1 // Copyright John Maddock 2010, 2012.
2 // Copyright Paul A. Bristow 2011, 2012.
4 // Use, modification and distribution are subject to the
5 // Boost Software License, Version 1.0. (See accompanying file
6 // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
8 #ifndef BOOST_MATH_CALCULATE_CONSTANTS_CONSTANTS_INCLUDED
9 #define BOOST_MATH_CALCULATE_CONSTANTS_CONSTANTS_INCLUDED
11 #include <boost/math/special_functions/trunc.hpp>
13 namespace boost{ namespace math{ namespace constants{ namespace detail{
17 inline T constant_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
21 return ldexp(acos(T(0)), 1);
24 // Although this code works well, it's usually more accurate to just call acos
25 // and access the number types own representation of PI which is usually calculated
26 // at slightly higher precision...
36 lim = boost::math::tools::epsilon<T>();
43 result = ldexp(result, -2);
51 bool neg = boost::math::sign(result) < 0;
58 result = ldexp(result, k - 1);
72 inline T constant_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
74 return 2 * pi<T, policies::policy<policies::digits2<N> > >();
77 template <class T> // 2 / pi
79 inline T constant_two_div_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
81 return 2 / pi<T, policies::policy<policies::digits2<N> > >();
84 template <class T> // sqrt(2/pi)
86 inline T constant_root_two_div_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
89 return sqrt((2 / pi<T, policies::policy<policies::digits2<N> > >()));
94 inline T constant_one_div_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
96 return 1 / two_pi<T, policies::policy<policies::digits2<N> > >();
101 inline T constant_root_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
104 return sqrt(pi<T, policies::policy<policies::digits2<N> > >());
109 inline T constant_root_half_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
112 return sqrt(pi<T, policies::policy<policies::digits2<N> > >() / 2);
117 inline T constant_root_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
120 return sqrt(two_pi<T, policies::policy<policies::digits2<N> > >());
125 inline T constant_log_root_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
128 return log(root_two_pi<T, policies::policy<policies::digits2<N> > >());
133 inline T constant_root_ln_four<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
136 return sqrt(log(static_cast<T>(4)));
141 inline T constant_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
144 // Although we can clearly calculate this from first principles, this hooks into
145 // T's own notion of e, which hopefully will more accurate than one calculated to
149 return exp(static_cast<T>(1));
154 inline T constant_half<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
156 return static_cast<T>(1) / static_cast<T>(2);
161 inline T constant_euler<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<M>))
165 // This is the method described in:
166 // "Some New Algorithms for High-Precision Computation of Euler's Constant"
167 // Richard P Brent and Edwin M McMillan.
168 // Mathematics of Computation, Volume 34, Number 149, Jan 1980, pages 305-312.
169 // See equation 17 with p = 2.
171 T n = 3 + (M ? (std::min)(M, tools::digits<T>()) : tools::digits<T>()) / 4;
172 T lim = M ? ldexp(T(1), 1 - (std::min)(M, tools::digits<T>())) : tools::epsilon<T>();
180 for(unsigned k = 1;; ++k)
185 N += term * (Hk - lnn);
196 inline T constant_euler_sqr<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
199 return euler<T, policies::policy<policies::digits2<N> > >()
200 * euler<T, policies::policy<policies::digits2<N> > >();
205 inline T constant_one_div_euler<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
208 return static_cast<T>(1)
209 / euler<T, policies::policy<policies::digits2<N> > >();
215 inline T constant_root_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
218 return sqrt(static_cast<T>(2));
224 inline T constant_root_three<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
227 return sqrt(static_cast<T>(3));
232 inline T constant_half_root_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
235 return sqrt(static_cast<T>(2)) / 2;
240 inline T constant_ln_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
243 // Although there are good ways to calculate this from scratch, this hooks into
244 // T's own notion of log(2) which will hopefully be accurate to the full precision
248 return log(static_cast<T>(2));
253 inline T constant_ln_ten<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
256 return log(static_cast<T>(10));
261 inline T constant_ln_ln_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
264 return log(log(static_cast<T>(2)));
269 inline T constant_third<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
272 return static_cast<T>(1) / static_cast<T>(3);
277 inline T constant_twothirds<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
280 return static_cast<T>(2) / static_cast<T>(3);
285 inline T constant_two_thirds<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
288 return static_cast<T>(2) / static_cast<T>(3);
293 inline T constant_three_quarters<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
296 return static_cast<T>(3) / static_cast<T>(4);
301 inline T constant_sixth<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
304 return static_cast<T>(1) / static_cast<T>(6);
307 // Pi and related constants.
