Lines Matching refs:phi
20 ``__sf_result`` ellint_1(T1 k, T2 phi);
23 ``__sf_result`` ellint_1(T1 k, T2 phi, const ``__Policy``&);
36 ['F([phi], k)] and its complete counterpart ['K(k) = F([pi]/2, k)].
45 ``__sf_result`` ellint_1(T1 k, T2 phi);
48 ``__sf_result`` ellint_1(T1 k, T2 phi, const ``__Policy``&);
50 Returns the incomplete elliptic integral of the first kind ['F([phi], k)]:
54 Requires k[super 2]sin[super 2](phi) < 1, otherwise returns the result of __domain_error.
121 ``__sf_result`` ellint_2(T1 k, T2 phi);
124 ``__sf_result`` ellint_2(T1 k, T2 phi, const ``__Policy``&);
137 ['E([phi], k)] and its complete counterpart ['E(k) = E([pi]/2, k)].
146 ``__sf_result`` ellint_2(T1 k, T2 phi);
149 ``__sf_result`` ellint_2(T1 k, T2 phi, const ``__Policy``&);
151 Returns the incomplete elliptic integral of the second kind ['E([phi], k)]:
155 Requires k[super 2]sin[super 2](phi) < 1, otherwise returns the result of __domain_error.
223 ``__sf_result`` ellint_3(T1 k, T2 n, T3 phi);
226 ``__sf_result`` ellint_3(T1 k, T2 n, T3 phi, const ``__Policy``&);
239 ['[Pi](n, [phi], k)] and its complete counterpart ['[Pi](n, k) = E(n, [pi]/2, k)].
248 ``__sf_result`` ellint_3(T1 k, T2 n, T3 phi);
251 ``__sf_result`` ellint_3(T1 k, T2 n, T3 phi, const ``__Policy``&);
253 Returns the incomplete elliptic integral of the third kind ['[Pi](n, [phi], k)]:
257 Requires ['k[super 2]sin[super 2](phi) < 1] and ['n < 1/sin[super 2]([phi])], otherwise
297 The implementation for [Pi](n, [phi], k) first siphons off the special cases:
299 [expression ['[Pi](0, [phi], k) = F([phi], k)]]
315 [expression ['[Pi](n, -[phi], k) = -[Pi](n, [phi], k)]]
317 [expression ['[Pi](n, [phi]+m[pi], k) = [Pi](n, [phi], k) + 2m[Pi](n, k) ; n <= 1]]
319 [expression ['[Pi](n, [phi]+m[pi], k) = [Pi](n, [phi], k) ; n > 1] [indent] [indent]
325 are used to move [phi] to the range \[0, [pi]\/2\].
348 ``__sf_result`` ellint_d(T1 k, T2 phi);
351 ``__sf_result`` ellint_d(T1 k, T2 phi, const ``__Policy``&);
364 ['D([phi], k)] and its complete counterpart ['D(k) = D([pi]/2, k)].
371 ``__sf_result`` ellint_d(T1 k, T2 phi);
374 ``__sf_result`` ellint_3(T1 k, T2 phi, const ``__Policy``&);
380 Requires ['k[super 2]sin[super 2](phi) < 1], otherwise
402 and generally have very low error rates (a few epsilon) unless parameter [phi]
428 The implementation for D([phi], k) first performs argument reduction using the relations:
430 [expression ['D(-[phi], k) = -D([phi], k)]]
434 [expression ['D(n[pi]+[phi], k) = 2nD(k) + D([phi], k)]]
436 to move [phi] to the range \[0, [pi]\/2\].
456 ``__sf_result`` jacobi_zeta(T1 k, T2 phi);
459 ``__sf_result`` jacobi_zeta(T1 k, T2 phi, const ``__Policy``&);
465 This function evaluates the Jacobi Zeta Function ['Z([phi], k)]
469 Please note the use of [phi], and /k/ as the parameters, the function is often defined as ['Z([phi]…
471 Or else as [@https://dlmf.nist.gov/22.16#E32 ['Z(x, k)]] with ['[phi] = am(x, k)],
484 Note that there is no complete analogue of this function (where [phi] = [pi] / 2)
490 and generally have very low error rates (a few epsilon) unless parameter [phi]
504 The implementation for Z([phi], k) first makes the argument [phi] positive using:
506 [expression ['Z(-[phi], k) = -Z([phi], k)]]
516 [expression ['Z([phi], 1) = sign(cos([phi])) sin([phi])]]
536 ``__sf_result`` heuman_lambda(T1 k, T2 phi);
539 ``__sf_result`` heuman_lambda(T1 k, T2 phi, const ``__Policy``&);
545 This function evaluates the Heuman Lambda Function ['[Lambda][sub 0]([phi], k)]
558 Note that there is no complete analogue of this function (where [phi] = [pi] / 2)
564 and generally have very low error rates (a few epsilon) unless parameter [phi]
583 This relation fails for ['|[phi]| >= [pi]/2] in which case the definition in terms of the