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_pi_minus_three<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
303 return pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(3);
308 inline T constant_four_minus_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
310 return static_cast<T>(4) - pi<T, policies::policy<policies::digits2<N> > >();
315 inline T constant_pow23_four_minus_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
318 return pow(four_minus_pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1.5));
323 inline T constant_exp_minus_half<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
326 return exp(static_cast<T>(-0.5));
332 inline T constant_one_div_root_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
334 return static_cast<T>(1) / root_two<T, policies::policy<policies::digits2<N> > >();
339 inline T constant_one_div_root_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
341 return static_cast<T>(1) / root_pi<T, policies::policy<policies::digits2<N> > >();
346 inline T constant_one_div_root_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
348 return static_cast<T>(1) / root_two_pi<T, policies::policy<policies::digits2<N> > >();
353 inline T constant_root_one_div_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
356 return sqrt(static_cast<T>(1) / pi<T, policies::policy<policies::digits2<N> > >());
362 inline T constant_four_thirds_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
365 return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(4) / static_cast<T>(3);
370 inline T constant_half_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
373 return pi<T, policies::policy<policies::digits2<N> > >() / static_cast<T>(2);
379 inline T constant_third_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
382 return pi<T, policies::policy<policies::digits2<N> > >() / static_cast<T>(3);
387 inline T constant_sixth_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
390 return pi<T, policies::policy<policies::digits2<N> > >() / static_cast<T>(6);
395 inline T constant_two_thirds_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
398 return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(2) / static_cast<T>(3);
403 inline T constant_three_quarters_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
406 return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(3) / static_cast<T>(4);
411 inline T constant_pi_pow_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
414 return pow(pi<T, policies::policy<policies::digits2<N> > >(), e<T, policies::policy<policies::digits2<N> > >()); //
419 inline T constant_pi_sqr<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
422 return pi<T, policies::policy<policies::digits2<N> > >()
423 * pi<T, policies::policy<policies::digits2<N> > >() ; //
428 inline T constant_pi_sqr_div_six<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
431 return pi<T, policies::policy<policies::digits2<N> > >()
432 * pi<T, policies::policy<policies::digits2<N> > >()
433 / static_cast<T>(6); //
439 inline T constant_pi_cubed<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
442 return pi<T, policies::policy<policies::digits2<N> > >()
443 * pi<T, policies::policy<policies::digits2<N> > >()
444 * pi<T, policies::policy<policies::digits2<N> > >()
450 inline T constant_cbrt_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
453 return pow(pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1)/ static_cast<T>(3));
458 inline T constant_one_div_cbrt_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
461 return static_cast<T>(1)
462 / pow(pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1)/ static_cast<T>(3));
469 inline T constant_e_pow_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
472 return pow(e<T, policies::policy<policies::digits2<N> > >(), pi<T, policies::policy<policies::digits2<N> > >()); //
477 inline T constant_root_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
480 return sqrt(e<T, policies::policy<policies::digits2<N> > >());
485 inline T constant_log10_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
488 return log10(e<T, policies::policy<policies::digits2<N> > >());
493 inline T constant_one_div_log10_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
496 return static_cast<T>(1) /
497 log10(e<T, policies::policy<policies::digits2<N> > >());
504 inline T constant_degree<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
507 return pi<T, policies::policy<policies::digits2<N> > >()
508 / static_cast<T>(180)
514 inline T constant_radian<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
517 return static_cast<T>(180)
518 / pi<T, policies::policy<policies::digits2<N> > >()
524 inline T constant_sin_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
527 return sin(static_cast<T>(1)) ; //
532 inline T constant_cos_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
535 return cos(static_cast<T>(1)) ; //
540 inline T constant_sinh_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
543 return sinh(static_cast<T>(1)) ; //
548 inline T constant_cosh_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
551 return cosh(static_cast<T>(1)) ; //
556 inline T constant_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
559 return (static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) ; //
564 inline T constant_ln_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
567 //return log(phi<T, policies::policy<policies::digits2<N> > >()); // ???
