1.042345,0.029461,
2308 -17,11,1.043212,0.032102,
2309 -16,11,1.043967,0.035034,
2310 -15,11,1.044559,0.038282,
2311
2312 -19,12,1.039177,0.027940,
2313 -18,12,1.039882,0.030245,
2314 -17,12,1.040490,0.032781,
2315 -16,12,1.040965,0.035567,
2316 -15,12,1.041259,0.038616,
2317
2318 -19,13,1.037018,0.028581,
2319 -18,13,1.037515,0.030796,
2320 -17,13,1.037901,0.033211,
2321 -16,13,1.038140,0.035836,
2322 -15,13,1.038192,0.038677,
2323
2324 -19,14,1.034942,0.029034,
2325 -18,14,1.035259,0.031148,
2326 -17,14,1.035456,0.033431,
2327 -16,14,1.035501,0.035888,
2328 -15,14,1.035359,0.038520,
2329
2330 -19,15,1.032959,0.029326,
2331 -18,15,1.033123,0.031330,
2332 -17,15,1.033162,0.033476,
2333 -16,15,1.033049,0.035765,
2334 -15,15,1.032754,0.038193,
2335
2336 -19,16,1.031076,0.029477,
2337 -18,16,1.031110,0.031368,
2338 -17,16,1.031019,0.033377,
2339 -16,16,1.030780,0.035502,
2340 -15,16,1.030365,0.037735,
2341
2342 -19,17,1.029296,0.029511,
2343 -18,17,1.029222,0.031288,
2344 -17,17,1.029025,0.033162,
2345 -16,17,1.028685,0.035129,
2346 -15,17,1.028180,0.037179,
2347
2348 -19,18,1.027620,0.029445,
2349 -18,18,1.027456,0.031110,
2350 -17,18,1.027174,0.032855,
2351 -16,18,1.026756,0.034672,
2352 -15,18,1.026183,0.036552,
2353
2354 -19,19,1.026046,0.029296,
2355 -18,19,1.025809,0.030854,
2356 -17,19,1.025459,0.032474,
2357 -16,19,1.024981,0.034150,
2358 -15,19,1.024360,0.035873,
2359
2360 -19,20,1.024570,0.029080,
2361 -18,20,1.024275,0.030534,
2362 -17,20,1.023872,0.032037,
2363 -16,20,1.023349,0.033582,
2364 -15,20,1.022695,0.035160,
2365
2366 -14,00,1.084892,0.000037,
2367 -13,00,1.093027,0.000092,
2368 -12,00,1.102975,0.000232,
2369 -11,00,1.115431,0.000577,
2370 -10,00,1.131470,0.001426,
2371
2372 -14,01,1.084200,0.007359,
2373 -13,01,1.092067,0.008913,
2374 -12,01,1.101566,0.011063,
2375 -11,01,1.113230,0.014169,
2376 -10,01,1.127796,0.018879,
2377
2378 -14,02,1.082276,0.014306,
2379 -13,02,1.089498,0.017161,
2380 -12,02,1.098025,0.020981,
2381 -11,02,1.108170,0.026241,
2382 -10,02,1.120286,0.033700,
2383
2384 -14,03,1.079313,0.020604,
2385 -13,03,1.085635,0.024471,
2386 -12,03,1.092873,0.029507,
2387 -11,03,1.101137,0.036189,
2388 -10,03,1.110462,0.045218,
2389
2390 -14,04,1.075560,0.026075,
2391 -13,04,1.080853,0.030637,
2392 -12,04,1.086686,0.036422,
2393 -11,04,1.093013,0.043843,
2394 -10,04,1.099666,0.053451,
2395
2396 -14,05,1.071279,0.030642,
2397 -13,05,1.075522,0.035599,
2398 -12,05,1.079985,0.041724,
2399 -11,05,1.084526,0.049336,
2400 -10,05,1.088877,0.058817,
2401
2402 -14,06,1.066708,0.034303,
2403 -13,06,1.069960,0.039405,
2404 -12,06,1.073185,0.045552,
2405 -11,06,1.076197,0.052967,
2406 -10,06,1.078701,0.061886,
2407
2408 -14,07,1.062046,0.037117,
2409 -13,07,1.064412,0.042169,
2410 -12,07,1.066578,0.048115,
2411 -11,07,1.068350,0.055093,
2412 -10,07,1.069450,0.063225,
2413
2414 -14,08,1.057448,0.039174,
2415 -13,08,1.059054,0.044041,
2416 -12,08,1.060352,0.049644,
2417 -11,08,1.061159,0.056057,
2418 -10,08,1.061235,0.063322,
2419
2420 -14,09,1.053021,0.040580,
2421 -13,09,1.053997,0.045176,
2422 -12,09,1.054606,0.050359,
2423 -11,09,1.054687,0.056158,
