# Eikonal Blog

## 2010.08.06

### Calculating e

Filed under: mathematics — Tags: , — sandokan65 @ 09:48

Related at this blog: Pi calculations – https://eikonal.wordpress.com/2010/01/05/pi-calculations/

## 2010.03.09

### Transcendental numbers: Euler’s constant (= Euler-Mascheroni constant) gamma

Filed under: mathematics — Tags: , , — sandokan65 @ 14:25

Source: http://www.wolframalpha.com/

Euler’s constant (= Euler-Mascheroni constant):
$\gamma =$ $0.577, 215, 664, 901, 532, 860, 606, 512, 090, 082, 402, 431, 042, 159, 335, 939, 923, 598, 805, 767, 234, 884, 867, 726, 777, 664, 670, 936, 947, 063, 291, 746, 749, 514, 631, 447, 249, 807, 082, 480, 960, 504, 014, 486, 542, 836, 224, 173, 997, 644, 923, 536, 253, 500, 333, 742, 937, 337, 737, 673, 942, 792, 595, 258, 247, 094, 916, 008, 735, 203, 948, 165, 670, 853, 233, 151, 776, 611, 528, 621, 199, 501, 507, 984, 793, 745, 085, 705, 740, 029, 921, 354, 786, 146, 694, 029, 604, 325, 421, 519, 058, 775, 535, 267, 331, 399, 254, 012, 967, 420, 513, 754, 139, 549, 111, 685, 102, 807, 984, 234, 877, 587, 205, 038, 431, 093, 997, 361, 372, 553, 060, 889, 331, 267, 600, 172, 479, 537, 836, 759, 271, 351, 577, 226, 102, 734, 929, 139, 407, 984, 301, 034, 177, 717, 780, 881, 549, 570, 661, 075, 010, 161, 916, 633, 401, 522, 789, 358, 679, 654, 972, 520, 362, 128, 792, 265, 559, 536, 696, 281, 763, 887, 927, 268, 013, 243, 101, 047, 650, 596, 370, 394, 739, 495, 763, 890, 657, 296, 792, 960, 100, 901, 512, 519, 595, 092, 224, 350, 140, 934, 987, 122, 824, 794, 974, 719, 564, 697, 631, 850, 667, 612, 906, 381, 105, 182, 419, 744, 486, 783, 638, 086, 174, 945, 516, 989, 279, 230, 187, 739, 107, 294, 578, 155, 431, 600, 500, 218, 284, 409, 605, 377, 243, 420, 328, 547, 836, 701, 517, 739, 439, 870, 030, 237, 033, 951, 832, 869, 000, 155, 819, 398, 804, 270, 741, 154, 222, 781, 971, 652, 301, 107, 356, 583, 396, 734, 871, 765, 049, 194, 181, 230, 004, 065, 469, 314, 299, 929, 777, 956, 930, 310, 050, 308, 630, 341, 856, 980, 323, 108, 369, 164, 002, 589, 297, 089, 098, 548, 682, 577, 736, 428, 825, 395, 492, 587, 362, 959, 613, 329, 857, 473, 930, 237, 343, 884, 707, 037, 028, 441, 292, 016, 641, 785, 024, 873, 337, 908, 056, 275, 499, 843, 459, 076, 164, 316, 710, 314, 671, 072, 237, 002, 181, 074, 504, 441, 866, 475, 913, 480, 366, 902, 553, 245, 862, 544, 222, 534, 518, 138, 791, 243, 457, 350, 136, 129, 778, 227, 828, 814, 894, 590, 986, 384, 600, 629, 316, 947, 188, 714, 958, 752, 549, 236, 649, 352, 047, 324, 364, 109, 726, 827, 616, 087, 759, 508, 809, 512, 620, 840, 454, 447, 799, 229, 915, 724, 829, 251, 625, 127, 842, 765, 965, 708, 321, 461, 029, 821, 461, 795, 195, 795, 909, 592, 270, 420, 898, 962, 797, 125, 536, 321, 794, 887, 376, 421, 066, 060, 706, 598, 256, 199, 010, 288, 075, 612, 519, 913, 751, 167, 821, 764, 361, 905, 705, 844, 078, 357, 350, 158, 005, 607, 745, 793, 421, 314, 498, 850, 078, 641, 517, 161, 519, 456, 570, 617, 043, 245, 075, 008, 168, 705, 230, 789, 093, 704, 614, 306, 684, 817, 916, 496, 842, 549, 150, 496, 724, 312, 183, 783, 875, 356, 489, 495, 086, 845, 410, 234, 060, 162, 250, 851, 558, 386, 723, 494, 418, 788, 044, 094, 077, 010, 688, 379, 511, 130, 787, 202, 342, 639, 522, 692, 097, 160, 885, 690, 838, 251, 137, 871, 283, 682, 049, 117, 892, 594, 478, 486, 199, 118, 529, 391, 029, 