\h'LSrSSKJr SSKJrJrJrJrJrJ r J r J r SSK J r Jr SSKJr SSKJr Sr\SS j5rS rSS jrS rS rSrSr\SSj5r\SSj5rSrSr\SSj5r\SSj5r Sr!\SSj5r"Sr#\SSj5r$Sr%Sr&SSjr'g)z:Efficient functions for generating orthogonal polynomials.)Dummy)dup_muldup_mul_ground dup_lshiftdup_subdup_add dup_sub_termdup_sub_grounddup_sqr)ZZQQ) named_poly)publicc US:a UR/$UR/X-U"S5- UR-X- U"S5- /pT[SUS-5GH2nU"U5X-U--X-U"S5U--U"S5- -nX-U"S5U--UR- X-X"-- -U"S5U-- nX-U"S5U--UR- X-U"S5U--U"S5- -X-U"S5U---U"S5U-- n X-UR- X&-UR- -X-U"S5U---U- n [XXU5n [[USU5X5n [XJU5n U[ [ XU5X5pTGM5 U$)z/Low-level implementation of Jacobi polynomials.)onerangerrrr)nabKm2m1idenf0f1f2p0p1p2s N/var/www/auris/envauris/lib/python3.13/site-packages/sympy/polys/orthopolys.py dup_jacobir$ s1uweeWQqTzAEE)AC1:6 1ac]dAEAI!Q1 56ead1fnquu$qs 3qtCx @ead1fnquu$1a!A$)> ?151Q4PQ6> RVWXYVZ[^V^ _eaeemaeaeem ,aead1fn = C BA & Jr1a0" 8 BA &WWRQ/7B INc .[U[SSX1U4U5$)a^Generates the Jacobi polynomial `P_n^{(a,b)}(x)`. Parameters ========== n : int Degree of the polynomial. a Lower limit of minimal domain for the list of coefficients. b Upper limit of minimal domain for the list of coefficients. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. NzJacobi polynomial)rr$)rrrxpolyss r# jacobi_polyr)s" aT+>q 5 QQr%cUS:a UR/$UR/U"S5U-UR/pC[SUS-5Hn[[ USU5U"S5XR- -U"U5- U"S5-U5n[X2"S5XR- -U"U5- UR-U5nU[ XgU5pCM U$)z3Low-level implementation of Gegenbauer polynomials.rrrzerorrrr)rrrrrrr!r"s r#dup_gegenbauerr-,s1uweeWqtAvqvv& 1ac] Jr1a0!A$%%.12E!2La P B!agqt 3aee ;Q ?WRQ'B Ir%c,[U[SSX!4U5$)aGenerates the Gegenbauer polynomial `C_n^{(a)}(x)`. Parameters ========== n : int Degree of the polynomial. x : optional a Decides minimal domain for the list of coefficients. polys : bool, optional If True, return a Poly, otherwise (default) return an expression. NzGegenbauer polynomial)rr-)rrr'r(s r#gegenbauer_polyr/7s a/FPU VVr%c`US:a UR/$US:a [X5$[X5$)zDLow-level implementation of Chebyshev polynomials of the first kind.r@)r_dup_chebyshevt_rec_dup_chebyshevt_prod)rrs r#dup_chebyshevtr4Gs11uw2v"1((  %%r%c UR/URUR/p2[US- 5H,nU[[ [ USU5U"S5U5X!5p2M. U$)aChebyshev polynomials of the first kind using recurrence. Explanation =========== Chebyshev polynomials of the first kind are defined by the recurrence relation: .. math:: T_0(x) &= 1\\ T_1(x) &= x\\ T_n(x) &= 2xT_{n-1}(x) - T_{n-2}(x) This function calculates the Chebyshev polynomial of the first kind using the above recurrence relation. Parameters ========== n : int n is a nonnegative integer. K : domain rrrr,rrrr)rrrr_s r#r2r2PsX2eeWquuaffo 1q5\W^Jr1a,@!A$JBRB Ir%c URUR/U"S5URUR*/p2[U5SSHn[[ [ X2U5U"S5U5URSU5nUS:Xa4U[ [ [X15U"S5U5URU5p2Mp[ [ [X!5U"S5U5URU5Up2M U$)arChebyshev polynomials of the first kind using recursive products. Explanation =========== Computes Chebyshev polynomials of the first kind using .. math:: T_{2n}(x) &= 2T_n^2(x) - 1\\ T_{2n+1}(x) &= 2T_{n+1}(x)T_n(x) - x This is faster than ``_dup_chebyshevt_rec`` for large ``n``. Parameters ========== n : int n is a nonnegative integer. K : domain rNr1)rr,binr rrr r )rrrrrcs r#r3r3ns,eeQVV_qtQVVaeeV4 VABZ (:AaD!DaeeQPQ R #I~gbnadA'NPQPUPUWXY#N72>1Q4$KQUUTUVXY  Ir%c US:a UR/$UR/U"S5UR/p2[SUS-5H,nU[[ [ USU5U"S5U5X!5p2M. U$)zELow-level implementation of Chebyshev polynomials of the second kind.rrr6rrrrrs r#dup_chebyshevur?sj1uweeWqtQVVn 1ac]W^Jr1a,@!A$JBRB Ir%c4[U[[SU4U5$)zGenerates the Chebyshev