310 inline T constant_pi_minus_three<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
312 return pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(3);
317 inline T constant_four_minus_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
319 return static_cast<T>(4) - pi<T, policies::policy<policies::digits2<N> > >();
324 //inline T constant_pow23_four_minus_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
326 // BOOST_MATH_STD_USING
327 // return pow(four_minus_pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1.5));
332 inline T constant_exp_minus_half<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
335 return exp(static_cast<T>(-0.5));
340 inline T constant_exp_minus_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
343 return exp(static_cast<T>(-1.));
348 inline T constant_one_div_root_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
350 return static_cast<T>(1) / root_two<T, policies::policy<policies::digits2<N> > >();
355 inline T constant_one_div_root_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
357 return static_cast<T>(1) / root_pi<T, policies::policy<policies::digits2<N> > >();
362 inline T constant_one_div_root_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
364 return static_cast<T>(1) / root_two_pi<T, policies::policy<policies::digits2<N> > >();
369 inline T constant_root_one_div_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
372 return sqrt(static_cast<T>(1) / pi<T, policies::policy<policies::digits2<N> > >());
377 inline T constant_four_thirds_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
380 return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(4) / static_cast<T>(3);
385 inline T constant_half_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
388 return pi<T, policies::policy<policies::digits2<N> > >() / static_cast<T>(2);
394 inline T constant_third_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
397 return pi<T, policies::policy<policies::digits2<N> > >() / static_cast<T>(3);
402 inline T constant_sixth_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
405 return pi<T, policies::policy<policies::digits2<N> > >() / static_cast<T>(6);
410 inline T constant_two_thirds_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
413 return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(2) / static_cast<T>(3);
418 inline T constant_three_quarters_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
421 return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(3) / static_cast<T>(4);
426 inline T constant_pi_pow_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
429 return pow(pi<T, policies::policy<policies::digits2<N> > >(), e<T, policies::policy<policies::digits2<N> > >()); //
434 inline T constant_pi_sqr<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
437 return pi<T, policies::policy<policies::digits2<N> > >()
438 * pi<T, policies::policy<policies::digits2<N> > >() ; //
443 inline T constant_pi_sqr_div_six<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
446 return pi<T, policies::policy<policies::digits2<N> > >()
447 * pi<T, policies::policy<policies::digits2<N> > >()
448 / static_cast<T>(6); //
454 inline T constant_pi_cubed<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
457 return pi<T, policies::policy<policies::digits2<N> > >()
458 * pi<T, policies::policy<policies::digits2<N> > >()
459 * pi<T, policies::policy<policies::digits2<N> > >()
465 inline T constant_cbrt_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
468 return pow(pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1)/ static_cast<T>(3));
473 inline T constant_one_div_cbrt_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
476 return static_cast<T>(1)
477 / pow(pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1)/ static_cast<T>(3));
484 inline T constant_e_pow_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
487 return pow(e<T, policies::policy<policies::digits2<N> > >(), pi<T, policies::policy<policies::digits2<N> > >()); //
492 inline T constant_root_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
495 return sqrt(e<T, policies::policy<policies::digits2<N> > >());
500 inline T constant_log10_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
503 return log10(e<T, policies::policy<policies::digits2<N> > >());
508 inline T constant_one_div_log10_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
511 return static_cast<T>(1) /
512 log10(e<T, policies::policy<policies::digits2<N> > >());
519 inline T constant_degree<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
522 return pi<T, policies::policy<policies::digits2<N> > >()
523 / static_cast<T>(180)
529 inline T constant_radian<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
532 return static_cast<T>(180)
533 / pi<T, policies::policy<policies::digits2<N> > >()
539 inline T constant_sin_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
542 return sin(static_cast<T>(1)) ; //
547 inline T constant_cos_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
550 return cos(static_cast<T>(1)) ; //
555 inline T constant_sinh_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
558 return sinh(static_cast<T>(1)) ; //
563 inline T constant_cosh_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
566 return cosh(static_cast<T>(1)) ; //
571 inline T constant_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
574 return (static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) ; //
579 inline T constant_ln_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
582 //return log(phi<T, policies::policy<policies::digits2<N> > >()); // ???