309, 905, 925, 526, 691, 727, 446, 892, 044, 386, 971, 114, 717, 457, 157, 457, 320, 393, 520, 912, 231, 608, 508, 682, 755, 889, 010, 945, 168, 118, 101, 687, 497, 547, 096, 936, 667, 121, 020, 630, 482, 716, 589, 504, 932, 731, 486, 087, 494, 020, 700, 674, 259, 091, 824, 875, 962, 137, 384, 231, 144, 265, 313, 502, 923, 031, 751, 722, 572, 216, 283, 248, 838, 112, 458, 957, 438, 623, 987, 037, 576, 628, 551, 303, 314, 392, 999, 540, 185, 313, 414, 158, 621, 278, 864, 807, 611, 003, 015, 211, 965, 780, 068, 117, 773, 763, 501, 681, 838, 973, 389, 663, 986, 895, 793, 299, 145, 638, 864, 431, 037, 060, 807, 817, 448, 995, 795, 832, 457, 941, 896, 202, 604, 984, 104, 392, 250, 786, 046, 036, 252, 772, 602, 291, 968, 299, 586, 098, 833, 901, 378, 717, 142, 269, 178, 838, 195, 298, 445, 607, 916, 051, 972, 797, 360, 475, 910, 251, 099, 577, 913, 351, 579, 177, 225, 150, 254, 929, 324, 632, 502, 874, 767, 794, 842, 158, 405, 075, 992, 904, 018, 557, 645, 990, 186, 269, 267, 764, 372, 660, 571, 176, 813, 365, 590, 881, 554, 810, 747, 000, 062, 336, 372, 528, 894, 955, 463, 697, 143, 301, 200, 791, 308, 555, 263, 959, 549, 782, 302, 314, 403, 914, 974, 049, 474, 682, 594, 732, 084, 618, 524, 605, 877, 669, 488, 287, 953, 010, 406, 349, 172, 292, 185, 800, 870, 677, 069, 042, 792, 674, 328, 444, 696, 851, 497, 182, 567, 809, 584, 165, 449, 185, 145, 753, 319, 640, 633, 119, 937, 382, 157, 345, 087, 498, 832, 556, 088, 887, 352, 801, 901, 915, 508, 968, 855, 468, 259, 245, 444, 527, 728, 173, 057, 301, 080, 606, 177, 011, 363, 773, 182, 462, 924, 660, 081, 277, 162, 101, 867, 744, 684, 959, 514, 281, 790, 145, 111, 948, 934, 228, 834, 482, 530, 753, 118, 701, 860, 976, 122, 462, 317, 674, 977, 556, 412, 461, 983, 856, 401, 484, 123, 587, 177, 249, 554, 224, 820, 161, 517, 657, 994, 080, 629, 683, 424, 289, 057, 259, 473, 926, 963, 863, 383, 874, 380, 547, 131, 967, 642, 926, 837, 249, 076, 087, 507, 378, 528, 370, 230, 468, 650, 349, 051, 203, 422, 721, 743, 668, 979, 284, 862, 972, 908, 892, 678, 977, 703, 262, 462, 391, 226, 188, 876, 530, 057, 786, 274, 360, 609, 444, 360, 392, 809, 770, 813, 383, 693, 423, 550, 858, 394, 112, 670, 921, 873, 441, 451, 218, 780, 327, 615, 050, 947, 805, 546, 630, 058, 684, 556, 315, 245, 460, 531, 511, 325, 281, 889, 107, 923, 149, 131, 103, 234, 430, 245, 093, 345, 000, 307, 655, 864, 874, 222, 971, 770, 033, 178, 453, 915, 056, 694, 015, 998, 849, 291, 609, 114, 002, 948, 690, 208, 848, 538, 169, 700, 955, 156, 634, 705, 544, 522, 176, 403, 586, 293, 982, 865, 813, 123, 870, 132, 535, 880, 062, 568, 662, 692, 699, 776, 773, 773, 068, 322, 690, 091, 608, 510, 451, 500, 226, 107, 180, 255, 465, 928, 493, 894, 927, 759, 589, 754, 076, 155, 993, 378, 264, 824, 197, 950, 641, 868, 143, 788, 171, 850, 885, 408, 036, 799, 631, 423, 954, 009, 196, 438, 875, 007, 890, 000, 062, 799, 794, 280, 988, 637, 299, 259, 197, 776, 504, 040, 992, 203, 794, 042, 761, 681, 783, 715, 668, 653, 066, 939, 830, 916, 524, 322, 705, 955, 304, 176, 673, 664, 011, 679, 295, 901, 293, 053, 744, 971, 830, 800, 427, 584, 863, 508, 380, 804, 246, 673, 509, 355, 983, 232, 411, 696, 921, 486, 064, 989, 276, 362, 443, 295, 885, 487, 378, 970, 148, 971, 334, 353, 844, 800, 289, 046, 665, 090, 284, 537, 689, 622, 398, 304, 881, 406, 273, 054, 087, 959, 118, 967, 05...$