568 return log((static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) );
572 inline T constant_one_div_ln_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
575 return static_cast<T>(1) /
576 log((static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) );
583 inline T constant_zeta_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
587 return pi<T, policies::policy<policies::digits2<N> > >()
588 * pi<T, policies::policy<policies::digits2<N> > >()
594 inline T constant_zeta_three<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
596 // http://mathworld.wolfram.com/AperysConstant.html
597 // http://en.wikipedia.org/wiki/Mathematical_constant
599 // http://oeis.org/A002117/constant
600 //T zeta3("1.20205690315959428539973816151144999076"
601 // "4986292340498881792271555341838205786313"
602 // "09018645587360933525814619915");
604 //"1.202056903159594285399738161511449990, 76498629234049888179227155534183820578631309018645587360933525814619915" A002117
605 // 1.202056903159594285399738161511449990, 76498629234049888179227155534183820578631309018645587360933525814619915780, +00);
606 //"1.2020569031595942 double
607 // http://www.spaennare.se/SSPROG/ssnum.pdf // section 11, Algorithm for Apery's constant zeta(3).
608 // Programs to Calculate some Mathematical Constants to Large Precision, Document Version 1.50
610 // by Stefan Spannare September 19, 2007
611 // zeta(3) = 1/64 * sum
613 T n_fact=static_cast<T>(1); // build n! for n = 0.
614 T sum = static_cast<double>(77); // Start with n = 0 case.
615 // for n = 0, (77/1) /64 = 1.203125
616 //double lim = std::numeric_limits<double>::epsilon();
617 T lim = N ? ldexp(T(1), 1 - (std::min)(N, tools::digits<T>())) : tools::epsilon<T>();
618 for(unsigned int n = 1; n < 40; ++n)
619 { // three to five decimal digits per term, so 40 should be plenty for 100 decimal digits.
620 //cout << "n = " << n << endl;
622 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
623 T num = ((205 * n * n) + (250 * n) + 77) * n_fact_p10; // 205n^2 + 250n + 77
624 // int nn = (2 * n + 1);
625 // T d = factorial(nn); // inline factorial.
627 for(unsigned int i = 1; i <= (n+n + 1); ++i) // (2n + 1)
631 T den = d * d * d * d * d; // [(2n+1)!]^5
632 //cout << "den = " << den << endl;
642 //cout << "term = " << term << endl;
643 //cout << "sum/64 = " << sum/64 << endl;
654 inline T constant_catalan<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
655 { // http://oeis.org/A006752/constant
656 //T c("0.915965594177219015054603514932384110774"
657 //"149374281672134266498119621763019776254769479356512926115106248574");
659 // 9.159655941772190150546035149323841107, 74149374281672134266498119621763019776254769479356512926115106248574422619, -01);
661 // This is equation (entry) 31 from
662 // http://www-2.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm
663 // See also http://www.mpfr.org/algorithms.pdf
669 T lim = N ? ldexp(T(1), 1 - (std::min)(N, tools::digits<T>())) : tools::epsilon<T>();
671 for(unsigned k = 1;; ++k)
674 tk_fact *= (2 * k) * (2 * k - 1);
675 term = k_fact * k_fact / (tk_fact * (2 * k + 1) * (2 * k + 1));
682 return boost::math::constants::pi<T, boost::math::policies::policy<> >()
683 * log(2 + boost::math::constants::root_three<T, boost::math::policies::policy<> >())
688 namespace khinchin_detail{
691 T zeta_polynomial_series(T s, T sc, int digits)
695 // This is algorithm 3 from:
697 // "An Efficient Algorithm for the Riemann Zeta Function", P. Borwein,
698 // Canadian Mathematical Society, Conference Proceedings, 2000.
