EiK SSKrSSKrSSKrSSKJr SSKJrJrJ r J r SSK J r J r JrJr /SQr"SS5r\"\ S S SS 9r \ R$S 5r\ R(S 5r\"\SSSS 9r\R$S5r\R*S5r\R,S5r\R(S5r\"\SSSSSS9r\R$S5r\R.S5r\R(S5r\"\ SSSSSS9r \ R$S5r\ R.S5r\ R*S 5r\ R,S!5r\ R(S"5r\"\S#S$SS 9r\R$S%5r\R(S&5r\"\ S'S(SS 9r \ R$S)5r\ R*S*5r\ R,S+5r\ R(S,5r\"\ S-S.S/SS09r \ R$S15r\ R(S25r\"\S3S4S/SS09r\R$S55r\R*S65r\R,S75r\R(S85rg)9N_nonneg_int_or_fail) legendre_passoc_legendre_psph_legendre_p sph_harm_y)legendre_p_allassoc_legendre_p_allsph_legendre_p_allsph_harm_y_all)rr rr r r rr c`\rSrSrSSS.Sjjr\S5rSrSrS r S r S r S r S r Srg) MultiUFuncNF)force_complex_outputc [U[R5(d[U[RR 5(aUR 5nO7[U[RR5(aUnO [S5e[5nUH[n[U[R5(d[SU35eUR[SUR555 M] [U5S:a [S5eX lXlX0lX@lXPlSUlSUlSUlSUlSUlg)Nz7ufunc_or_ufuncs should be a ufunc or a ufunc collectionz2All ufuncs must have type `numpy.ufunc`. Received c3H# UHoRS5Sv M g7f)z->rN)split).0xs W/home/mande/repo/quber/.venv/lib/python3.13/site-packages/scipy/special/_multiufuncs.py &MultiUFunc.__init__..+s.UAwwt}Q/?s "rz*All ufuncs must take the same input types.cg)Nrargskwargss r%MultiUFunc.__init__..7s2c0$Nrrs rrr 8sRr!) isinstancenpufunc collectionsabcMappingvaluesIterable ValueErrorsetadd frozensettypeslen__name___ufunc_or_ufuncs_MultiUFunc__doc!_MultiUFunc__force_complex_output_default_kwargs_resolve_out_shapes _finalize_out_key_ufunc_default_args_ufunc_default_kwargs) selfufunc_or_ufuncsnamedocrdefault_kwargs ufuncs_iterseen_input_typesr&s r__init__MultiUFunc.__init__s'/28844/;??+B+BCC-446 O[__-E-EFF-  "566 #u $!%22$&22A1B&DEE $$Y.U.U%UV % #$q( !MNN / &:#-#' ! #= %?"r!cUR$r#)r4)r<s r__doc__MultiUFunc.__doc__:s zzr!cXlg)z3Set `key` method by decorating a function. 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((2622##-f-26yyjk1BC1B#bjjo1B C11;F;  $ $ 0$$U*$UU #77 Bk z9J B9I BKP:: B:@ B  %%B6@%B B u.///**w2FF $33LA #/ #= "$..2B"C orzzBB&(jjO#(::0B#B **#()R@P)R$R D/BDDC#& Z0<0    *$$S)C ODs G() __doc__force_complex_outputr2r6r8r9r7r:r;r3)NN)r2 __module__ __qualname____firstlineno__rCpropertyrFrKrNrQrUrXr\r__static_attributes__rr!rrrsI@&+@B (*("0/r!rrasph_legendre_p(n, m, theta, *, diff_n=0) Spherical Legendre polynomial of the first kind. Parameters ---------- n : ArrayLike[int] Degree of the spherical Legendre polynomial. Must have ``n >= 0``. m : ArrayLike[int] Order of the spherical Legendre polynomial. theta : ArrayLike[float] Input value. diff_n : Optional[int] A non-negative integer. Compute and return all derivatives up to order ``diff_n``. Default is 0. Returns ------- p : ndarray or tuple[ndarray] Spherical Legendre polynomial with ``diff_n`` derivatives. Notes ----- The spherical counterpart of an (unnormalized) associated Legendre polynomial has the additional factor .. math:: \sqrt{\frac{(2 n + 1) (n - m)!}{4 \pi (n + m)!}} It is the same as the spherical harmonic :math:`Y_{n}^{m}(\theta, \phi)` with :math:`\phi = 0`. diff_ncX[USSS9nSUs=::aS::dO [SUS35eU$NrFstrictrGdiff_n is currently only implemented for orders 0, 1, and 2, received: .rr,rs r_rB % @F  !    $   Mr!c2[R"USS5$Nrr%moveaxisrps rrr ;;sB ""r!r asph_legendre_p_all(n, m, theta, *, diff_n=0) All spherical Legendre polynomials of the first kind up to the specified degree ``n``, order ``m``, and all derivatives up to order ``diff_n``. Output shape is ``(diff_n + 1, n + 1, 2 * m + 1, ...)