2424 -10,09,1.054046,0.062566,
2425
2426 -14,10,1.048834,0.041444,
2427 -13,10,1.049303,0.045719,
2428 -12,10,1.049380,0.050452,
2429 -11,10,1.048933,0.055640,
2430 -10,10,1.047807,0.061249,
2431
2432 -14,11,1.044928,0.041867,
2433 -13,11,1.044997,0.045801,
2434 -12,11,1.044674,0.050084,
2435 -11,11,1.043853,0.054695,
2436 -10,11,1.042417,0.059584,
2437
2438 -14,12,1.041320,0.041938,
2439 -13,12,1.041080,0.045531,
2440 -12,12,1.040464,0.049384,
2441 -11,12,1.039389,0.053465,
2442 -10,12,1.037766,0.057719,
2443
2444 -14,13,1.038010,0.041734,
2445 -13,13,1.037537,0.044999,
2446 -12,13,1.036713,0.048452,
2447 -11,13,1.035473,0.052056,
2448 -10,13,1.033752,0.055758,
2449
2450 -14,14,1.034989,0.041321,
2451 -13,14,1.034344,0.044277,
2452 -12,14,1.033378,0.047365,
2453 -11,14,1.032040,0.050547,
2454 -10,14,1.030282,0.053773,
2455
2456 -14,15,1.032241,0.040751,
2457 -13,15,1.031474,0.043422,
2458 -12,15,1.030414,0.046180,
2459 -11,15,1.029026,0.048991,
2460 -10,15,1.027274,0.051808,
2461
2462 -14,16,1.029747,0.040066,
2463 -13,16,1.028895,0.042477,
2464 -12,16,1.027781,0.044941,
2465 -11,16,1.026377,0.047428,
2466 -10,16,1.024658,0.049894,
2467
2468 -14,17,1.027486,0.039301,
2469 -13,17,1.026579,0.041475,
2470 -12,17,1.025438,0.043679,
2471 -11,17,1.024043,0.045883,
2472 -10,17,1.022375,0.048049,
2473
2474 -14,18,1.025437,0.038481,
2475 -13,18,1.024499,0.040444,
2476 -12,18,1.023352,0.042417,
2477 -11,18,1.021981,0.044374,
2478 -10,18,1.020375,0.046282,
2479
2480 -14,19,1.023580,0.037629,
2481 -13,19,1.022628,0.039401,
2482 -12,19,1.021489,0.041170,
2483 -11,19,1.020155,0.042912,
2484 -10,19,1.018617,0.044599,
2485
2486 -14,20,1.021896,0.036759,
2487 -13,20,1.020942,0.038361,
2488 -12,20,1.019824,0.039950,
2489 -11,20,1.018533,0.041505,
2490 -10,20,1.017066,0.043001,
2491
2492 -09,00,1.152759,0.003489,
2493 -08,00,1.181848,0.008431,
2494 -07,00,1.222408,0.020053,
2495 -06,00,1.278884,0.046723,
2496 -05,00,1.353831,0.105839,
2497
2498 -09,01,1.146232,0.026376,
2499 -08,01,1.169677,0.038841,
2500 -07,01,1.199049,0.060219,
2501 -06,01,1.233798,0.097331,
2502 -05,01,1.268723,0.160826,
2503
2504 -09,02,1.134679,0.044579,
2505 -08,02,1.151385,0.060814,
2506 -07,02,1.169639,0.085335,
2507 -06,02,1.186778,0.122162,
2508 -05,02,1.196351,0.175646,
2509
2510 -09,03,1.120694,0.057595,
2511 -08,03,1.131255,0.074701,
2512 -07,03,1.140733,0.098259,
2513 -06,03,1.146266,0.130005,
2514 -05,03,1.142853,0.170672,
2515
2516 -09,04,1.106249,0.065948,
2517 -08,04,1.111968,0.082156,
2518 -07,04,1.115404,0.102861,
2519 -06,04,1.114273,0.128440,
2520 -05,04,1.105376,0.158134,
2521
2522 -09,05,1.092564,0.070592,
2523 -08,05,1.094818,0.085055,
2524 -07,05,1.094475,0.102411,
2525 -06,05,1.089952,0.122397,
2526 -05,05,1.079407,0.143879,
2527
2528 -09,06,1.080246,0.072520,
2529 -08,06,1.080188,0.084987,
2530 -07,06,1.077672,0.099188,
2531 -06,06,1.071684,0.114638,
2532 -05,06,1.061236,0.130280,
2533
2534 -09,07,1.069494,0.072580,
2535 -08,07,1.067987,0.083120,
2536 -07,07,1.064339,0.094618,
2537 -06,07,1.057935,0.106568,
2538 -05,07,1.048279,0.118116,
2539
2540 -09,08,1.060276,0.071425,