699 // See: http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf
702 int n = (digits * 19) / 53;
704 T two_n = ldexp(T(1), n);
706 for(int j = 0; j < n; ++j)
708 sum += ej_sign * -two_n / pow(T(j + 1), s);
713 for(int j = n; j <= 2 * n - 1; ++j)
715 sum += ej_sign * (ej_sum - two_n) / pow(T(j + 1), s);
717 ej_term *= 2 * n - j;
718 ej_term /= j - n + 1;
721 return -sum / (two_n * (1 - pow(T(2), sc)));
725 T khinchin(int digits)
730 T lim = ldexp(T(1), 1-digits);
734 for(unsigned n = 1;; ++n)
736 for(unsigned k = last_k; k <= 2 * n - 1; ++k)
742 term = (zeta_polynomial_series(T(2 * n), T(1 - T(2 * n)), digits) - 1) * factor / n;
747 return exp(sum / boost::math::constants::ln_two<T, boost::math::policies::policy<> >());
754 inline T constant_khinchin<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
756 int n = N ? (std::min)(N, tools::digits<T>()) : tools::digits<T>();
757 return khinchin_detail::khinchin<T>(n);
762 inline T constant_extreme_value_skewness<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
763 { // from e_float constants.cpp
764 // Mathematica: N[12 Sqrt[6] Zeta[3]/Pi^3, 1101]
766 T ev(12 * sqrt(static_cast<T>(6)) * zeta_three<T, policies::policy<policies::digits2<N> > >()
767 / pi_cubed<T, policies::policy<policies::digits2<N> > >() );
770 //"1.1395470994046486574927930193898461120875997958365518247216557100852480077060706857071875468869385150"
771 //"1894272048688553376986765366075828644841024041679714157616857834895702411080704529137366329462558680"
772 //"2015498788776135705587959418756809080074611906006528647805347822929577145038743873949415294942796280"
773 //"0895597703063466053535550338267721294164578901640163603544404938283861127819804918174973533694090594"
774 //"3094963822672055237678432023017824416203652657301470473548274848068762500300316769691474974950757965"
775 //"8640779777748741897542093874605477776538884083378029488863880220988107155275203245233994097178778984"
776 //"3488995668362387892097897322246698071290011857605809901090220903955815127463328974447572119951192970"
777 //"3684453635456559086126406960279692862247058250100678008419431185138019869693206366891639436908462809"
778 //"9756051372711251054914491837034685476095423926553367264355374652153595857163724698198860485357368964"
779 //"3807049634423621246870868566707915720704996296083373077647528285782964567312903914752617978405994377"
780 //"9064157147206717895272199736902453130842229559980076472936976287378945035706933650987259357729800315");
787 // Calculation of the Glaisher constant depends upon calculating the
788 // derivative of the zeta function at 2, we can then use the relation:
789 // zeta'(2) = 1/6 pi^2 [euler + ln(2pi)-12ln(A)]
790 // To get the constant A.
791 // See equation 45 at http://mathworld.wolfram.com/RiemannZetaFunction.html.
793 // The derivative of the zeta function is computed by direct differentiation
795 // (1-2^(1-s))zeta(s) = SUM(n=0, INF){ (-n)^n / (n+1)^s }
796 // Which gives us 2 slowly converging but alternating sums to compute,
797 // for this we use Algorithm 1 from "Convergent Acceleration of Alternating Series",
798 // Henri Cohen, Fernando Rodriguez Villegas and Don Zagier, Experimental Mathematics 9:1 (1999).
799 // See http://www.math.utexas.edu/users/villegas/publications/conv-accel.pdf
802 T zeta_series_derivative_2(unsigned digits)
804 // Derivative of the series part, evaluated at 2:
806 int n = digits * 301 * 13 / 10000;
807 boost::math::itrunc((std::numeric_limits<T>::digits10 + 1) * 1.3);
808 T d = pow(3 + sqrt(T(8)), n);
813 for(int k = 0; k < n; ++k)
815 T a = -log(T(k+1)) / ((k+1) * (k+1));
818 b = (k + n) * (k - n) * b / ((k + T(0.5f)) * (k + 1));
824 T zeta_series_2(unsigned digits)
826 // Series part of zeta at 2:
828 int n = digits * 301 * 13 / 10000;
829 T d = pow(3 + sqrt(T(8)), n);
834 for(int k = 0; k < n; ++k)
836 T a = T(1) / ((k + 1) * (k + 1));
839 b = (k + n) * (k - n) * b / ((k + T(0.5f)) * (k + 1));
845 inline T zeta_series_lead_2()
852 inline T zeta_series_derivative_lead_2()
854 // derivative of lead part at 2:
855 return -2 * boost::math::constants::ln_two<T>();
859 inline T zeta_derivative_2(unsigned n)
861 // zeta derivative at 2:
862 return zeta_series_derivative_2<T>(n) * zeta_series_lead_2<T>()
863 + zeta_series_derivative_lead_2<T>() * zeta_series_2<T>(n);
866 } // namespace detail
870 inline T constant_glaisher<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
874 typedef policies::policy<policies::digits2<N> > forwarding_policy;
875 int n = N ? (std::min)(N, tools::digits<T>()) : tools::digits<T>();
876 T v = detail::zeta_derivative_2<T>(n);
878 v /= boost::math::constants::pi<T, forwarding_policy>() * boost::math::constants::pi<T, forwarding_policy>();
879 v -= boost::math::constants::euler<T, forwarding_policy>();
880 v -= log(2 * boost::math::constants::pi<T, forwarding_policy>());
885 // from http://mpmath.googlecode.com/svn/data/glaisher.txt
886 // 20,000 digits of the Glaisher-Kinkelin constant A = exp(1/2 - zeta'(-1))
887 // Computed using A = exp((6 (-zeta'(2))/pi^2 + log 2 pi + gamma)/12)
888 // with Euler-Maclaurin summation for zeta'(2).
