U L?h'@s:dZddlmZddlmZmZmZmZmZm Z m Z m Z ddl m Z mZddlmZddlmZddZed3d d Zd dZd4ddZddZddZddZddZed5ddZed6ddZddZdd Zed7d!d"Zed8d#d$Z d%d&Z!ed9d'd(Z"d)d*Z#ed:d+d,Z$d-d.Z%d/d0Z&d;d1d2Z'd S)Low-level implementation of probabilist's Hermite polynomials.rrr8r9rrr dup_hermite_probs r;cCst|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:r r5rrr hermite_polys r<cCst|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 r;r r5rrr hermite_prob_polys r=cCs|dkr|jgS|jg|j|jg}}td|dD]N}tt|d||d|d||}t|||d||}|t|||}}q4|S)z1Low-level implementation of Legendre polynomials.rrr%r9rrr dup_legendres"r>cCst|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 r>r r5rrr legendre_polys r?cCs|jg|jg}}td|dD]h}t||j ||||j|||dg|}t|||j|||j|}|t|||}}q |S)z1Low-level implementation of Laguerre polynomials.rr)r&rrrrr)ralpharrrrrrrrr dup_laguerres 2 rAcCst|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 rA)rr"r@r#rrr laguerre_polysrBcCsx|dkr|j|jgS|jg|j|jg}}td|dD]2}|ttt|d||d|d|||}}q8t|d|S)z%Low-level implementation of fn(n, x).rrr-r3rrr dup_spherical_bessel_fns  0rCc Cs\|j|jg|jg}}td|dD]2}|ttt|d||dd||||}}q$|S)z&Low-level implementation of fn(-n, x).rrr/r-r3rrr dup_spherical_bessel_fn_minus(s0rDcCs@|dkrtd}|dkrtnt}tt||tdtd|f|S)a Coefficients for the spherical Bessel functions. These are only needed in the jn() function. The coefficients are calculated from: fn(0, z) = 1/z fn(1, z) = 1/z**2 fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z) Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. Examples ======== >>> from sympy.polys.orthopolys import spherical_bessel_fn as fn >>> from sympy import Symbol >>> z = Symbol("z") >>> fn(1, z) z**(-2) >>> fn(2, z) -1/z + 3/z**3 >>> fn(3, z) -6/z**2 + 15/z**4 >>> fn(4, z) 1/z - 45/z**3 + 105/z**5 Nr"rr)rrDrCr absr r )rr"r#frrr spherical_bessel_fn/s%rH)NF)NF)NF)NF)NF)NF)NF)NrF)NF)(__doc__Zsympy.core.symbolrZsympy.polys.densearithrrrrrrr r Zsympy.polys.domainsr r Zsympy.polys.polytoolsr Zsympy.utilitiesrr!r$r'r(r,r*r+r4r6r7r:r;r<r=r>r?rArBrCrDrHrrrr sB (