For definition and some properties see the separate posting https://eikonal.wordpress.com/2010/01/04/euler-mascheroni-constant/.

### Transcendental numbers: pi

Filed under: mathematics — Tags: , — sandokan65 @ 14:23

Source: http://www.wolframalpha.com/

$\pi =$ $3.141, 592, 653, 589, 793, 238, 462, 643, 383, 279, 502, 884, 197, 169, 399, 375, 105, 820, 974, 944, 592, 307, 816, 406, 286, 208, 998, 628, 034, 825, 342, 117, 067, 982, 148, 086, 513, 282, 306, 647, 093, 844, 609, 550, 582, 231, 725, 359, 408, 128, 481, 117, 450, 284, 102, 701, 938, 521, 105, 559, 644, 622, 948, 954, 930, 381, 964, 428, 810, 975, 665, 933, 446, 128, 475, 648, 233, 786, 783, 165, 271, 201, 909, 145, 648, 566, 923, 460, 348, 610, 454, 326, 648, 213, 393, 607, 260, 249, 141, 273, 724, 587, 006, 606, 315, 588, 174, 881, 520, 920, 962, 829, 254, 091, 715, 364, 367, 892, 590, 360, 011, 330, 530, 548, 820, 466, 521, 384, 146, 951, 941, 511, 609, 433, 057, 270, 365, 759, 591, 953, 092, 186, 117, 381, 932, 611, 793, 105, 118, 548, 074, 462, 379, 962, 749, 567, 351, 885, 752, 724, 891, 227, 938, 183, 011, 949, 129, 833, 673, 362, 440, 656, 643, 086, 021, 394, 946, 395, 224, 737, 190, 702, 179, 860, 943, 702, 770, 539, 217, 176, 293, 176, 752, 384, 674, 818, 467, 669, 405, 132, 000, 568, 127, 145, 263, 560, 827, 785, 771, 342, 757, 789, 609, 173, 637, 178, 721, 468, 440, 901, 224, 953, 430, 146, 549, 585, 371, 050, 792, 279, 689, 258, 923, 542, 019, 956, 112, 129, 021, 960, 864, 034, 418, 159, 813, 629, 774, 771, 309, 960, 518, 707, 211, 349, 999, 998, 372, 978, 049, 951, 059, 731, 732, 816, 096, 318, 595, 024, 459, 455, 346, 908, 302, 642, 522, 308, 253, 344, 685, 035, 261, 931, 188, 171, 010, 003, 137, 838, 752, 886, 587, 533, 208, 381, 420, 617, 177, 669, 147, 303, 598, 253, 490, 428, 755, 468, 731, 159, 562, 863, 882, 353, 787, 593, 751, 957, 781, 857, 780, 532, 171, 226, 806, 613, 001, 927, 876, 611, 195, 909, 216, 420, 198, 938, 095, 257, 201, 065, 485, 863, 278, 865, 936, 153, 381, 827, 968, 230, 301, 952, 035, 301, 852, 968, 995, 773, 622, 599, 413, 891, 249, 721, 775, 283, 479, 131, 515, 574, 857, 242, 454, 150, 695, 950, 829, 533, 116, 861, 727, 855, 889, 075, 098, 381, 754, 637, 464, 939, 319, 255, 060, 400, 927, 701, 671, 139, 009, 848, 824, 012, 858, 361, 603, 563, 707, 660, 104, 710, 181, 942, 955, 596, 198, 946, 767, 837, 449, 448, 255, 379, 774, 726, 847, 104, 047, 534, 646, 208, 046, 684, 259, 069, 491, 293, 313, 677, 028, 989, 152, 104, 752, 162, 056, 966, 024, 058, 038, 150, 193, 511, 253, 382, 430, 035, 587, 640, 247, 496, 473, 263, 914, 199, 272, 604, 269, 922, 796, 782, 354, 781, 636, 009, 341, 721, 641, 219, 924, 586, 315, 030, 286, 182, 974, 555, 706, 749, 838, 505, 494, 588, 586, 926, 995, 690, 927, 210, 797, 509, 302, 955, 321, 165, 344, 987, 202, 755, 960, 236, 480, 665, 499, 119, 881, 834, 797, 753, 566, 369, 807, 426, 542, 527, 862, 551, 818, 417, 574, 672, 890, 977, 772, 793, 800, 081, 647, 060, 016, 145, 249, 192, 173, 217, 214, 772, 350, 141, 441, 973, 568, 548, 161, 361, 157, 352, 552, 133, 475, 741, 849, 468, 438, 523, 323, 907, 394, 143, 334, 547, 762, 416, 862, 518, 983, 569, 485, 562, 099, 219, 222, 184, 272, 550, 254, 256, 887, 671, 790, 494, 601, 653, 466, 804, 