890 "1.282427129100622636875342568869791727767688927325001192063740021740406308858826"
891 "46112973649195820237439420646120399000748933157791362775280404159072573861727522"
892 "14334327143439787335067915257366856907876561146686449997784962754518174312394652"
893 "76128213808180219264516851546143919901083573730703504903888123418813674978133050"
894 "93770833682222494115874837348064399978830070125567001286994157705432053927585405"
895 "81731588155481762970384743250467775147374600031616023046613296342991558095879293"
896 "36343887288701988953460725233184702489001091776941712153569193674967261270398013"
897 "52652668868978218897401729375840750167472114895288815996668743164513890306962645"
898 "59870469543740253099606800842447417554061490189444139386196089129682173528798629"
899 "88434220366989900606980888785849587494085307347117090132667567503310523405221054"
900 "14176776156308191919997185237047761312315374135304725819814797451761027540834943"
901 "14384965234139453373065832325673954957601692256427736926358821692159870775858274"
902 "69575162841550648585890834128227556209547002918593263079373376942077522290940187");
910 inline T constant_rayleigh_skewness<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
912 // 1100 digits of the Rayleigh distribution skewness
913 // Mathematica: N[2 Sqrt[Pi] (Pi - 3)/((4 - Pi)^(3/2)), 1100]
916 T rs(2 * root_pi<T, policies::policy<policies::digits2<N> > >()
917 * pi_minus_three<T, policies::policy<policies::digits2<N> > >()
918 / pow(four_minus_pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(3./2))
920 // 6.31110657818937138191899351544227779844042203134719497658094585692926819617473725459905027032537306794400047264,
922 //"0.6311106578189371381918993515442277798440422031347194976580945856929268196174737254599050270325373067"
923 //"9440004726436754739597525250317640394102954301685809920213808351450851396781817932734836994829371322"
924 //"5797376021347531983451654130317032832308462278373358624120822253764532674177325950686466133508511968"
925 //"2389168716630349407238090652663422922072397393006683401992961569208109477307776249225072042971818671"
926 //"4058887072693437217879039875871765635655476241624825389439481561152126886932506682176611183750503553"
927 //"1218982627032068396407180216351425758181396562859085306247387212297187006230007438534686340210168288"
928 //"8956816965453815849613622117088096547521391672977226658826566757207615552041767516828171274858145957"
929 //"6137539156656005855905288420585194082284972984285863898582313048515484073396332610565441264220790791"
930 //"0194897267890422924599776483890102027823328602965235306539844007677157873140562950510028206251529523"
931 //"7428049693650605954398446899724157486062545281504433364675815915402937209673727753199567661561209251"
932 //"4695589950526053470201635372590001578503476490223746511106018091907936826431407434894024396366284848"); ;
938 inline T constant_rayleigh_kurtosis_excess<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
939 { // - (6 Pi^2 - 24 Pi + 16)/((Pi - 4)^2)
940 // Might provide and calculate this using pi_minus_four.
942 return - (((static_cast<T>(6) * pi<T, policies::policy<policies::digits2<N> > >()
943 * pi<T, policies::policy<policies::digits2<N> > >())
944 - (static_cast<T>(24) * pi<T, policies::policy<policies::digits2<N> > >()) + static_cast<T>(16) )
946 ((pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4))
947 * (pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4)))
953 inline T constant_rayleigh_kurtosis<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
954 { // 3 - (6 Pi^2 - 24 Pi + 16)/((Pi - 4)^2)
955 // Might provide and calculate this using pi_minus_four.
957 return static_cast<T>(3) - (((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 #endif // BOOST_MATH_CALCULATE_CONSTANTS_CONSTANTS_INCLUDED