``. The entry at ``(i, j, k)`` corresponds to the ``i``-th derivative, degree ``j``, and order ``k`` for all ``0 <= i <= diff_n``, ``0 <= j <= n``, and ``-m <= k <= m``. See Also -------- sph_legendre_p cX[USSS9nSUs=::aS::dO [SUS35eU$rrrs rrrrr!cSS/S/-0$Naxesr)rrrrrs rrrs RDJ<' ((r!c[U[R5(aUS:a [S5eUS-S[ U5-S-4U-US-4-4$)Nr!n must be a non-negative integer.rr)r$numbersIntegralr,abs)nm theta_shapertrs rrrsU a)) * *q1u<== UAAJN #k 1VaZM A CCr!c2[R"USS5$rrrs rrrrr!raassoc_legendre_p(n, m, z, *, branch_cut=2, norm=False, diff_n=0) Associated Legendre polynomial of the first kind. Parameters ---------- n : ArrayLike[int] Degree of the associated Legendre polynomial. Must have ``n >= 0``. m : ArrayLike[int] order of the associated Legendre polynomial. z : ArrayLike[float | complex] Input value. branch_cut : Optional[ArrayLike[int]] Selects branch cut. Must be 2 (default) or 3. 2: cut on the real axis ``|z| > 1`` 3: cut on the real axis ``-1 < z < 1`` norm : Optional[bool] If ``True``, compute the normalized associated Legendre polynomial. Default is ``False``. diff_n : Optional[int] A non-negative integer. Compute and return all derivatives up to order ``diff_n``. Default is 0. Returns ------- p : ndarray or tuple[ndarray] Associated Legendre polynomial with ``diff_n`` derivatives. Notes ----- The normalized counterpart of an (unnormalized) associated Legendre polynomial has the additional factor .. math:: \sqrt{\frac{(2 n + 1) (n - m)!}{2 (n + m)!}} rF branch_cutnormrcZ[USSS9nSUs=::aS::dO [SUS35eX4$rrrs rrr#sE % @F  !    $   <r!cU4$r#rrs rrr. ;r!c2[R"USS5$rrrs rrr3rr!r aassoc_legendre_p_all(n, m, z, *, branch_cut=2, norm=False, diff_n=0) All associated Legendre polynomials of the first kind up to the specified degree ``n``, order ``m``, and all derivatives up to order ``diff_n``. Output shape is ``(diff_n + 1, n + 1, 2 * m + 1, ...)``. The entry at ``(i, j, k)`` corresponds to the ``i``-th derivative, degree ``j``, and order ``k`` for all ``0 <= i <= diff_n``, ``0 <= j <= n``, and ``-m <= k <= m``. See Also -------- assoc_legendre_p c[U[R5(aUS:d[SUS35eSUs=::aS::dO [SUS35eX4$Nrz1diff_n must be a non-negative integer, received: rrr)r$rrr,rs rrrMsl  0 0 1 1! ?xq I    !    $   <r!cU4$r#rrs rrr\rr!cSSS/S/-0$rrrs rrras RH |+ ,,r!c 6USn[U[R5(aUS:a [S5e[U[R5(aUS:a [S5eUS-S[ U5-S-4[ R "X#5-US-4-4$)Nrrrz!m must be a non-negative integer.rrr$rrr,rr%broadcast_shapes)rrz_shapebranch_cut_shapertrrs rrrfs H F a)) * *q1u<== a)) * *q1u<== UAAJN # G6 7:@1* G IIr!c2[R"USS5$rrrs rrrsrr!ralegendre_p(n, z, *, diff_n=0) Legendre polynomial of the first kind. Parameters ---------- n : ArrayLike[int] Degree of the Legendre polynomial. Must have ``n >= 0``. z : ArrayLike[float] Input value. diff_n : Optional[int] A non-negative integer. Compute and return all derivatives up to order ``diff_n``. Default is 0. Returns ------- p : ndarray or tuple[ndarray] Legendre polynomial with ``diff_n`` derivatives. See Also -------- legendre References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html c[U[R5(aUS:a[SUS35eSUs=::aS::dO [ SUS35eU$r)r$rrr,NotImplementedErrorrs rrrsh vw// 0 0fqj?xq I    ! !   $   Mr!c2[R"USS5$rrrs rrrrr!r alegendre_p_all(n, z, *, diff_n=0) All Legendre polynomials of the first kind up to the specified degree ``n`` and all derivatives up to order ``diff_n``. Output shape is ``(diff_n + 1, n + 1, ...)``. The entry at ``(i, j)`` corresponds to the ``i``-th derivative and degree ``j`` for all ``0 <= i <= diff_n`` and ``0 <= j <= n``. See Also -------- legendre_p cX[USSS9nSUs=::aS::dO [SUS35eU$rrrs rrrrr!cSSS/0$)Nrr)rrrrs rrrs RM ""r!c>[USSS9nX S-4U-US-4-4-$)NrFrrr)rrrtrs rrrs2As51A E8g%! 