988, 627, 232, 791, 786, 085, 784, 383, 827, 967, 976, 681, 454, 100, 953, 883, 786, 360, 950, 680, 064, 225, 125, 205, 117, 392, 984, 896, 084, 128, 488, 626, 945, 604, 241, 965, 285, 022, 210, 661, 186, 306, 744, 278, 622, 039, 194, 945, 047, 123, 713, 786, 960, 956, 364, 371, 917, 287, 467, 764, 657, 573, 962, 413, 890, 865, 832, 645, 995, 813, 390, 478, 027, 590, 099, 465, 764, 078, 951, 269, 468, 398, 352, 595, 709, 825, 822, 620, 522, 489, 407, 726, 719, 478, 268, 482, 601, 476, 990, 902, 640, 136, 394, 437, 455, 305, 068, 203, 496, 252, 451, 749, 399, 651, 431, 429, 809, 190, 659, 250, 937, 221, 696, 461, 515, 709, 858, 387, 410, 597, 885, 959, 772, 975, 498, 930, 161, 753, 928, 468, 138, 268, 683, 868, 942, 774, 155, 991, 855, 925, 245, 953, 959, 431, 049, 972, 524, 680, 845, 987, 273, 644, 695, 848, 653, 836, 736, 222, 626, 099, 124, 608, 051, 243, 884, 390, 451, 244, 136, 549, 762, 780, 797, 715, 691, 435, 997, 700, 129, 616, 089, 441, 694, 868, 555, 848, 406, 353, 422, 072, 225, 828, 488, 648, 158, 456, 028, 506, 016, 842, 739, 452, 267, 467, 678, 895, 252, 138, 522, 549, 954, 666, 727, 823, 986, 456, 596, 116, 354, 886, 230, 577, 456, 498, 035, 593, 634, 568, 174, 324, 112, 515, 076, 069, 479, 451, 096, 596, 094, 025, 228, 879, 710, 893, 145, 669, 136, 867, 228, 748, 940, 560, 101, 503, 308, 617, 928, 680, 920, 874, 760, 917, 824, 938, 589, 009, 714, 909, 675, 985, 261, 365, 549, 781, 893, 129, 784, 821, 682, 998, 948, 722, 658, 804, 857, 564, 014, 270, 477, 555, 132, 379, 641, 451, 523, 746, 234, 364, 542, 858, 444, 795, 265, 867, 821, 051, 141, 354, 735, 739, 523, 113, 427, 166, 102, 135, 969, 536, 231, 442, 952, 484, 937, 187, 110, 145, 765, 403, 590, 279, 934, 403, 742, 007, 310, 578, 539, 062, 198, 387, 447, 808, 478, 489, 683, 321, 445, 713, 868, 751, 943, 506, 430, 218, 453, 191, 048, 481, 005, 370, 614, 680, 674, 919, 278, 191, 197, 939, 952, 061, 419, 663, 428, 754, 440, 643, 745, 123, 718, 192, 179, 998, 391, 015, 919, 561, 814, 675, 142, 691, 239, 748, 940, 907, 186, 494, 231, 961, 567, 945, 208, 095, 146, 550, 225, 231, 603, 881, 930, 142, 093, 762, 137, 855, 956, 638, 937, 787, 083, 039, 069, 792, 077, 346, 722, 182, 562, 599, 661, 501, 421, 503, 068, 038, 447, 734, 549, 202, 605, 414, 665, 925, 201, 497, 442, 850, 732, 518, 666, 002, 132, 434, 088, 190, 710, 486, 331, 734, 649, 651, 453, 905, 796, 268, 561, 005, 508, 106, 658, 796, 998, 163, 574, 736, 384, 052, 571, 459, 102, 897, 064, 140, 110, 971, 206, 280, 439, 039, 759, 515, 677, 157, 700, 420, 337, 869, 936, 007, 230, 558, 763, 176, 359, 421, 873, 125, 147, 120, 532, 928, 191, 826, 186, 125, 867, 321, 579, 198, 414, 848, 829, 164, 470, 609, 575, 270, 695, 722, 091, 756, 711, 672, 291, 098, 169, 091, 528, 017, 350, 671, 274, 858, 322, 287, 183, 520, 935, 396, 572, 512, 108, 357, 915, 136, 988, 209, 144, 421, 006, 751, 033, 467, 110, 314, 126, 711, 136, 990, 865, 851, 639, 831, 501, 970, 165, 151, 168, 517, 143, 765, 761, 835, 155, 650, 884, 909, 989, 859, 982, 387, 345, 528, 331, 635, 507, 647, 918, 535, 893, 226, 185, 489, 632, 132, 933, 089, 857, 064, 204, 675, 259, 070, 915, 481, 416, 549, 859, 461, 637, 180, 270, 981, 994, 309, 924, 488, 957, 571, 282, 890, 592, 323, 326, 097, 29...$