2541 -08,08,1.057920,0.080250,
2542 -07,08,1.053778,0.089537,
2543 -06,08,1.047493,0.098840,
2544 -05,08,1.038838,0.107508,
2545
2546 -09,09,1.052450,0.069523,
2547 -08,09,1.049645,0.076885,
2548 -07,09,1.045382,0.084405,
2549 -06,09,1.039464,0.091717,
2550 -05,09,1.031806,0.098337,
2551
2552 -09,10,1.045832,0.067197,
2553 -08,10,1.042834,0.073340,
2554 -07,10,1.038659,0.079462,
2555 -06,10,1.033205,0.085271,
2556 -05,10,1.026459,0.090413,
2557
2558 -09,11,1.040241,0.064664,
2559 -08,11,1.037210,0.069803,
2560 -07,11,1.033231,0.074821,
2561 -06,11,1.028260,0.079488,
2562 -05,11,1.022317,0.083544,
2563
2564 -09,12,1.035508,0.062063,
2565 -08,12,1.032539,0.066381,
2566 -07,12,1.028808,0.070524,
2567 -06,12,1.024300,0.074315,
2568 -05,12,1.019052,0.077561,
2569
2570 -09,13,1.031490,0.059482,
2571 -08,13,1.028638,0.063128,
2572 -07,13,1.025171,0.066576,
2573 -06,13,1.021090,0.069688,
2574 -05,13,1.016439,0.072320,
2575
2576 -09,14,1.028065,0.056975,
2577 -08,14,1.025359,0.060070,
2578 -07,14,1.022152,0.062962,
2579 -06,14,1.018458,0.065542,
2580 -05,14,1.014319,0.067702,
2581
2582 -09,15,1.025132,0.054573,
2583 -08,15,1.022583,0.057215,
2584 -07,15,1.019626,0.059658,
2585 -06,15,1.016277,0.061817,
2586 -05,15,1.012577,0.063610,
2587
2588 -09,16,1.022608,0.052291,
2589 -08,16,1.020219,0.054559,
2590 -07,16,1.017494,0.056638,
2591 -06,16,1.014452,0.058460,
2592 -05,16,1.011130,0.059962,
2593
2594 -09,17,1.020426,0.050135,
2595 -08,17,1.018192,0.052094,
2596 -07,17,1.015681,0.053874,
2597 -06,17,1.012912,0.055424,
2598 -05,17,1.009915,0.056694,
2599
2600 -09,18,1.018530,0.048106,
2601 -08,18,1.016444,0.049806,
2602 -07,18,1.014129,0.051341,
2603 -06,18,1.011600,0.052670,
2604 -05,18,1.008887,0.053752,
2605
2606 -09,19,1.016874,0.046201,
2607 -08,19,1.014929,0.047684,
2608 -07,19,1.012790,0.049015,
2609 -06,19,1.010476,0.050161,
2610 -05,19,1.008009,0.051092,
2611
2612 -09,20,1.015422,0.044413,
2613 -08,20,1.013607,0.045714,
2614 -07,20,1.011629,0.046875,
2615 -06,20,1.009505,0.047870,
2616 -05,20,1.007254,0.048675,
2617
2618 /* Table 5.6 p. 250 */
2619
2620 -4,00,1.438208,0.230161,
2621 -3,00,1.483729,0.469232,
2622 -2,00,1.340965,0.850337,
2623 -1,00,0.697175,1.155727,
2624 /* -0,00,0.577216,0.000000, den Test auf Null verbessern */
2625
2626 -4,01,1.287244,0.263705,
2627 -3,01,1.251069,0.410413,
2628 -2,01,1.098808,0.561916,
2629 -1,01,0.813486,0.578697,
2630 -0,01,0.621450,0.343378,
2631
2632 -4,02,1.185758,0.247356,
2633 -3,02,1.136171,0.328439,
2634 -2,02,1.032990,0.388428,
2635 -1,02,0.896419,0.378838,
2636 -0,02,0.798042,0.289091,
2637
2638 -4,03,1.123282,0.217835,
2639 -3,03,1.080316,0.262814,
2640 -2,03,1.013205,0.289366,
2641 -1,03,0.936283,0.280906,
2642 -0,03,0.875873,0.237665,
2643
2644 -4,04,1.085153,0.189003,
2645 -3,04,1.051401,0.215118,
2646 -2,04,1.006122,0.228399,
2647 -1,04,0.957446,0.222612,
2648 -0,04,0.916770,0.198713,
2649
2650 -4,05,1.061263,0.164466,
2651 -3,05,1.035185,0.180487,
2652 -2,05,1.003172,0.187857,
2653 -1,05,0.969809,0.183963,
2654 -0,05,0.940714,0.169481,
2655
2656 -4,06,1.045719,0.144391,
2657 -3,06,1.025396,0.154746,
