U 9%eN!@sdZddlmZddlmZmZmZmZmZddl m Z m Z ddl m Z ddlmZddZed/d d Zd dZd0ddZddZddZed1ddZed2ddZddZddZed3ddZed4dd Zd!d"Zed5d#d$Zd%d&Zed6d'd(Zd)d*Z d+d,Z!d7d-d.Z"d S)8z:Efficient functions for generating orthogonal polynomials.)Dummy)dup_muldup_mul_ground dup_lshiftdup_subdup_add)ZZQQ) named_poly)publiccCs|dkr|jgS|jg|||d|j|||dg}}td|dD]:}||||||||d||d}|||d||j|||||d|}|||d||j|||d||d|||d||d|} |||j|||j|||d||} t|||} tt|d|| |} t|| |} |tt| | || |}}qN|S)z/Low-level implementation of Jacobi polynomials.)onerangerrrr)nabKm2m1iZdenZf0f1f2p0p1p2rU/var/www/html/Darija-Ai-API/env/lib/python3.8/site-packages/sympy/polys/orthopolys.py dup_jacobi s006V4  rNFcCst|tdd|||f|S)aGenerates the Jacobi polynomial `P_n^{(a,b)}(x)`. Parameters ========== n : int Degree of the polynomial. a Lower limit of minimal domain for the list of coefficients. b Upper limit of minimal domain for the list of coefficients. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. NzJacobi polynomial)r r)rrrxpolysrrr jacobi_polysr!cCs|dkr|jgS|jg|d||jg}}td|dD]p}tt|d||d||j|||d|}t||d||j|||j|}|t|||}}q:|S)z3Low-level implementation of Gegenbauer polynomials.r r rzerorrrr)rrrrrrrrrrrdup_gegenbauer,s2(r$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$)rrrr rrrgegenbauer_poly7sr%cCsd|dkr|jgS|jg|j|jg}}td|dD]*}|ttt|d||d|||}}q4|S)zDLow-level implementation of Chebyshev polynomials of the first kind.r r rr#rrrrrrrrrrrrdup_chebyshevtGs (r(cCsf|dkr|jgS|jg|d|jg}}td|dD]*}|ttt|d||d|||}}q6|S)zELow-level implementation of Chebyshev polynomials of the second kind.r r r&r'rrrdup_chebyshevuPs (r)cCst|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(rrrr rrrchebyshevt_polyYs r+cCst|ttd|f|S)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 r)rr*rrrchebyshevu_polyis r,cCs~|dkr|jgS|jg|d|jg}}td|dD]B}t|d|}t|||d|}|tt||||d|}}q6|S)z0Low-level implementation of Hermite polynomials.r r rr#rrrrrrrrrrrrrr dup_hermiteys  r/cCsp|dkr|jgS|jg|j|jg}}td|dD]6}t|d|}t|||d|}|t|||}}q4|S)z>Low-level implementation of probabilist's Hermite polynomials.r r r-r.rrrdup_hermite_probs r0cCst|ttd|f|S)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/rr*rrr hermite_polys r1cCst|ttd|f|S)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 r0rr*rrrhermite_prob_polys r2cCs|dkr|jgS|jg|j|jg}}td|dD]N}tt|d||d|d||}t|||d||}|t|||}}q4|S)z1Low-level implementation of Legendre polynomials.r r r"r.rrr dup_legendres"r3cCst|ttd|f|S)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 r3r r*rrr legendre_polys r4cCs|jg|jg}}td|dD]h}t||j ||||j|||dg|}t|||j|||j|}|t|||}}q |S)z1Low-level implementation of Laguerre polynomials.r r )r#rrrrr)ralpharrrrrrrrr dup_laguerres 2 r6cCst|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 r6)rrr5r rrr laguerre_polysr7cCsx|dkr|j|jgS|jg|j|jg}}td|dD]2}|ttt|d||d|d|||}}q8t|d|S)z%Low-level implementation of fn(n, x).r r r&r'rrrdup_spherical_bessel_fns  0r8c Cs\|j|jg|jg}}td|dD]2}|ttt|d||dd||||}}q$|S)z&Low-level implementation of fn(-n, x).r r r&r'rrrdup_spherical_bessel_fn_minuss0r:cCs@|dkrtd}|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 Nrrr )rr:r8r absrr )rrr frrrspherical_bessel_fns%r>)NF)NF)NF)NF)NF)NF)NF)NrF)NF)#__doc__Zsympy.core.symbolrZsympy.polys.densearithrrrrrZsympy.polys.domainsrr Zsympy.polys.polytoolsr Zsympy.utilitiesr rr!r$r%r(r)r+r,r/r0r1r2r3r4r6r7r8r:r>rrrrs>