### Transcedental numbers: e

Filed under: mathematics — Tags: , — sandokan65 @ 14:21

Source: http://www.wolframalpha.com/

$e = 2.718, 281, 828, 459, 045, 235, 360, 287, 471, 352, 662, 497, 757, 247, 093, 699, 959, 574, 966, 967, 627, 724, 076, 630, 353, 547, 594, 571, 382, 178, 525, 166, 427, 427, 466, 391, 932, 003, 059, 921, 817, 413, 596, 629, 043, 572, 900, 334, 295, 260, 595, 630, 738, 132, 328, 627, 943, 490, 763, 233, 829, 880, 753, 195, 251, 019, 011, 573, 834, 187, 930, 702, 154, 089, 149, 934, 884, 167, 509, 244, 761, 460, 668, 082, 264, 800, 168, 477, 411, 853, 742, 345, 442, 437, 107, 539, 077, 744, 992, 069, 551, 702, 761, 838, 606, 261, 331, 384, 583, 000, 752, 044, 933, 826, 560, 297, 606, 737, 113, 200, 709, 328, 709, 127, 443, 747, 047, 230, 696, 977, 209, 310, 141, 692, 836, 819, 025, 515, 108, 657, 463, 772, 111, 252, 389, 784, 425, 056, 953, 696, 770, 785, 449, 969, 967, 946, 864, 454, 905, 987, 931, 636, 889, 230, 098, 793, 127, 736, 178, 215, 424, 999, 229, 576, 351, 482, 208, 269, 895, 193, 668, 033, 182, 528, 869, 398, 496, 465, 105, 820, 939, 239, 829, 488, 793, 320, 362, 509, 443, 117, 301, 238, 197, 068, 416, 140, 397, 019, 837, 679, 320, 683, 282, 376, 464, 804, 295, 311, 802, 328, 782, 509, 819, 455, 815, 301, 756, 717, 361, 332, 069, 811, 250, 996, 181, 881, 593, 041, 690, 351, 598, 888, 519, 345, 807, 273, 866, 738, 589, 422, 879, 228, 499, 892, 086, 805, 825, 749, 279, 610, 484, 198, 444, 363, 463, 244, 968, 487, 560, 233, 624, 827, 041, 978, 623, 209, 002, 160, 990, 235, 304, 369, 941, 849, 146, 314, 093, 431, 738, 143, 640, 546, 253, 152, 096, 183, 690, 888, 707, 016, 768, 396, 424, 378, 140, 592, 714, 563, 549, 061, 303, 107, 208, 510, 383, 750, 510, 115, 747, 704, 171, 898, 610, 687, 396, 965, 521, 267, 154, 688, 957, 035, 035, 402, 123, 407, 849, 819, 334, 321, 068, 170, 121, 005, 627, 880, 235, 193, 033, 224, 745, 015, 853, 904, 730, 419, 957, 777, 093, 503, 660, 416, 997, 329, 725, 088, 687, 696, 640, 355, 570, 716, 226, 844, 716, 256, 079, 882, 651, 787, 134, 195, 124, 665, 201, 030, 592, 123, 667, 719, 432, 527, 867, 539, 855, 894, 489, 697, 096, 409, 754, 591, 856, 956, 380, 236, 370, 162, 112, 047, 742, 722, 836, 489, 613, 422, 516, 445, 078, 182, 442, 352, 948, 636, 372, 141, 740, 238, 893, 441, 247, 963, 574, 370, 263, 755, 294, 448, 337, 998, 016, 125, 492, 278, 509, 257, 782, 562, 092, 622, 648, 326, 277, 933, 386, 566, 481, 627, 725, 164, 019, 105, 900, 491, 644, 998, 289, 315, 056, 604, 725, 802, 778, 631, 864, 155, 195, 653, 244, 258, 698, 294, 695, 930, 801, 915, 298, 721, 172, 556, 347, 546, 396, 447, 910, 145, 904, 090, 586, 298, 496, 791, 287, 406, 870, 504, 895, 858, 671, 747, 985, 466, 775, 757, 320, 568, 128, 845, 920, 541, 334, 053, 922, 000, 113, 786, 300, 945, 560, 688, 166, 740, 016, 984, 205, 580, 403, 363, 795, 376, 452, 030, 402, 432, 256, 613, 527, 836, 951, 177, 883, 863, 874, 439, 662, 532, 249, 850, 654, 995, 886, 234, 281, 899, 707, 733, 276, 171, 783, 928, 034, 946, 501, 434, 558, 897, 071, 942, 586, 398, 772, 754, 710, 962, 953, 741, 521, 115, 136, 835, 062, 752, 602, 326, 484, 728, 703, 920, 764, 310, 059, 584, 116, 612, 054, 529, 