+ nDjX ^RIt^RIt^RIt^RIHt^RIHtHtH t H t ^RI H t H t HtHt.R6Ot!R R4t]!] R R^R7t ] P$R4t] P(R4t]!]R R^R7t]P$R4t]P*R4t]P,R4t]P(R4t]!]RR^R^R7t]P$R4t]P.R4t]P(R4t]!] RR^R^R7t ] P$R4t] P.R 4t] P*R!4t] P,R"4t] P(R#4t]!]RR$^R7t]P$R%4t]P(R&4t]!] RR'^R7t ] P$R(4t] P*R)4t] P,R*4t] P(R+4t]!] R R,R-^R.7t ] P$R/4t] P(R04t]!]R R1R-^R.7t]P$R24t]P*R34t]P,R44t]P(R54tR#)7N_nonneg_int_or_fail) legendre_passoc_legendre_psph_legendre_p sph_harm_y)legendre_p_allassoc_legendre_p_allsph_legendre_p_allsph_harm_y_allrr rr rr rr cla]tRt^toRRR/Rllt]R4tRtRtRt R t R t R t R t R tVtR#) MultiUFuncNforce_complex_outputFc \V\P4'g\V\PP 4'dVP 4pM8\V\PP4'dTpM \R4h\4pVF\p\V\P4'g\RV 24hVP\RVP444K^ \V4^8d \R4hW nWnW0nW@nWPnRVnRVnRVnRVnRVnV'd VP1R4^,VnR#RVnR#)z7ufunc_or_ufuncs should be a ufunc or a ufunc collectionz2All ufuncs must have type `numpy.ufunc`. Received c3P"TFqPR4^,xK R#5i)z->N)split).0xs& T/data/cameron/venvs/s3viz/lib/python3.14/site-packages/scipy/special/_multiufuncs.py &MultiUFunc.__init__..+s.UAwwt}Q/?/?$&z*All ufuncs must take the same input types.NcR#)Nrargskwargss*,r%MultiUFunc.__init__..7s2c/#Nrrs*,rrr8sRr z ) 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_kwargsr__text_signature__) selfufunc_or_ufuncsnamedocrdefault_kwargs ufuncs_iterseen_input_typesr%s &&&&$, r__init__MultiUFunc.__init__sF/28844/;??+B+BCC-446 O[__-E-EFF-  "566 #u $!%22$&22A1B&DEE $$Y.U.U%UV % #$q( !MNN / &:#-#' ! #= %?":=#))F"3A"64r cVP#r")r3)r<s&r__doc__MultiUFunc.__doc__;s zzr c WnR#)z3Set `key` method by decorating a function. N)r8r<funcs&&r _override_keyMultiUFunc._override_key?s  r cWnR#r")r9rIs&&r_override_ufunc_default_args'MultiUFunc._override_ufunc_default_argsDs#' r cWnR#r")r:rIs&&r_override_ufunc_default_kwargs)MultiUFunc._override_ufunc_default_kwargsGs%)"r c JVPfRVnRVnWnR#)z9Set `resolve_out_shapes` method by decorating a function.Nz2Resolve to output shapes based on relevant inputs.resolve_out_shapes)rFr1r6rIs&&r_override_resolve_out_shapes'MultiUFunc._override_resolve_out_shapesJs# << H L, #' r cWnR#r")r7rIs&&r_override_finalize_out!MultiUFunc._override_finalize_outRs!r c \VP\P4'd VP#VP!R/VBpVPV,#)z.Resolve to a ufunc based on keyword arguments.r)r#r2r$r%r8)r<r ufunc_keys&, r_resolve_ufuncMultiUFunc._resolve_ufuncUsI d++RXX 6 6(( (II'' $$Y//r cVPV,pWP!R/VB, pVP!R/VBpWP)RUu.uFp\P !V4NK ppVP !R/VBpVPEe\;QJd.RV4FNK 5M !RV44pVP!.VRVP)OVOVPN5/VBp\;QJd.RV4FNK 5M !RV44p \VR4'd;WPR,,p VPV 4p WP)Rp Mc\P!V !p \P!V \P4'g\Pp VPV 3,p VP 'd-\;QJd.RV 4FNK 5M !RV 44p \;QJd.R\#W44FNK 5M!R\#W444p WR&V!V/VBp VP$eVP%V 4p V #uupi) Nc3N"TFp\P!V4xK R#5ir")r$shaper ufunc_args& rr&MultiUFunc.__call__..ks$U*YRXXi%8%8*s#%c3"TFBp\VR4'd VPM\P!\V44xKD R#5i)dtypeN)hasattrrer$typeras& rrrcps?%B6@9@ 78S8SY__*,((4 ?