[heading Implementation]
-These functions are implemented in terms of Carlson's integrals
-using the relations:
+These functions are implemented in terms of Carlson's integrals using the relations:
[equation ellint19]
[equation ellint20]
-
-[endsect]
+[endsect] [/section:ellint_1 Elliptic Integrals of the First Kind - Legendre Form]
[section:ellint_2 Elliptic Integrals of the Second Kind - Legendre Form]
[equation ellint22]
-
-[endsect]
+[endsect] [/section:ellint_2 Elliptic Integrals of the Second Kind - Legendre Form]
[section:ellint_3 Elliptic Integrals of the Third Kind - Legendre Form]
The implementation for [Pi](n, [phi], k) first siphons off the special cases:
-['[Pi](0, [phi], k) = F([phi], k)]
+[expression ['[Pi](0, [phi], k) = F([phi], k)]]
-['[Pi](n, [pi]/2, k) = [Pi](n, k)]
+[expression ['[Pi](n, [pi]/2, k) = [Pi](n, k)]]
and
Then the relations:
-['[Pi](n, -[phi], k) = -[Pi](n, [phi], k)]
+[expression ['[Pi](n, -[phi], k) = -[Pi](n, [phi], k)]]
-['[Pi](n, [phi]+m[pi], k) = [Pi](n, [phi], k) + 2m[Pi](n, k) ; n <= 1]
+[expression ['[Pi](n, [phi]+m[pi], k) = [Pi](n, [phi], k) + 2m[Pi](n, k) ; n <= 1]]
-['[Pi](n, [phi]+m[pi], k) = [Pi](n, [phi], k) ; n > 1]
+[expression ['[Pi](n, [phi]+m[pi], k) = [Pi](n, [phi], k) ; n > 1] [indent] [indent]
[footnote I haven't been able to find a literature reference for this
relation, but it appears to be the convention used by Mathematica.
Intuitively the first ['2 * m * [Pi](n, k)] terms cancel out as the
-derivative alternates between +[infin] and -[infin].]
+derivative alternates between +[infin] and -[infin].]]
-are used to move [phi][space] to the range \[0, [pi]\/2\].
+are used to move [phi] to the range \[0, [pi]\/2\].
-The functions are then implemented in terms of Carlson's integrals
-using the relations:
+The functions are then implemented in terms of Carlson's integrals using the relations:
[equation ellint25]
[equation ellint26]
-[endsect]
+[endsect] [/section:ellint_3 Elliptic Integrals of the Third Kind - Legendre Form]
[section:ellint_d Elliptic Integral D - Legendre Form]
[graph elliptic_integral_d____float128]
-
[heading Testing]
The tests use a mixture of spot test values calculated using
-values calculated at wolframalpha.com, and random test data generated using
+values calculated at __WolframAlpha, and random test data generated using
MPFR at 1000-bit precision and a deliberately naive implementation in terms of
the Legendre integrals.
The implementation for D([phi], k) first performs argument reduction using the relations:
-['D(-[phi], k) = -D([phi], k)]
+[expression ['D(-[phi], k) = -D([phi], k)]]
and
-['D(n[pi]+[phi], k) = 2nD(k) + D([phi], k)]
+[expression ['D(n[pi]+[phi], k) = 2nD(k) + D([phi], k)]]
-to move [phi][space] to the range \[0, [pi]\/2\].
+to move [phi] to the range \[0, [pi]\/2\].
The functions are then implemented in terms of Carlson's integral R[sub D]
using the relation:
[equation ellint_d]
-[endsect]
+[endsect] [/section:ellint_d Elliptic Integral D - Legendre Form]
[section:jacobi_zeta Jacobi Zeta Function]
Please note the use of [phi], and /k/ as the parameters, the function is often defined as ['Z([phi], m)]
with ['m = k[super 2]], see for example [@http://mathworld.wolfram.com/JacobiZetaFunction.html Weisstein, Eric W. "Jacobi Zeta Function." From MathWorld--A Wolfram Web Resource.]
-Or else as [@https://dlmf.nist.gov/22.16#E32 ['Z(x, k)]] with ['[phi] = am(x, k)], where ['am] is the [@https://dlmf.nist.gov/22.16#E1 Jacobi amplitude function] which is equivalent to
-['asin(jacobi_elliptic(k, x))].
+Or else as [@https://dlmf.nist.gov/22.16#E32 ['Z(x, k)]] with ['[phi] = am(x, k)],
+where ['am] is the [@https://dlmf.nist.gov/22.16#E1 Jacobi amplitude function]
+which is equivalent to ['asin(jacobi_elliptic(k, x))].
The return type of this function is computed using the __arg_promotion_rules
when the arguments are of different types: when they are the same type then the result
is the same type as the arguments.
Requires ['-1 <= k <= 1], otherwise
-returns the result of __domain_error (outside this range the result
-would be complex).
+returns the result of __domain_error (outside this range the result would be complex).
[optional_policy]
[heading Testing]
The tests use a mixture of spot test values calculated using
-values calculated at wolframalpha.com, and random test data generated using
+values calculated at __WolframAlpha, and random test data generated using
MPFR at 1000-bit precision and a deliberately naive implementation in terms of
the Legendre integrals.
The implementation for Z([phi], k) first makes the argument [phi] positive using:
-['Z(-[phi], k) = -Z([phi], k)]
+[expression ['Z(-[phi], k) = -Z([phi], k)]]
The function is then implemented in terms of Carlson's integral R[sub J]
using the relation:
There is one special case where the above relation fails: when ['k = 1], in that case
the function simplifies to
-['Z([phi], 1) = sign(cos([phi])) sin([phi])]
+[expression ['Z([phi], 1) = sign(cos([phi])) sin([phi])]]
+
+[h5:jacobi_zeta_example Example]
+
+A simple example comparing use of __WolframAlpha with Boost.Math (including much higher precision using Boost.Multiprecision)
+is [@../../example/jacobi_zeta_example.cpp jacobi_zeta_example.cpp].
-[endsect]
+[endsect] [/section:jacobi_zeta Jacobi Zeta Function]
[section:heuman_lambda Heuman Lambda Function]
is the same type as the arguments.
Requires ['-1 <= k <= 1], otherwise
-returns the result of __domain_error (outside this range the result
-would be complex).
+returns the result of __domain_error (outside this range the result would be complex).
[optional_policy]
[heading Testing]
The tests use a mixture of spot test values calculated using
-values calculated at wolframalpha.com, and random test data generated using
+values calculated at __WolframAlpha, and random test data generated using
MPFR at 1000-bit precision and a deliberately naive implementation in terms of
the Legendre integrals.
This relation fails for ['|[phi]| >= [pi]/2] in which case the definition in terms of the
Jacobi Zeta is used.
-[endsect]
+[endsect] [/section:heuman_lambda Heuman Lambda Function]
+
projects / platform / upstream / boost.git / blobdiff