703, 023, 647, 254, 929, 666, 938, 115, 137, 322, 753, 645, 098, 889, 031, 360, 205, 724, 817, 658, 511, 806, 303, 644, 281, 231, 496, 550, 704, 751, 025, 446, 501, 172, 721, 155, 519, 486, 685, 080, 036, 853, 228, 183, 152, 196, 003, 735, 625, 279, 449, 515, 828, 418, 829, 478, 761, 085, 263, 981, 395, 599, 006, 737, 648, 292, 244, 375, 287, 184, 624, 578, 036, 192, 981, 971, 399, 147, 564, 488, 262, 603, 903, 381, 441, 823, 262, 515, 097, 482, 798, 777, 996, 437, 308, 997, 038, 886, 778, 227, 138, 360, 577, 297, 882, 412, 561, 190, 717, 663, 946, 507, 063, 304, 527, 954, 661, 855, 096, 666, 185, 664, 709, 711, 344, 474, 016, 070, 462, 621, 568, 071, 748, 187, 784, 437, 143, 698, 821, 855, 967, 095, 910, 259, 686, 200, 235, 371, 858, 874, 856, 965, 220, 005, 031, 173, 439, 207, 321, 139, 080, 329, 363, 447, 972, 735, 595, 527, 734, 907, 178, 379, 342, 163, 701, 205, 005, 451, 326, 383, 544, 000, 186, 323, 991, 490, 705, 479, 778, 056, 697, 853, 358, 048, 966, 906, 295, 119, 432, 473, 099, 587, 655, 236, 812, 859, 041, 383, 241, 160, 722, 602, 998, 330, 535, 370, 876, 138, 939, 639, 177, 957, 454, 016, 137, 223, 618, 789, 365, 260, 538, 155, 841, 587, 186, 925, 538, 606, 164, 779, 834, 025, 435, 128, 439, 612, 946, 035, 291, 332, 594, 279, 490, 433, 729, 908, 573, 158, 029, 095, 863, 138, 268, 329, 147, 711, 639, 633, 709, 240, 031, 689, 458, 636, 060, 645, 845, 925, 126, 994, 655, 724, 839, 186, 564, 209, 752, 685, 082, 307, 544, 254, 599, 376, 917, 041, 977, 780, 085, 362, 730, 941, 710, 163, 434, 907, 696, 423, 722, 294, 352, 366, 125, 572, 508, 814, 779, 223, 151, 974, 778, 060, 569, 672, 538, 017, 180, 776, 360, 346, 245, 927, 877, 846, 585, 065, 605, 078, 084, 421, 152, 969, 752, 189, 087, 401, 966, 090, 665, 180, 351, 650, 179, 250, 461, 950, 136, 658, 543, 663, 271, 254, 963, 990, 854, 914, 420, 001, 457, 476, 081, 930, 221, 206, 602, 433, 009, 641, 270, 489, 439, 039, 717, 719, 518, 069, 908, 699, 860, 663, 658, 323, 227, 870, 937, 650, 226, 014, 929, 101, 151, 717, 763, 594, 460, 202, 324, 930, 028, 040, 186, 772, 391, 028, 809, 786, 660, 565, 118, 326, 004, 368, 850, 881, 715, 723, 866, 984, 224, 220, 102, 495, 055, 188, 169, 480, 322, 100, 251, 542, 649, 463, 981, 287, 367, 765, 892, 768, 816, 359, 831, 247, 788, 652, 014, 117, 411, 091, 360, 116, 499, 507, 662, 907, 794, 364, 600, 585, 194, 199, 856, 016, 264, 790, 761, 532, 103, 872, 755, 712, 699, 251, 827, 568, 798, 930, 276, 176, 114, 616, 254, 935, 649, 590, 379, 804, 583, 818, 232, 336, 861, 201, 624, 373, 656, 984, 670, 378, 585, 330, 527, 583, 333, 793, 990, 752, 166, 069, 238, 053, 369, 887, 956, 513, 728, 559, 388, 349, 989, 470, 741, 618, 155, 012, 539, 706, 464, 817, 194, 670, 834, 819, 721, 448, 889, 879, 067, 650, 379, 590, 366, 967, 249, 499, 254, 527, 903, 372, 963, 616, 265, 897, 603, 949, 857, 674, 139, 735, 944, 102, 374, 432, 970, 935, 547, 798, 262, 961, 459, 144, 293, 645, 142, 861, 715, 858, 733, 974, 679, 189, 757, 121, 195, 618, 738, 578, 364, 475, 844, 842, 355, 558, 105, 002, 561, 149, 239, 151, 889, 309, 946, 342, 841, 393, 608, 038, 309, 166, 281, 881, 150, 371, 528, 496, 705, 974, 162, 562, 823, 609, 21...$