2658 -2,06,1.001788,0.159189,
2659 -1,06,0.977582,0.156511,
2660 -0,06,0.955833,0.147129,
2661
2662 -4,07,1.035205,0.128073,
2663 -3,07,1.019109,0.135079,
2664 -2,07,1.001077,0.137939,
2665 -1,07,0.982756,0.136042,
2666 -0,07,0.965937,0.129646,
2667
2668 -4,08,1.027834,0.114732,
2669 -3,08,1.014861,0.119660,
2670 -2,08,1.000684,0.121599,
2671 -1,08,0.986356,0.120218,
2672 -0,08,0.972994,0.115678,
2673
2674 -4,09,1.022501,0.103711,
2675 -3,09,1.011869,0.107294,
2676 -2,09,1.000454,0.108665,
2677 -1,09,0.988955,0.107634,
2678 -0,09,0.978103,0.104303,
2679
2680 -4,10,1.018534,0.094502,
2681 -3,10,1.009688,0.097181,
2682 -2,10,1.000312,0.098184,
2683 -1,10,0.990887,0.097396,
2684 -0,10,0.981910,0.094885,
2685
2686 -4,11,1.015513,0.086718,
2687 -3,11,1.008052,0.088770,
2688 -2,11,1.000221,0.089525,
2689 -1,11,0.992361,0.088911,
2690 -0,11,0.984819,0.086975,
2691
2692 -4,12,1.013163,0.080069,
2693 -3,12,1.006795,0.081673,
2694 -2,12,1.000161,0.082255,
2695 -1,12,0.993508,0.081769,
2696 -0,12,0.987088,0.080245,
2697
2698 -4,13,1.011303,0.074333,
2699 -3,13,1.005809,0.075609,
2700 -2,13,1.000119,0.076067,
2701 -1,13,0.994418,0.075676,
2702 -0,13,0.988891,0.074457,
2703
2704 -4,14,1.009806,0.069340,
2705 -3,14,1.005022,0.070371,
2706 -2,14,1.000090,0.070738,
2707 -1,14,0.995151,0.070419,
2708 -0,14,0.990345,0.069429,
2709
2710 -4,15,1.008585,0.064959,
2711 -3,15,1.004384,0.065803,
2712 -2,15,1.000070,0.066102,
2713 -1,15,0.995751,0.065838,
2714 -0,15,0.991534,0.065024,
2715
2716 -4,16,1.007577,0.061086,
2717 -3,16,1.003859,0.061786,
2718 -2,16,1.000055,0.062032,
2719 -1,16,0.996246,0.061812,
2720 -0,16,0.992518,0.061135,
2721
2722 -4,17,1.006735,0.057640,
2723 -3,17,1.003423,0.058227,
2724 -2,17,1.000043,0.058432,
2725 -1,17,0.996661,0.058246,
2726 -0,17,0.993342,0.057677,
2727
2728 -4,18,1.006025,0.054555,
2729 -3,18,1.003057,0.055052,
2730 -2,18,1.000035,0.055224,
2731 -1,18,0.997011,0.055066,
2732 -0,18,0.994038,0.054583,
2733
2734 -4,19,1.005420,0.051779,
2735 -3,19,1.002747,0.052202,
2736 -2,19,1.000028,0.052349,
2737 -1,19,0.997309,0.052214,
2738 -0,19,0.994631,0.051801,
2739
2740 -4,20,1.004902,0.049267,
2741 -3,20,1.002481,0.049631,
2742 -2,20,1.000023,0.049757,
2743 -1,20,0.997565,0.049640,
2744 -0,20,0.995140,0.049284,
2745
2746 01,00,0.596347,0.000000,
2747 02,00,0.722657,0.000000,
2748 03,00,0.786251,0.000000,
2749 04,00,0.825383,0.000000,
2750 05,00,0.852111,0.000000,
2751
2752 01,01,0.673321,0.147864,
2753 02,01,0.747012,0.075661,
2754 03,01,0.797036,0.045686,
2755 04,01,0.831126,0.030619,
2756 05,01,0.855544,0.021985,
2757
2758 01,02,0.777514,0.186570,
2759 02,02,0.796965,0.118228,
2760 03,02,0.823055,0.078753,
2761 04,02,0.846097,0.055494,
2762 05,02,0.864880,0.040999,
2763
2764 01,03,0.847468,0.181226,
2765 02,03,0.844361,0.132252,
2766 03,03,0.853176,0.096659,
2767 04,03,0.865521,0.072180,
2768 05,03,0.877860,0.055341,
2769
2770 01,04,0.891460,0.165207,
2771 02,04,0.881036,0.131686,
2772 03,04,0.880584,0.103403,
2773 04,04,0.885308,0.081408,
2774 05,04,0.892143,0.064825,
2775
2776 01,05,0.919826,0.148271,
2777 02,05,0.907873,0.125136,