583 return log((static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) );
587 inline T constant_one_div_ln_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
590 return static_cast<T>(1) /
591 log((static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) );
598 inline T constant_zeta_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
602 return pi<T, policies::policy<policies::digits2<N> > >()
603 * pi<T, policies::policy<policies::digits2<N> > >()
609 inline T constant_zeta_three<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
611 // http://mathworld.wolfram.com/AperysConstant.html
612 // http://en.wikipedia.org/wiki/Mathematical_constant
614 // http://oeis.org/A002117/constant
615 //T zeta3("1.20205690315959428539973816151144999076"
616 // "4986292340498881792271555341838205786313"
617 // "09018645587360933525814619915");
619 //"1.202056903159594285399738161511449990, 76498629234049888179227155534183820578631309018645587360933525814619915" A002117
620 // 1.202056903159594285399738161511449990, 76498629234049888179227155534183820578631309018645587360933525814619915780, +00);
621 //"1.2020569031595942 double
622 // http://www.spaennare.se/SSPROG/ssnum.pdf // section 11, Algorithm for Apery's constant zeta(3).
623 // Programs to Calculate some Mathematical Constants to Large Precision, Document Version 1.50
625 // by Stefan Spannare September 19, 2007
626 // zeta(3) = 1/64 * sum
628 T n_fact=static_cast<T>(1); // build n! for n = 0.
629 T sum = static_cast<double>(77); // Start with n = 0 case.
630 // for n = 0, (77/1) /64 = 1.203125
631 //double lim = std::numeric_limits<double>::epsilon();
632 T lim = N ? ldexp(T(1), 1 - (std::min)(N, tools::digits<T>())) : tools::epsilon<T>();
633 for(unsigned int n = 1; n < 40; ++n)
634 { // three to five decimal digits per term, so 40 should be plenty for 100 decimal digits.
635 //cout << "n = " << n << endl;
637 T n_fact_p10 = n_fact * n_fact * n_fact * n_fact * n_fact * n_fact * n_fact * n_fact * n_fact * n_fact; // (n!)^10
638 T num = ((205 * n * n) + (250 * n) + 77) * n_fact_p10; // 205n^2 + 250n + 77
639 // int nn = (2 * n + 1);
640 // T d = factorial(nn); // inline factorial.
642 for(unsigned int i = 1; i <= (n+n + 1); ++i) // (2n + 1)
646 T den = d * d * d * d * d; // [(2n+1)!]^5
647 //cout << "den = " << den << endl;
657 //cout << "term = " << term << endl;
658 //cout << "sum/64 = " << sum/64 << endl;
669 inline T constant_catalan<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
670 { // http://oeis.org/A006752/constant
671 //T c("0.915965594177219015054603514932384110774"
672 //"149374281672134266498119621763019776254769479356512926115106248574");
674 // 9.159655941772190150546035149323841107, 74149374281672134266498119621763019776254769479356512926115106248574422619, -01);
676 // This is equation (entry) 31 from
677 // http://www-2.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm
678 // See also http://www.mpfr.org/algorithms.pdf
684 T lim = N ? ldexp(T(1), 1 - (std::min)(N, tools::digits<T>())) : tools::epsilon<T>();
686 for(unsigned k = 1;; ++k)
689 tk_fact *= (2 * k) * (2 * k - 1);
690 term = k_fact * k_fact / (tk_fact * (2 * k + 1) * (2 * k + 1));
697 return boost::math::constants::pi<T, boost::math::policies::policy<> >()
698 * log(2 + boost::math::constants::root_three<T, boost::math::policies::policy<> >())
703 namespace khinchin_detail{
706 T zeta_polynomial_series(T s, T sc, int digits)
710 // This is algorithm 3 from:
712 // "An Efficient Algorithm for the Riemann Zeta Function", P. Borwein,
713 // Canadian Mathematical Society, Conference Proceedings, 2000.
