o GZh'@s:dZddlmZddlmZmZmZmZmZm Z m Z m Z ddl m Z mZddlmZddlmZddZed3d d Zd dZd3ddZddZddZddZddZed3ddZed3ddZddZdd Zed3d!d"Zed3d#d$Z d%d&Z!ed3d'd(Z"d)d*Z#ed4d+d,Z$d-d.Z%d/d0Z&d3d1d2Z'd S)5z:Efficient functions for generating orthogonal polynomials.)Dummy)dup_muldup_mul_ground dup_lshiftdup_subdup_add dup_sub_termdup_sub_grounddup_sqr)ZZQQ) named_poly)publiccCs|dkr|jgS|jg|||d|j|||dg}}td|dD]}||||||||d||d}|||d||j|||||d|}|||d||j|||d||d|||d||d|} |||j|||j|||d||} t|||} tt|d|| |} t|| |} |tt| | || |}}q'|S)z/Low-level implementation of Jacobi polynomials.)onerangerrrr)nabKm2m1iZdenZf0f1f2p0p1p2rE/var/www/auris/lib/python3.10/site-packages/sympy/polys/orthopolys.py dup_jacobi s006V4  r!NFcCst|tdd|||f|S)aGenerates 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)r r!)rrrxpolysrrr jacobi_polysr$cCs|dkr|jgS|jg|d||jg}}td|dD]8}tt|d||d||j|||d|}t||d||j|||j|}|t|||}}q|S)z3Low-level implementation of Gegenbauer polynomials.rrrzerorrrr)rrrrrrrrrrr dup_gegenbauer,s2(r'cCst|tdd||f|S)a?Generates the Gegenbauer polynomial `C_n^{(a)}(x)`. Parameters ========== n : int Degree of the polynomial. x : optional a Decides minimal domain for the list of coefficients. polys : bool, optional If True, return a Poly, otherwise (default) return an expression. NzGegenbauer polynomial)r r')rrr"r#rrr gegenbauer_poly7sr(cCs,|dkr|jgS|dkrt||St||S)zDLow-level implementation of Chebyshev polynomials of the first kind.r@)r_dup_chebyshevt_rec_dup_chebyshevt_prod)rrrrr dup_chebyshevtGs   r,cCsR|jg|j|jg}}t|dD]}|ttt|d||d|||}}q|S)a Chebyshev polynomials of the first kind using recurrence. Explanation =========== Chebyshev polynomials of the first kind are defined by the recurrence relation: .. math:: T_0(x) &= 1\\ T_1(x) &= x\\ T_n(x) &= 2xT_{n-1}(x) - T_{n-2}(x) This function calculates the Chebyshev polynomial of the first kind using the above recurrence relation. Parameters ========== n : int n is a nonnegative integer. K : domain rrrr&rrrr)rrrr_rrr r*Ps(r*cCs|j|jg|d|j|j g}}t|ddD]?}ttt||||d||jd|}|dkrE|ttt|||d||j|}}qttt|||d||j||}}q|S)a Chebyshev polynomials of the first kind using recursive products. Explanation =========== Computes Chebyshev polynomials of the first kind using .. math:: T_{2n}(x) &= 2T_n^2(x) - 1\\ T_{2n+1}(x) &= 2T_{n+1}(x)T_n(x) - x This is faster than ``_dup_chebyshevt_rec`` for large ``n``. Parameters ========== n : int n is a nonnegative integer. K : domain rNr1)rr&binrrrr r )rrrrrcrrr r+ns"$((r+cCsf|dkr|jgS|jg|d|jg}}td|dD]}|ttt|d||d|||}}q|S)zELow-level implementation of Chebyshev polynomials of the second kind.rrr-rrrrrrrr dup_chebyshevus (r4cCt|ttd|f|S)aGenerates 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)r r,r rr"r#rrr chebyshevt_poly r7cCr5)aGenerates 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)r r4r r6rrr chebyshevu_polyr8r9cCs~|dkr|jgS|jg|d|jg}}td|dD]!}t|d|}t|||d|}|tt||||d|}}q|S)z0Low-level implementation of Hermite polynomials.rrrr&rrrrrrrrrrrrrr dup_hermites  r<cCsp|dkr|jgS|jg|j|jg}}td|dD]}t|d|}t|||d|}|t|||}}q|S)z>Low-level implementation of probabilist's Hermite polynomials.rrr:r;rrr dup_hermite_probs r=cCr5)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)r r<r r6rrr hermite_poly r>cCr5)aGenerates 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)r r=r r6rrr hermite_prob_polyr8r@cCs|dkr|jgS|jg|j|jg}}td|dD]'}tt|d||d|d||}t|||d||}|t|||}}q|S)z1Low-level implementation of Legendre polynomials.rrr%r;rrr dup_legendres"rAcCr5)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)r rAr r6rrr legendre_polyr?rBcCs|jg|jg}}td|dD]4}t||j ||||j|||dg|}t|||j|||j|}|t|||}}q|S)z1Low-level implementation of Laguerre polynomials.rr)r&rrrrr)ralpharrrrrrrrr dup_laguerres 2 rDcCst|tdd||f|S)aQGenerates 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)r rD)rr"rCr#rrr laguerre_polysrEcCsx|dkr |j|jgS|jg|j|jg}}td|dD]}|ttt|d||d|d|||}}qt|d|S)z%Low-level implementation of fn(n, x).rrr-r3rrr dup_spherical_bessel_fns  0 rFc Cs\|j|jg|jg}}td|dD]}|ttt|d||dd||||}}q|S)z&Low-level implementation of fn(-n, x).rrr/r-r3rrr dup_spherical_bessel_fn_minus(s0rGcCs@|durtd}|dkrtnt}tt||tdtd|f|S)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 Nr"rr)rrGrFr absr r )rr"r#frrr spherical_bessel_fn/s% rK)NF)NrF)(__doc__Zsympy.core.symbolrZsympy.polys.densearithrrrrrrr r Zsympy.polys.domainsr r Zsympy.polys.polytoolsr Zsympy.utilitiesrr!r$r'r(r,r*r+r4r7r9r<r=r>r@rArBrDrErFrGrKrrrr sD (