*C&D6@sA A resolve_dtypesc3P"TFp\P!RV4xK R#5i)y?N)r$ result_type)rufunc_out_dtypes& rrrcs&)R@P_*,O)L)L@Prc3T"TFwr\P!WR7xK R#5i))reN)r$empty)rufunc_out_shaperks& rrrcs)DB=OHHBs&(outrr")r5r9r\ninr$asarrayr:r6tuplenoutrfrhrj issubdtypeinexactfloat64r4zipr7)r<rrr%arg ufunc_args ufunc_kwargsufunc_arg_shapesufunc_out_shapesufunc_arg_dtypes ufunc_dtypesufunc_out_dtypesrkros&*, r__call__MultiUFunc.__call__^sW%%. ((2622##-f-26yyjk1BC1B#bjjo1B C11;F;  $ $ 0$u$U*$Uuu$U*$UU #77 Bk z9J B9I BKP:: B:@ B  %u%B6@%Buu%B6@%B B u.///**w2FF $33LA #/ #= "$..2B"C orzzBB&(jjO#(::0B#B ***#(5)R@P)R55)R@P)R$R %D/BD%%D/BDDC#& Z0<0    *$$S)C ODsI2) __doc__force_complex_outputr1r;r5r7r8r6r9r:r2)NN)r1 __module__ __qualname____firstlineno__rCpropertyrFrKrNrQrUrXr\r__static_attributes____classdictcell__) __classdict__s@rrrsQ H&+ HD (*("0//r rasph_legendre_p(n, m, theta, *, diff_n=0) Spherical Legendre polynomial of the first kind. Parameters ---------- n : array_like of ints Degree of the spherical Legendre polynomial. Must have ``n >= 0``. m : array_like of ints Order of the spherical Legendre polynomial. theta : array_like Input value. diff_n : int, optional A non-negative integer. Compute and return all derivatives up to order `diff_n`. Default is 0. Returns ------- ndarray or tuple of 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_ncb\VRRR7p^Tu;8:d^8:gM\RV R24hV#rFstrictGdiff_n is currently only implemented for orders 0, 1, and 2, received: .rr+rs&r_rB % @F  !    $   Mr c2\P!VR^4#r$moveaxisros&rrr ;;sB ""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`. Parameters ---------- n : int Degree of the spherical Legendre polynomials. Must have ``n >= 0``. m : int Order of the spherical Legendre polynomials. theta : array_like Input value. diff_n : int, optional A non-negative integer. Compute and return all derivatives up to order `diff_n`. Default is 0. Returns ------- ndarray 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 cb\VRRR7p^Tu;8:d^8:gM\RV R24hV#rrrs&rrrrr cRR.R.,/#axesr)rrrrrs&rrrs RDJ<' ((r c\V\P4'dV^8d \R4hV^,^\ V4,^,3V,V^,3,3#)r!n must be a non-negative integer.)r#numbersIntegralr+abs)nm theta_shapersrs&&&&&rrrsU a)) * *q1u<== UAAJN #k 1VaZM A CCr c2\P!VR^4#rrrs&rrrrr aYassoc_legendre_p(n, m, z, *, branch_cut=2, norm=False, diff_n=0) Associated Legendre polynomial of the first kind. Defined as .. math:: P_n^m(z) = (-1)^m (1 - z^2)^{m/2} \frac{d^m}{dz^m} P_n(z) where :math:`P_n` is the Legendre polynomial. This definition includes the Condon-Shortley phase :math:`(-1)^m`. Parameters ---------- n : array_like of ints Degree of the associated Legendre polynomial. Must have ``n >= 0``. m : array_like of ints Order of the associated Legendre polynomial. z : array_like Input value. branch_cut : array_like of ints, optional 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 : bool, optional If ``True``, compute the normalized associated Legendre polynomial. Default is ``False``. diff_n : int, optional A non-negative integer. Compute and return all derivatives up to order ``diff_n``. Default is 0. Returns ------- ndarray or tuple of 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)!