a kº”hÅ'ã@s:dZddlmZddlmZmZmZmZmZm Z m Z m Z ddl m Z mZddlmZddlmZdd„Zed3d d „ƒZd d„Zd4dd„Zdd„Zdd„Zdd„Zdd„Zed5dd„ƒZed6dd„ƒZdd„Zdd „Zed7d!d"„ƒZed8d#d$„ƒZ d%d&„Z!ed9d'd(„ƒZ"d)d*„Z#ed:d+d,„ƒZ$d-d.„Z%d/d0„Z&d;d1d2„Z'd S)Low-level implementation of probabilist's Hermite polynomials.rrr7r8rrrÚdup_hermite_probÁs r:cCst|ttd|f|ƒS)zóGenerates 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 r9r r4rrrÚ hermite_polyÌs r;cCst|ttd|f|ƒS)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 r4rrrÚhermite_prob_polyÛs ÿr<cCsˆ|dkr|jgS|jg|j|jg}}td|dƒD]N}tt|d|ƒ|d|d|ƒ|ƒ}t|||d|ƒ|ƒ}|t|||ƒ}}q4|S)z1Low-level implementation of Legendre polynomials.rrr$r8rrrÚ dup_legendreës"r=cCst|ttd|f|ƒS)zôGenerates the Legendre polynomial `P_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. zLegendre polynomial)r r=r r4rrrÚ legendre_polyös r>cCsŽ|jg|jg}}td|dƒD]h}t||j ||ƒ||j||ƒ|dƒg|ƒ}t|||j||ƒ|j|ƒ}|t|||ƒ}}q |S)z1Low-level implementation of Laguerre polynomials.rr)r%rrrrr)rÚalpharrrrrrrrrÚ dup_laguerres 2 r@cCst|tdd||f|ƒS)aQGenerates the Laguerre polynomial `L_n^{(\alpha)}(x)`. Parameters ========== n : int Degree of the polynomial. x : optional alpha : optional Decides minimal domain for the list of coefficients. polys : bool, optional If True, return a Poly, otherwise (default) return an expression. NzLaguerre polynomial)r r@)rr!r?r"rrrÚ laguerre_polysrAcCsx|dkr|j|jgS|jg|j|jg}}td|dƒD]2}|ttt|d|ƒ|d|dƒ|ƒ||ƒ}}q8t|d|ƒS)z%Low-level implementation of fn(n, x).rrr,r2rrrÚdup_spherical_bessel_fns  0rBc Cs\|j|jg|jg}}td|dƒD]2}|ttt|d|ƒ|dd|ƒ|ƒ||ƒ}}q$|S)z&Low-level implementation of fn(-n, x).rrr.r,r2rrrÚdup_spherical_bessel_fn_minus(s0rCcCs@|durtdƒ}|dkrtnt}tt|ƒ|tdtdƒ|f|ƒS)aè 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!rÚr)rrCrBr Úabsr r )rr!r"ÚfrrrÚspherical_bessel_fn/s%rG)NF)NF)NF)NF)NF)NF)NF)NrF)NF)(Ú__doc__Zsympy.core.symbolrZsympy.polys.densearithrrrrrrr r Zsympy.polys.domainsr r Zsympy.polys.polytoolsr Zsympy.utilitiesrr r#r&r'r+r)r*r3r5r6r9r:r;r<r=r>r@rArBrCrGrrrrÚsB (