714 // See: http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf
717 int n = (digits * 19) / 53;
719 T two_n = ldexp(T(1), n);
721 for(int j = 0; j < n; ++j)
723 sum += ej_sign * -two_n / pow(T(j + 1), s);
728 for(int j = n; j <= 2 * n - 1; ++j)
730 sum += ej_sign * (ej_sum - two_n) / pow(T(j + 1), s);
732 ej_term *= 2 * n - j;
733 ej_term /= j - n + 1;
736 return -sum / (two_n * (1 - pow(T(2), sc)));
740 T khinchin(int digits)
745 T lim = ldexp(T(1), 1-digits);
749 for(unsigned n = 1;; ++n)
751 for(unsigned k = last_k; k <= 2 * n - 1; ++k)
757 term = (zeta_polynomial_series(T(2 * n), T(1 - T(2 * n)), digits) - 1) * factor / n;
762 return exp(sum / boost::math::constants::ln_two<T, boost::math::policies::policy<> >());
769 inline T constant_khinchin<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
771 int n = N ? (std::min)(N, tools::digits<T>()) : tools::digits<T>();
772 return khinchin_detail::khinchin<T>(n);
777 inline T constant_extreme_value_skewness<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
778 { // from e_float constants.cpp
779 // Mathematica: N[12 Sqrt[6] Zeta[3]/Pi^3, 1101]
781 T ev(12 * sqrt(static_cast<T>(6)) * zeta_three<T, policies::policy<policies::digits2<N> > >()
782 / pi_cubed<T, policies::policy<policies::digits2<N> > >() );
785 //"1.1395470994046486574927930193898461120875997958365518247216557100852480077060706857071875468869385150"
786 //"1894272048688553376986765366075828644841024041679714157616857834895702411080704529137366329462558680"
787 //"2015498788776135705587959418756809080074611906006528647805347822929577145038743873949415294942796280"
788 //"0895597703063466053535550338267721294164578901640163603544404938283861127819804918174973533694090594"
789 //"3094963822672055237678432023017824416203652657301470473548274848068762500300316769691474974950757965"
790 //"8640779777748741897542093874605477776538884083378029488863880220988107155275203245233994097178778984"
791 //"3488995668362387892097897322246698071290011857605809901090220903955815127463328974447572119951192970"
792 //"3684453635456559086126406960279692862247058250100678008419431185138019869693206366891639436908462809"
793 //"9756051372711251054914491837034685476095423926553367264355374652153595857163724698198860485357368964"
794 //"3807049634423621246870868566707915720704996296083373077647528285782964567312903914752617978405994377"
795 //"9064157147206717895272199736902453130842229559980076472936976287378945035706933650987259357729800315");
802 // Calculation of the Glaisher constant depends upon calculating the
803 // derivative of the zeta function at 2, we can then use the relation:
804 // zeta'(2) = 1/6 pi^2 [euler + ln(2pi)-12ln(A)]
805 // To get the constant A.
806 // See equation 45 at http://mathworld.wolfram.com/RiemannZetaFunction.html.
808 // The derivative of the zeta function is computed by direct differentiation
810 // (1-2^(1-s))zeta(s) = SUM(n=0, INF){ (-n)^n / (n+1)^s }
811 // Which gives us 2 slowly converging but alternating sums to compute,
812 // for this we use Algorithm 1 from "Convergent Acceleration of Alternating Series",
813 // Henri Cohen, Fernando Rodriguez Villegas and Don Zagier, Experimental Mathematics 9:1 (1999).
814 // See http://www.math.utexas.edu/users/villegas/publications/conv-accel.pdf
817 T zeta_series_derivative_2(unsigned digits)
819 // Derivative of the series part, evaluated at 2:
821 int n = digits * 301 * 13 / 10000;
822 boost::math::itrunc((std::numeric_limits<T>::digits10 + 1) * 1.3);
823 T d = pow(3 + sqrt(T(8)), n);
828 for(int k = 0; k < n; ++k)
830 T a = -log(T(k+1)) / ((k+1) * (k+1));
833 b = (k + n) * (k - n) * b / ((k + T(0.5f)) * (k + 1));
839 T zeta_series_2(unsigned digits)
841 // Series part of zeta at 2:
843 int n = digits * 301 * 13 / 10000;
844 T d = pow(3 + sqrt(T(8)), n);
849 for(int k = 0; k < n; ++k)
851 T a = T(1) / ((k + 1) * (k + 1));
854 b = (k + n) * (k - n) * b / ((k + T(0.5f)) * (k + 1));
860 inline T zeta_series_lead_2()
867 inline T zeta_series_derivative_lead_2()
869 // derivative of lead part at 2:
870 return -2 * boost::math::constants::ln_two<T>();
874 inline T zeta_derivative_2(unsigned n)
876 // zeta derivative at 2:
877 return zeta_series_derivative_2<T>(n) * zeta_series_lead_2<T>()
878 + zeta_series_derivative_lead_2<T>() * zeta_series_2<T>(n);
881 } // namespace detail
885 inline T constant_glaisher<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
889 typedef policies::policy<policies::digits2<N> > forwarding_policy;
890 int n = N ? (std::min)(N, tools::digits<T>()) : tools::digits<T>();
891 T v = detail::zeta_derivative_2<T>(n);
893 v /= boost::math::constants::pi<T, forwarding_policy>() * boost::math::constants::pi<T, forwarding_policy>();
894 v -= boost::math::constants::euler<T, forwarding_policy>();
895 v -= log(2 * boost::math::constants::pi<T, forwarding_policy>());
900 // from http://mpmath.googlecode.com/svn/data/glaisher.txt
901 // 20,000 digits of the Glaisher-Kinkelin constant A = exp(1/2 - zeta'(-1))
902 // Computed using A = exp((6 (-zeta'(2))/pi^2 + log 2 pi + gamma)/12)
903 // with Euler-Maclaurin summation for zeta'(2).