polynomial of the first kind `T_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. z&Chebyshev polynomial of the first kind)rr4r rr'r(s r#chebyshevt_polyrBs" a 4qdE CCr%c4[U[[SU4U5$)zGenerates the Chebyshev polynomial of the second kind `U_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. z'Chebyshev polynomial of the second kind)rr?r rAs r#chebyshevu_polyrDs" a 5tU DDr%c US:a UR/$UR/U"S5UR/p2[SUS-5HCn[USU5n[ X!"US- 5U5nU[ [ XVU5U"S5U5p2ME U$)z0Low-level implementation of Hermite polynomials.rrrr,rrrrrrrrrrrs r# dup_hermiterHs1uweeWqtQVVn 1ac] r1a  2q1vq )^GA!$4adA>B Ir%cUS:a UR/$UR/URUR/p2[SUS-5H2n[USU5n[ X!"US- 5U5nU[ XVU5p2M4 U$)z>Low-level implementation of probabilist's Hermite polynomials.rrrFrGs r#dup_hermite_probrJsz1uweeWquuaffo 1ac] r1a  2q1vq )WQ1%B Ir%c4[U[[SU4U5$)zGenerates the Hermite polynomial `H_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. zHermite polynomial)rrHr rAs r# hermite_polyrLs ab*>e LLr%c4[U[[SU4U5$)zGenerates the probabilist's Hermite polynomial `He_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. z probabilist's Hermite polynomial)rrJr rAs r#hermite_prob_polyrNs! a)2 .e ==r%c0US:a UR/$UR/URUR/p2[SUS-5HKn[[ USU5U"SU-S- U5U5n[X!"US- U5U5nU[ XVU5p2MM U$)z1Low-level implementation of Legendre polynomials.rrr+rGs r# dup_legendrerPs1uweeWquuaffo 1ac] :b!Q/1Q3q5!a @ 2q1ay! ,WQ1%B Ir%c4[U[[SU4U5$)zGenerates the Legendre polynomial `P_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. zLegendre polynomial)rrPr rAs r# legendre_polyrRs ar+@1$ NNr%c \UR/UR/pC[SUS-5Hn[XBR*U"U5- XR- U"U5- U"S5-/U5n[ X1UR- U"U5- UR-U5nU[ XgU5pCM U$)z1Low-level implementation of Laguerre polynomials.rr)r,rrrrr)ralpharrrrrrs r# dup_laguerrerUsffXw 1ac] B%%!uUU{AaD&81Q4&?@! D 2aee QqT1AEE91 =WQ1%B Ir%c,[U[SSX4U5$)a)Generates the Laguerre polynomial `L_n^{(\alpha)}(x)`. Parameters ========== n : int Degree of the polynomial. x : optional alpha : optional Decides minimal domain for the list of coefficients. polys : bool, optional If True, return a Poly, otherwise (default) return an expression. NzLaguerre polynomial)rrU)rr'rTr(s r# laguerre_polyrWs at-BQJPU VVr%c *US:aURUR/$UR/URUR/p2[SUS-5H2nU[[ [ USU5U"SU-S- 5U5X!5p2M4 [ USU5$)z%Low-level implementation of fn(n, x).rrr6r>s r#dup_spherical_bessel_fnrYs1uqvveeWquuaffo 1ac]W^Jr1a,@!AaCE(ANPRVB b!Q r%c URUR/UR/p2[SUS-5H2nU[[ [ USU5U"SSU-- 5U5X!5p2M4 U$)z&Low-level implementation of fn(-n, x).rrr9r6r>s r#dup_spherical_bessel_fn_minusr[(saeeQVV_qvvh 1ac]W^Jr1a,@!AacE(ANPRVB Ir%c Uc [S5nUS:a[O[n[[ U5U[ S[ S5U- 4U5$)a| Coefficients for the spherical Bessel functions. These are only needed in the jn() function. The coefficients are calculated from: fn(0, z) = 1/z fn(1, z) = 1/z**2 fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z) Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. Examples ======== >>> from sympy.polys.orthopolys import spherical_bessel_fn as fn >>> from sympy import Symbol >>> z = Symbol("z") >>> fn(1, z) z**(-2) >>> fn(2, z) -1/z + 3/z**3 >>> fn(3, z) -6/z**2 + 15/z**4 >>> fn(4, z) 1/z - 45/z**3 + 105/z**5 r'rr)rr[rYrabsr r )rr'r(fs r#spherical_bessel_fnr`/sEJ y #J)*Q%4KA c!faR"Q%'U ;;r%)NF)NrF)(__doc__sympy.core.symbolrsympy.polys.densearithrrrrrr r r sympy.polys.domainsr r sympy.polys.polytoolsrsympy.utilitiesrr$r)r-r/r4r2r3r?rBrDrHrJrLrNrPrRrUrWrYr[r`r%r#rhs@#III&," RR$ W &<> C C D D   M M = =  O OWW  (