## 2010.01.05

### Pi calculations

Filed under: mathematics — Tags: , , , — sandokan65 @ 12:00

Bailey-Borwein-Plouffe formula:
$\pi = \sum_{n=0}^\infty \frac1{16^n} \left( \frac{4}{8n+1} - \frac{2}{8n+4} - \frac1{8n+5} - \frac1{8n+6} \right)$. (Sources: [2], [3]).

Plouffe:
$\frac{\pi}{8} = \sum_{n=0}^\infty \frac1{n 2^{\left[\frac{n+1}2\right]}} \left( \left[\frac{n+7}8\right] - \left[\frac{n+6}8\right] + \left[\frac{n+1}8\right] - \left[\frac{n+4}8\right]\right)$. (Source: [3])

Belard:
$\pi = \frac1{2^6} \sum_{n=0}^\infty \frac{(-)^n}{2^{10n}} \left( -\frac{2^5}{4n+1} - \frac1{4n+3} + \frac{2^8}{10n+1} - \frac{2^6}{10n+3} - \frac{2^2}{10n+5} - \frac{2^2}{10n+7} + \frac1{10n+9} \right)$. (Source: [1])

Belard:
$\pi = \frac1{740,025}\left(\sum_{n=1}^\infty \frac{3P(n)}{\binom{7n}{2n} 2^{n-1}} - 20,379,280 \right)$ where:
$P(n) = - 885,673,181 n^5 + 3,125,347,237 n^4 - 2,942,969,225 n^3 + 1,031,962,795 n^2 - 196,882,274 n + 10,996,648$. (Source: [1])

Chudnovsky:
$\frac1\pi = \frac{\sqrt{10005}}{4270934400} \sum_{k=0}^\infty (-)^k \frac{(6k)!}{(k!)^3 (3k)!} \frac{(13591409+545140134 k)}{640320^{3k}}$