2778 03,05,0.903152,0.103577,
2779 04,05,0.903231,0.085187,
2780 05,05,0.906058,0.070209,
2781
2782 01,06,0.938827,0.132986,
2783 02,06,0.927384,0.116656,
2784 03,06,0.921006,0.100357,
2785 04,06,0.918527,0.085460,
2786 05,06,0.918708,0.072544,
2787
2788 01,07,0.952032,0.119807,
2789 02,07,0.941722,0.107990,
2790 03,07,0.934958,0.095598,
2791 04,07,0.931209,0.083666,
2792 05,07,0.929765,0.072792,
2793
2794 01,08,0.961512,0.108589,
2795 02,08,0.952435,0.099830,
2796 03,08,0.945868,0.090303,
2797 04,08,0.941594,0.080755,
2798 05,08,0.939221,0.071700,
2799
2800 01,09,0.968512,0.099045,
2801 02,09,0.960582,0.092408,
2802 03,09,0.954457,0.084986,
2803 04,09,0.950072,0.077313,
2804 05,09,0.947219,0.069799,
2805
2806 01,10,0.973810,0.090888,
2807 02,10,0.966885,0.085758,
2808 03,10,0.961283,0.079898,
2809 04,10,0.957007,0.073688,
2810 05,10,0.953955,0.067447,
2811
2812 01,11,0.977904,0.083871,
2813 02,11,0.971842,0.079836,
2814 03,11,0.966766,0.075147,
2815 04,11,0.962708,0.070080,
2816 05,11,0.959626,0.064878,
2817
2818 01,12,0.981127,0.077790,
2819 02,12,0.975799,0.074567,
2820 03,12,0.971216,0.070769,
2821 04,12,0.967423,0.066599,
2822 05,12,0.964412,0.062242,
2823
2824 01,13,0.983706,0.072484,
2825 02,13,0.979000,0.069873,
2826 03,13,0.974865,0.066762,
2827 04,13,0.971351,0.063300,
2828 05,13,0.968464,0.059630,
2829
2830 01,14,0.985799,0.067822,
2831 02,14,0.981621,0.065679,
2832 03,14,0.977888,0.063104,
2833 04,14,0.974646,0.060206,
2834 05,14,0.971911,0.057096,
2835
2836 01,15,0.987519,0.063698,
2837 02,15,0.983791,0.061921,
2838 03,15,0.980414,0.059767,
2839 04,15,0.977430,0.057322,
2840 05,15,0.974858,0.054671,
2841
2842 01,16,0.988949,0.060029,
2843 02,16,0.985606,0.058539,
2844 03,16,0.982544,0.056723,
2845 04,16,0.979799,0.054644,
2846 05,16,0.977391,0.052371,
2847
2848 01,17,0.990149,0.056745,
2849 02,17,0.987138,0.055485,
2850 03,17,0.984353,0.053941,
2851 04,17,0.981827,0.052162,
2852 05,17,0.979579,0.050200,
2853
2854 01,18,0.991167,0.053792,
2855 02,18,0.988442,0.052717,
2856 03,18,0.985902,0.051394,
2857 04,18,0.983574,0.049861,
2858 05,18,0.981478,0.048160,
2859
2860 01,19,0.992036,0.051122,
2861 02,19,0.989561,0.050199,
2862 03,19,0.987237,0.049057,
2863 04,19,0.985089,0.047728,
2864 05,19,0.983135,0.046245,
2865
2866 01,20,0.992784,0.048699,
2867 02,20,0.990527,0.047900,
2868 03,20,0.988395,0.046909,
2869 04,20,0.986410,0.045749,
2870 05,20,0.984587,0.044449,
2871
2872 06,00,0.871606,0.000000,
2873 07,00,0.886488,0.000000,
2874 08,00,0.898237,0.000000,
2875 09,00,0.907758,0.000000,
2876 10,00,0.915633,0.000000,
2877
2878 06,01,0.873827,0.016570,
2879 07,01,0.888009,0.012947,
2880 08,01,0.899327,0.010401,
2881 09,01,0.908565,0.008543,
2882 10,01,0.916249,0.007143,
2883
2884 06,02,0.880023,0.031454,
2885 07,02,0.892327,0.024866,
2886 08,02,0.902453,0.020140,
2887 09,02,0.910901,0.016639,
2888 10,02,0.918040,0.013975,
2889
2890 06,03,0.889029,0.043517,
2891 07,03,0.898793,0.034995,
2892 08,03,0.907236,0.028693,
2893 09,03,0.914531,0.023921,
2894 10,03,0.920856,0.020230,
2895
2896 06,04,0.899484,0.052380,
2897 07,04,0.906591,0.042967,
2898 08,04,0.913167,0.035755,