905 "1.282427129100622636875342568869791727767688927325001192063740021740406308858826"
906 "46112973649195820237439420646120399000748933157791362775280404159072573861727522"
907 "14334327143439787335067915257366856907876561146686449997784962754518174312394652"
908 "76128213808180219264516851546143919901083573730703504903888123418813674978133050"
909 "93770833682222494115874837348064399978830070125567001286994157705432053927585405"
910 "81731588155481762970384743250467775147374600031616023046613296342991558095879293"
911 "36343887288701988953460725233184702489001091776941712153569193674967261270398013"
912 "52652668868978218897401729375840750167472114895288815996668743164513890306962645"
913 "59870469543740253099606800842447417554061490189444139386196089129682173528798629"
914 "88434220366989900606980888785849587494085307347117090132667567503310523405221054"
915 "14176776156308191919997185237047761312315374135304725819814797451761027540834943"
916 "14384965234139453373065832325673954957601692256427736926358821692159870775858274"
917 "69575162841550648585890834128227556209547002918593263079373376942077522290940187");
925 inline T constant_rayleigh_skewness<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
927 // 1100 digits of the Rayleigh distribution skewness
928 // Mathematica: N[2 Sqrt[Pi] (Pi - 3)/((4 - Pi)^(3/2)), 1100]
931 T rs(2 * root_pi<T, policies::policy<policies::digits2<N> > >()
932 * pi_minus_three<T, policies::policy<policies::digits2<N> > >()
933 / pow(four_minus_pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(3./2))
935 // 6.31110657818937138191899351544227779844042203134719497658094585692926819617473725459905027032537306794400047264,
937 //"0.6311106578189371381918993515442277798440422031347194976580945856929268196174737254599050270325373067"
938 //"9440004726436754739597525250317640394102954301685809920213808351450851396781817932734836994829371322"
939 //"5797376021347531983451654130317032832308462278373358624120822253764532674177325950686466133508511968"
940 //"2389168716630349407238090652663422922072397393006683401992961569208109477307776249225072042971818671"
941 //"4058887072693437217879039875871765635655476241624825389439481561152126886932506682176611183750503553"
942 //"1218982627032068396407180216351425758181396562859085306247387212297187006230007438534686340210168288"
943 //"8956816965453815849613622117088096547521391672977226658826566757207615552041767516828171274858145957"
944 //"6137539156656005855905288420585194082284972984285863898582313048515484073396332610565441264220790791"
945 //"0194897267890422924599776483890102027823328602965235306539844007677157873140562950510028206251529523"
946 //"7428049693650605954398446899724157486062545281504433364675815915402937209673727753199567661561209251"
947 //"4695589950526053470201635372590001578503476490223746511106018091907936826431407434894024396366284848"); ;
953 inline T constant_rayleigh_kurtosis_excess<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
954 { // - (6 Pi^2 - 24 Pi + 16)/((Pi - 4)^2)
955 // Might provide and calculate this using pi_minus_four.
957 return - (((static_cast<T>(6) * pi<T, policies::policy<policies::digits2<N> > >()
958 * pi<T, policies::policy<policies::digits2<N> > >())
959 - (static_cast<T>(24) * pi<T, policies::policy<policies::digits2<N> > >()) + static_cast<T>(16) )
961 ((pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4))
962 * (pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4)))
968 inline T constant_rayleigh_kurtosis<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
969 { // 3 - (6 Pi^2 - 24 Pi + 16)/((Pi - 4)^2)
970 // Might provide and calculate this using pi_minus_four.
972 return static_cast<T>(3) - (((static_cast<T>(6) * pi<T, policies::policy<policies::digits2<N> > >()
973 * pi<T, policies::policy<policies::digits2<N> > >())
974 - (static_cast<T>(24) * pi<T, policies::policy<policies::digits2<N> > >()) + static_cast<T>(16) )
976 ((pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4))
977 * (pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4)))
983 inline T constant_log2_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
985 return 1 / boost::math::constants::ln_two<T>();
990 inline T constant_quarter_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
992 return boost::math::constants::pi<T>() / 4;
997 inline T constant_one_div_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
999 return 1 / boost::math::constants::pi<T>();
1004 inline T constant_two_div_root_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
1006 return 2 * boost::math::constants::one_div_root_pi<T>();
1014 #endif // BOOST_MATH_CALCULATE_CONSTANTS_CONSTANTS_INCLUDED