}} Examples -------- Evaluate :math:`P_2^1(z)` on :math:`[-1, 1]`: >>> import numpy as np >>> from scipy.special import assoc_legendre_p >>> z = np.linspace(-1, 1, 11) >>> expected = -3 * z * np.sqrt(1 - z**2) >>> np.allclose(assoc_legendre_p(2, 1, z), expected) True Compute the normalized associated Legendre polynomial: >>> scale = np.sqrt(5/12) >>> np.allclose(assoc_legendre_p(2, 1, z, norm=True), scale * expected) True F branch_cutnormrcd\VRRR7p^Tu;8:d^8:gM\RV R24hW3#rrrs&&&rrrOsE % @F  !    $   <r cV3#r"rrs&&&rrrZ ;r c2\P!VR^4#rrrs&rrr_rr 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`. Parameters ---------- n : int Degree of the associated Legendre polynomials. Must have ``n >= 0``. m : int Order of the associated Legendre polynomials. z : array_like Input value. branch_cut : array_like of ints, optional 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 : bool, optional If ``True``, compute the normalized associated Legendre polynomials. Default is ``False``. diff_n : int, optional A non-negative integer. Compute and return all derivatives up to order ``diff_n``. Default is 0. Returns ------- ndarray 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\V\P4'dV^8g\RV R24h^Tu;8:d^8:gM\RV R24hW3#rz1diff_n must be a non-negative integer, received: rr)r#rrr+rs&&&rrrsl  0 0 1 1! ?xq I    !    $   <r cV3#r"rrs&&&rrrrr cRRR.R.,/#rrrs&&&rrrs RH |+ ,,r c vVR,p\V\P4'dV^8d \R4h\V\P4'dV^8d \R4hV^,^\ V4,^,3\ P !W#4,V^,3,3#)rrz!m must be a non-negative integer.r#rrr+rr$broadcast_shapes)rrz_shapebranch_cut_shapersrrs&&&&&, rrrs H F a)) * *q1u<== a)) * *q1u<== UAAJN # G6 7:@1* G IIr c2\P!VR^4#rrrs&rrrrr alegendre_p(n, z, *, diff_n=0) Legendre polynomial of the first kind. Parameters ---------- n : array_like of ints Degree of the Legendre polynomial. Must have ``n >= 0``. z : array_like Input value. diff_n : int, optional A non-negative integer. Compute and return all derivatives up to order ``diff_n``. Default is 0. Returns ------- p : ndarray or tuple of 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 Examples -------- Evaluate the Legendre polynomial :math:`P_3` at :math:`z = 0.5`: >>> import numpy as np >>> from scipy.special import legendre_p >>> np.allclose(legendre_p(3, 0.5), -0.4375) True Compute the value and first derivative with respect to ``z``: >>> p, dp = legendre_p(3, 0.5, diff_n=1) >>> np.allclose([p, dp], [-0.4375, 0.375]) True c\V\P4'dV^8d\RV R24h^Tu;8:d^8:gM\ RV R24hV#r)r#rrr+NotImplementedErrorrs&rrrsh vw// 0 0fqj?xq I    ! !   $   Mr c2\P!VR^4#rrrs&rrrrr 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`. Parameters ---------- n : int Degree of the Legendre polynomials. Must have ``n >= 0``. z : array_like Input value. diff_n : int, optional A non-negative integer. Compute and return all derivatives up to order ``diff_n``. Default is 0. Returns ------- ndarray 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 cb\VRRR7p^Tu;8:d^8:gM\RV R24hV#rrrs&rrrrr cRRR./#)rr)rrrrs&rrr*s RM ""r cl\VRRR7pW ^,3V,V^,3,3,#)rFrr)rrrsrs&&&&rrr/s2As51A E8g%! 