2899 09,04,0.919127,0.030145,
2900 10,04,0.924479,0.025717,
2901
2902 06,05,0.910242,0.058259,
2903 07,05,0.914952,0.048780,
2904 08,05,0.919729,0.041242,
2905 09,05,0.924336,0.035208,
2906 10,05,0.928664,0.030334,
2907
2908 06,06,0.920534,0.061676,
2909 07,06,0.923283,0.052667,
2910 08,06,0.926481,0.045242,
2911 09,06,0.929836,0.039123,
2912 10,06,0.933175,0.034063,
2913
2914 06,07,0.929945,0.063220,
2915 07,07,0.931193,0.054971,
2916 08,07,0.933096,0.047942,
2917 09,07,0.935365,0.041986,
2918 10,07,0.937807,0.036944,
2919
2920 06,08,0.938313,0.063425,
2921 07,08,0.938469,0.056047,
2922 08,08,0.939359,0.049570,
2923 09,08,0.940731,0.043936,
2924 10,08,0.942398,0.039060,
2925
2926 06,09,0.945629,0.062714,
2927 07,09,0.945023,0.056211,
2928 08,09,0.945154,0.050349,
2929 09,09,0.945812,0.045128,
2930 10,09,0.946833,0.040514,
2931
2932 06,10,0.951965,0.061408,
2933 07,10,0.950850,0.055725,
2934 08,10,0.950427,0.050481,
2935 09,10,0.950535,0.045711,
2936 10,10,0.951035,0.041413,
2937
2938 06,11,0.957427,0.059735,
2939 07,11,0.955987,0.054790,
2940 08,11,0.955176,0.050135,
2941 09,11,0.954870,0.045818,
2942 10,11,0.954959,0.041861,
2943
2944 06,12,0.962128,0.057855,
2945 07,12,0.960495,0.053560,
2946 08,12,0.959421,0.049444,
2947 09,12,0.958814,0.045563,
2948 10,12,0.958586,0.041948,
2949
2950 06,13,0.966178,0.055877,
2951 07,13,0.964444,0.052146,
2952 08,13,0.963201,0.048514,
2953 09,13,0.962379,0.045038,
2954 10,13,0.961913,0.041755,
2955
2956 06,14,0.969673,0.053874,
2957 07,14,0.967903,0.050627,
2958 08,14,0.966559,0.047425,
2959 09,14,0.965591,0.044319,
2960 10,14,0.964949,0.041347,
2961
2962 06,15,0.972699,0.051894,
2963 07,15,0.970935,0.049062,
2964 08,15,0.969539,0.046236,
2965 09,15,0.968477,0.043463,
2966 10,15,0.967710,0.040780,
2967
2968 06,16,0.975326,0.049966,
2969 07,16,0.973597,0.047489,
2970 08,16,0.972185,0.044992,
2971 09,16,0.971067,0.042516,
2972 10,16,0.970214,0.040095,
2973
2974 06,17,0.977617,0.048109,
2975 07,17,0.975940,0.045935,
2976 08,17,0.974538,0.043724,
2977 09,17,0.973393,0.041512,
2978 10,17,0.972484,0.039329,
2979
2980 06,18,0.979622,0.046332,
2981 07,18,0.978009,0.044419,
2982 08,18,0.976632,0.042456,
2983 09,18,0.975481,0.040477,
2984 10,18,0.974540,0.038508,
2985
2986 06,19,0.981384,0.044641,
2987 07,19,0.979839,0.042951,
2988 08,19,0.978500,0.041205,
2989 09,19,0.977357,0.039431,
2990 10,19,0.976402,0.037653,
2991
2992 06,20,0.982938,0.043036,
2993 07,20,0.981465,0.041538,
2994 08,20,0.980169,0.039980,
2995 09,20,0.979047,0.038388,
2996 10,20,0.978090,0.036781,
2997
2998 /* Table 5.6 p. 251 */
2999
3000 11,00,0.922260,0.000000,
3001 12,00,0.927914,0.000000,
3002 13,00,0.932796,0.000000,
3003 14,00,0.937055,0.000000,
3004 15,00,0.940804,0.000000,
3005
3006 11,01,0.922740,0.006063,
3007 12,01,0.928295,0.005212,
3008 13,01,0.933105,0.004528,
3009 14,01,0.937308,0.003972,
3010 15,01,0.941014,0.003512,
3011
3012 11,02,0.924143,0.011902,
3013 12,02,0.929416,0.010258,
3014 13,02,0.934013,0.008932,
3015 14,02,0.938055,0.007847,
3016 15,02,0.941636,0.006949,
3017
3018 11,03,0.926370,0.017321,
3019 12,03,0.931205,0.014991,