57 77r!c2[R"USS5$rrrs rrrrr!r asph_harm_y(n, m, theta, phi, *, diff_n=0) Spherical harmonics. They are defined as .. math:: Y_n^m(\theta,\phi) = \sqrt{\frac{2 n + 1}{4 \pi} \frac{(n - m)!}{(n + m)!}} P_n^m(\cos(\theta)) e^{i m \phi} where :math:`P_n^m` are the (unnormalized) associated Legendre polynomials. Parameters ---------- n : ArrayLike[int] Degree of the harmonic. Must have ``n >= 0``. This is often denoted by ``l`` (lower case L) in descriptions of spherical harmonics. m : ArrayLike[int] Order of the harmonic. theta : ArrayLike[float] Polar (colatitudinal) coordinate; must be in ``[0, pi]``. phi : ArrayLike[float] Azimuthal (longitudinal) coordinate; must be in ``[0, 2*pi]``. diff_n : Optional[int] A non-negative integer. Compute and return all derivatives up to order ``diff_n``. Default is 0. Returns ------- y : ndarray[complex] or tuple[ndarray[complex]] Spherical harmonics with ``diff_n`` derivatives. Notes ----- There are different conventions for the meanings of the input arguments ``theta`` and ``phi``. In SciPy ``theta`` is the polar angle and ``phi`` is the azimuthal angle. It is common to see the opposite convention, that is, ``theta`` as the azimuthal angle and ``phi`` as the polar angle. Note that SciPy's spherical harmonics include the Condon-Shortley phase [2]_ because it is part of `sph_legendre_p`. With SciPy's conventions, the first several spherical harmonics are .. math:: Y_0^0(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{1}{\pi}} \\ Y_1^{-1}(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{3}{2\pi}} e^{-i\phi} \sin(\theta) \\ Y_1^0(\theta, \phi) &= \frac{1}{2} \sqrt{\frac{3}{\pi}} \cos(\theta) \\ Y_1^1(\theta, \phi) &= -\frac{1}{2} \sqrt{\frac{3}{2\pi}} e^{i\phi} \sin(\theta). References ---------- .. [1] Digital Library of Mathematical Functions, 14.30. https://dlmf.nist.gov/14.30 .. [2] https://en.wikipedia.org/wiki/Spherical_harmonics#Condon.E2.80.93Shortley_phase T)rrcX[USSS9nSUs=::aS::dO [SUS35eU$rrrs rrr!rr!cURSS:XaUS$URSS:XaUSUSSS/SS/44$URSS:Xa$USUSSS/SS/4USSS/SS//SS/SS//44$gNrr).rrr.rr`rs rrr, " 9~ " 9~s3AA#6777 " IC!Q!Q$7 8 q!fq!f%AA'77 8: : r!r asph_harm_y_all(n, m, theta, phi, *, diff_n=0) All spherical harmonics up to the specified degree ``n``, order ``m``, and all derivatives up to order ``diff_n``. Returns a tuple of length ``diff_n + 1`` (if ``diff_n > 0``). The first entry corresponds to the spherical harmonics, the second entry (if ``diff_n >= 1``) to the gradient, and the third entry (if ``diff_n >= 2``) to the Hessian matrix. Each entry is an array of shape ``(n + 1, 2 * m + 1, ...)``, where the entry at ``(i, j)`` corresponds to degree ``i`` and order ``j`` for all ``0 <= i <= n`` and ``-m <= j <= m``. See Also -------- sph_harm_y cX[USSS9nSUs=::aS::dO [SUS35eU$)NrFrrrz=diff_n is currently only implemented for orders 2, received: rrrs rrrPrr!cSSS/S/-0$)Nrr)rrrrrs rrr[s RH// 00r!c USn[U[R5(aUS:a [S5eUS-S[ U5-S-4[ R "X#5-US-US-4-4$)Nrrrrrr)rrr phi_shapertrrs rrr`sx H F a)) * *q1u<== UAAJN #b&9&9+&Q Q !VaZ  ! ##r!cURSS:XaUS$URSS:XaUSUSSS/SS/44$URSS:Xa$USUSSS/SS/4USSS/SS//SS/SS//44$grrrs rrrkrr!)r'rnumpyr%_input_validationr_special_ufuncsrrrr _gufuncsr r r r __all__rrKrrXrQrUrNrr!rrs2::;; ttn @G$N&&#'# #*!!"22)3)00D1D**#+#$HE!O(V ../((#)#"E!#*## $ 22344-5-22 I3 I,,#-#8?  F    ""### &..#/#,,8-8 &&#'#=z#1AA H  "" :# : #1'...1/1,,#-#&& :' :r!