57 77r c2\P!VR^4#rrrs&rrr6rr 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 : array_like of ints Degree of the harmonic. Must have ``n >= 0``. This is often denoted by ``l`` (lower case L) in descriptions of spherical harmonics. m : array_like of ints Order of the harmonic. theta : array_like Polar (colatitudinal) coordinate; must be in ``[0, pi]``. phi : array_like Azimuthal (longitudinal) coordinate; must be in ``[0, 2*pi]``. diff_n : int, optional A non-negative integer. Compute and return all derivatives up to order `diff_n`. Default is 0. Returns ------- ndarray or tuple of ndarray 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)rrcb\VRRR7p^Tu;8:d^8:gM\RV R24hV#rrrs&rrrrr c:VPR,^8Xd VR,#VPR,^8XdVR,VR^^.^^.3,3#VPR,^8Xd1VR,VR^^.^^.3,VR^^.^^..^^.^^..3,3#R#r.Nr).rrr`rs&rrr " 9~ " 9~s3AA#6777 " IC!Q!Q$7 8 q!fq!f%AA'77 8: : r asph_harm_y_all(n, m, theta, phi, *, diff_n=0) All spherical harmonics up to the specified degree `n`, order `m`, and all derivatives up to order `diff_n`. Parameters ---------- n : int Degree of the harmonics. Must have ``n >= 0``. This is often denoted by ``l`` (lower case L) in descriptions of spherical harmonics. m : int Order of the harmonics. theta : array_like Polar (colatitudinal) coordinate; must be in ``[0, pi]``. phi : array_like Azimuthal (longitudinal) coordinate; must be in ``[0, 2*pi]``. diff_n : int, optional A non-negative integer. Compute and return all derivatives up to order `diff_n`. Default is 0. Returns ------- ndarray or tuple of ndarray Returns a tuple of length ``diff_n + 1`` (if ``diff_n > 0``). The first entry corresponds to the spherical harmonics, the second entry (if ``diff_n >= 1``) to the gradient, and the third entry (if ``diff_n >= 2``) to the Hessian matrix. Each entry is an array of shape ``(n + 1, 2 * m + 1, ...)``, where the entry at ``(i, j)`` corresponds to degree ``i`` and order ``j`` for all ``0 <= i <= n`` and ``-m <= j <= m``. See Also -------- sph_harm_y cb\VRRR7p^Tu;8:d^8:gM\RV R24hV#)rFrz=diff_n is currently only implemented for orders 2, received: rrrs&rrrrr cRRR.R.,/#)rr)rrrrrs&rrrs RH// 00r c "VR,p\V\P4'dV^8d \R4hV^,^\ V4,^,3\ P !W#4,V^,V^,3,3#)rrr)rrr phi_shapersrrs&&&&&, rrrsx H F a)) * *q1u<== UAAJN #b&9&9+&Q Q !VaZ  ! ##r c:VPR,^8Xd VR,#VPR,^8XdVR,VR^^.^^.3,3#VPR,^8Xd1VR,VR^^.^^.3,VR^^.^^..^^.^^..3,3#R#rrrs&rrrrr )rr rr rr rr )r&rnumpyr$_input_validationr_special_ufuncsrrrr_gufuncsr r r r __all__rrKrrXrQrUrNrr rrs2::;; uup @G$N&&#'# :A!H!!"22)3)00D1D**#+#@@E!GDN ../((#)#"$HE!O(V## $ 22344-5-22 I3 I,,#-#+V]/ d    ""###29@..#/#,,8-8 &&#'#?~#1EC L  "" :# :#F#1M'T..1/1,,#-#&& :' :r