3020 13,03,0.935473,0.013098,
3021 14,03,0.939261,0.011540,
3022 15,03,0.942643,0.010242,
3023
3024 11,04,0.929270,0.022171,
3025 12,04,0.933560,0.019295,
3026 13,04,0.937408,0.016934,
3027 14,04,0.940870,0.014974,
3028 15,04,0.943994,0.013331,
3029
3030 11,05,0.932672,0.026361,
3031 12,05,0.936356,0.023091,
3032 13,05,0.939729,0.020373,
3033 14,05,0.942816,0.018095,
3034 15,05,0.945640,0.016169,
3035
3036 11,06,0.936400,0.029857,
3037 12,06,0.939462,0.026339,
3038 13,06,0.942338,0.023378,
3039 14,06,0.945024,0.020867,
3040 15,06,0.947522,0.018725,
3041
3042 11,07,0.940297,0.032670,
3043 12,07,0.942757,0.029036,
3044 13,07,0.945140,0.025934,
3045 14,07,0.947419,0.023273,
3046 15,07,0.949582,0.020980,
3047
3048 11,08,0.944229,0.034847,
3049 12,08,0.946132,0.031205,
3050 13,08,0.948047,0.028052,
3051 14,08,0.949933,0.025315,
3052 15,08,0.951765,0.022931,
3053
3054 11,09,0.948093,0.036453,
3055 12,09,0.949500,0.032887,
3056 13,09,0.950985,0.029756,
3057 14,09,0.952502,0.027004,
3058 15,09,0.954018,0.024582,
3059
3060 11,10,0.951816,0.037566,
3061 12,10,0.952792,0.034134,
3062 13,10,0.953895,0.031081,
3063 14,10,0.955075,0.028365,
3064 15,10,0.956296,0.025949,
3065
3066 11,11,0.955347,0.038261,
3067 12,11,0.955958,0.035004,
3068 13,11,0.956729,0.032068,
3069 14,11,0.957610,0.029426,
3070 15,11,0.958563,0.027052,
3071
3072 11,12,0.958659,0.038612,
3073 12,12,0.958968,0.035552,
3074 13,12,0.959454,0.032761,
3075 14,12,0.960073,0.030221,
3076 15,12,0.960787,0.027915,
3077
3078 11,13,0.961739,0.038684,
3079 12,13,0.961800,0.035833,
3080 13,13,0.962049,0.033201,
3081 14,13,0.962443,0.030781,
3082 15,13,0.962947,0.028564,
3083
3084 11,14,0.964583,0.038534,
3085 12,14,0.964447,0.035893,
3086 13,14,0.964499,0.033428,
3087 14,14,0.964702,0.031140,
3088 15,14,0.965026,0.029024,
3089
3090 11,15,0.967199,0.038211,
3091 12,15,0.966907,0.035775,
3092 13,15,0.966799,0.033479,
3093 14,15,0.966843,0.031327,
3094 15,15,0.967011,0.029320,
3095
3096 11,16,0.969597,0.037756,
3097 12,16,0.969184,0.035515,
3098 13,16,0.968947,0.033384,
3099 14,16,0.968860,0.031370,
3100 15,16,0.968897,0.029476,
3101
3102 11,17,0.971789,0.037200,
3103 12,17,0.971285,0.035144,
3104 13,17,0.970946,0.033172,
3105 14,17,0.970752,0.031293,
3106 15,17,0.970680,0.029512,
3107
3108 11,18,0.973792,0.036572,
3109 12,18,0.973220,0.034687,
3110 13,18,0.972802,0.032865,
3111 14,18,0.972521,0.031117,
3112 15,18,0.972359,0.029448,
3113
3114 11,19,0.975621,0.035893,
3115 12,19,0.974999,0.034166,
3116 13,19,0.974521,0.032485,
3117 14,19,0.974172,0.030862,
3118 15,19,0.973936,0.029301,
3119
3120 11,20,0.977290,0.035179,
3121 12,20,0.976634,0.033597,
3122 13,20,0.976112,0.032049,
3123 14,20,0.975709,0.030542,
3124 15,20,0.975414,0.029086,
3125
3126 16,00,0.944130,0.000000,
3127 17,00,0.947100,0.000000,
3128 18,00,0.949769,0.000000,
3129 19,00,0.952181,0.000000,
3130 20,00,0.954371,0.000000,
3131
3132 16,01,0.944306,0.003128,
3133 17,01,0.947250,0.002804,
3134 18,01,0.949897,0.002527,
3135 19,01,0.952291,0.002290,
3136 20,01,0.954467,0.002085,
3137
3138 16,02,0.944829,0.006196,
3139 17,02,0.947693,0.005560,
3140 18,02,0.950277,0.005016,
3141 19,02,0.952619,0.004549,
3142 20,02,0.954752,0.004144,
3143
3144 16,03,0.945678,0.009150,
3145 17,03,0.948416,0.008223,
3146 18,03,0.950898,0.007430,
3147 19,03,0.953156,0.006745,
3148 20,03,0.955219,0.006151,
3149
3150 16,04,0.946824,0.011940,
3151 17,04,0.949395,0.010754,
3152 18,04,0.951741,0.009735,
3153 19,04,0.953887,0.008853,
3154 20,04,0.955856,0.008084,
3155
3156 16,05,0.948226,0.014529,
3157 17,05,0.950600,0.013121,
3158 18,05,0.952782,0.011904,
3159 19,05,0.954793,0.010847,
3160 20,05,0.956650,0.009922,
3161
3162 16,06,0.949842,0.016886,
3163 17,06,0.951995,0.015296,
3164 18,06,0.953995,0.013916,
3165 19,06,0.955853,0.012709,
3166 20,06,0.957581,0.011649,
3167
3168 16,07,0.951624,0.018994,
3169 17,07,0.953545,0.017265,
3170 18,07,0.955349,0.015753,
3171 19,07,0.957043,0.014425,
3172 20,07,0.958631,0.013253,
3173
3174 16,08,0.953527,0.020847,
3175 17,08,0.955212,0.019019,
3176 18,08,0.956815,0.017409,
3177 19,08,0.958337,0.015986,
3178 20,08,0.959779,0.014723,
3179
3180 16,09,0.955509,0.022445,
3181 17,09,0.956960,0.020555,
3182 18,09,0.958363,0.018878,
3183 19,09,0.959712,0.017387,
3184 20,09,0.961004,0.016056,
3185
3186 16,10,0.957530,0.023797,
3187 17,10,0.958758,0.021878,
3188 18,10,0.959966,0.020163,
3189 19,10,0.961144,0.018628,
3190 20,10,0.962288,0.017250,
3191
3192 16,11,0.959559,0.024917,
3193 17,11,0.960576,0.022998,
3194 18,11,0.961598,0.021270,
3195 19,11,0.962612,0.019712,
3196 20,11,0.963611,0.018305,
3197
3198 16,12,0.961568,0.025823,
3199 17,12,0.962391,0.023927,
3200 18,12,0.963238,0.022207,
3201 19,12,0.964097,0.020645,
3202 20,12,0.964956,0.019227,
3203
3204 16,13,0.963534,0.026534,
3205 17,13,0.964181,0.024679,
3206 18,13,0.964868,0.022984,
3207 19,13,0.965582,0.021436,
3208 20,13,0.966310,0.020021,
3209
3210 16,14,0.965443,0.027070,
3211 17,14,0.965931,0.025271,
3212 18,14,0.966472,0.023616,
3213 19,14,0.967052,0.022094,
3214 20,14,0.967658,0.020694,
3215
3216 16,15,0.967280,0.027453,
3217 17,15,0.967628,0.025720,
3218 18,15,0.968039,0.024114,
3219 19,15,0.968496,0.022629,
3220 20,15,0.968990,0.021255,
3221
3222 16,16,0.969038,0.027700,
3223 17,16,0.969264,0.026041,
3224 18,16,0.969558,0.024493,
3225 19,16,0.969906,0.023052,
3226 20,16,0.970297,0.021712,
3227
3228 16,17,0.970712,0.027831,
3229 17,17,0.970832,0.026249,
3230 18,17,0.971023,0.024765,
3231 19,17,0.971273,0.023375,
3232 20,17,0.971571,0.022075,
3233
3234 16,18,0.972300,0.027862,
3235 17,18,0.972328,0.026361,
3236 18,18,0.972430,0.024943,
3237 19,18,0.972594,0.023607,
3238 20,18,0.972808,0.022352,
3239
3240 16,19,0.973800,0.027809,
3241 17,19,0.973751,0.026388,
3242 18,19,0.973775,0.025038,
3243 19,19,0.973863,0.023760,
3244 20,19,0.974004,0.022552,
3245
3246 16,20,0.975215,0.027685,
3247 17,20,0.975099,0.026343,
3248 18,20,0.975057,0.025062,
3249 19,20,0.975079,0.023842,
3250 20,20,0.975155,0.022684
3251 ]),done);
3252 done;
3253
3254 test_complex_table(lambda([z],z*%e^z*expintegral_e1(z)),'ec_1,838,1.25e-6);
3255 [];
3256
3257 kill(all);
3258 done;
3259
3260 /*****************************************************************************/