ó ¡¼™\c@ sCdZddlmZmZddlmZddlmZmZddl m Z ddl m Z ddl mZmZmZddlmZdd lmZdd lmZdd lmZdd lmZdd lmZddlmZddlm Z m!Z!m"Z"m#Z#m$Z$m%Z%m&Z&e dƒZ'defd„ƒYZ(de(fd„ƒYZ)d„Z*de(fd„ƒYZ+de(fd„ƒYZ,de(fd„ƒYZ-defd„ƒYZ.defd„ƒYZ/d e(fd!„ƒYZ0d"efd#„ƒYZ1d$e(fd%„ƒYZ2d&e(fd'„ƒYZ3d(e(fd)„ƒYZ4d*S(+s™ This module mainly implements special orthogonal polynomials. See also functions.combinatorial.numbers which contains some combinatorial polynomials. iÿÿÿÿ(tprint_functiontdivision(tRational(tFunctiontArgumentIndexError(tS(tDummy(tbinomialt factorialtRisingFactorial(tre(texp(tfloor(tsqrt(tcos(tgamma(thyper(t jacobi_polytgegenbauer_polytchebyshevt_polytchebyshevu_polyt laguerre_polyt hermite_polyt legendre_polytxtOrthogonalPolynomialcB s&eZdZed„ƒZd„ZRS(s+Base class for orthogonal polynomials. cC s;|jr7|dkr7|jt|ƒtƒjt|ƒSdS(Ni(t is_integert _ortho_polytintt_xtsubs(tclstnR((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyt_eval_at_order)scC s$|j|jd|jdjƒƒS(Nii(tfunctargst conjugate(tself((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyt_eval_conjugate.s(t__name__t __module__t__doc__t classmethodR!R&(((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR%stjacobicB s;eZdZed„ƒZdd„Zd„Zd„ZRS(sÜ Jacobi polynomial :math:`P_n^{\left(\alpha, \beta\right)}(x)` jacobi(n, alpha, beta, x) gives the nth Jacobi polynomial in x, :math:`P_n^{\left(\alpha, \beta\right)}(x)`. The Jacobi polynomials are orthogonal on :math:`[-1, 1]` with respect to the weight :math:`\left(1-x\right)^\alpha \left(1+x\right)^\beta`. Examples ======== >>> from sympy import jacobi, S, conjugate, diff >>> from sympy.abc import n,a,b,x >>> jacobi(0, a, b, x) 1 >>> jacobi(1, a, b, x) a/2 - b/2 + x*(a/2 + b/2 + 1) >>> jacobi(2, a, b, x) # doctest:+SKIP (a**2/8 - a*b/4 - a/8 + b**2/8 - b/8 + x**2*(a**2/8 + a*b/4 + 7*a/8 + b**2/8 + 7*b/8 + 3/2) + x*(a**2/4 + 3*a/4 - b**2/4 - 3*b/4) - 1/2) >>> jacobi(n, a, b, x) jacobi(n, a, b, x) >>> jacobi(n, a, a, x) RisingFactorial(a + 1, n)*gegenbauer(n, a + 1/2, x)/RisingFactorial(2*a + 1, n) >>> jacobi(n, 0, 0, x) legendre(n, x) >>> jacobi(n, S(1)/2, S(1)/2, x) RisingFactorial(3/2, n)*chebyshevu(n, x)/factorial(n + 1) >>> jacobi(n, -S(1)/2, -S(1)/2, x) RisingFactorial(1/2, n)*chebyshevt(n, x)/factorial(n) >>> jacobi(n, a, b, -x) (-1)**n*jacobi(n, b, a, x) >>> jacobi(n, a, b, 0) 2**(-n)*gamma(a + n + 1)*hyper((-b - n, -n), (a + 1,), -1)/(factorial(n)*gamma(a + 1)) >>> jacobi(n, a, b, 1) RisingFactorial(a + 1, n)/factorial(n) >>> conjugate(jacobi(n, a, b, x)) jacobi(n, conjugate(a), conjugate(b), conjugate(x)) >>> diff(jacobi(n,a,b,x), x) (a/2 + b/2 + n/2 + 1/2)*jacobi(n - 1, a + 1, b + 1, x) See Also ======== gegenbauer, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly, sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Jacobi_polynomials .. [2] http://mathworld.wolfram.com/JacobiPolynomial.html .. [3] http://functions.wolfram.com/Polynomials/JacobiP/ cC sÔ||krÝ|tj krCttj|ƒt|ƒt||ƒS|tjkr_t||ƒS|tjkrtdtj|ƒt|dƒt||ƒSt|d|ƒtd|d|ƒt||tj|ƒSn»|| kr;t ||dƒt |dƒd||dd||dt || |ƒS|| kr˜t ||dƒt |dƒd||dd||dt |||ƒS|j s½|j ƒrÌtj |t|||| ƒS|tjkr1d| t ||dƒt |dƒt|ƒt| || g|dgdƒS|tjkr[t|d|ƒt|ƒS|tjkrÐ|jrº||d|jr—tdƒ‚nt|||d|ƒtjSqÐnt||||ƒSdS(Niiiiÿÿÿÿs,Error. a + b + 2*n should not be an integer.(RtHalfR Rt chebyshevttZerotlegendret chebyshevut gegenbauerRtassoc_legendret is_Numbertcould_extract_minus_signt NegativeOneR+RtOnetInfinityt is_positiveRt ValueErrorR(RR tatbR((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyteval„s4 ' /@ Q P  2$ &ic C sbddlm}|dkr.t||ƒ‚n0|dkr |j\}}}}tdƒ}d||||d}||d|dt||d||ƒ||t|||d||ƒ} ||t||||ƒ| t||||ƒ|d|dfƒS|dkrø|j\}}}}tdƒ}d||||d}d||||d|dt||d||ƒ||t|||d||ƒ} ||t||||ƒ| t||||ƒ|d|dfƒS|dkrO|j\}}}}tj|||dt|d|d|d|ƒSt||ƒ‚dS( Niÿÿÿÿ(tSumiitkiii( tsympyR=RR#RR R+RR,( R%targindexR=R R:R;RR>tf1tf2((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pytfdiff­s*   ,(D  7)D 6c K sÃddlm}|js(|jtkr7tdƒ‚ntdƒ}t| |ƒt|||d|ƒt||d||ƒt|ƒd|d|}dt|ƒ|||d|fƒS(Niÿÿÿÿ(R=s*Error: n should be a non-negative integer.R>iii( R?R=t is_negativeRtFalseR9RR R( R%R R:R;RtkwargsR=R>tkern((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyt_eval_rewrite_as_polynomialÉs  \cC s=|j\}}}}|j||jƒ|jƒ|jƒƒS(N(R#R"R$(R%R R:R;R((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR&Ós(R'R(R)R*R<RCRHR&(((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR+6s L)  cC stdƒ||dt||dƒt||dƒd|||dt|ƒt|||dƒ}t||||ƒt|ƒS(s Jacobi polynomial :math:`P_n^{\left(\alpha, \beta\right)}(x)` jacobi_normalized(n, alpha, beta, x) gives the nth Jacobi polynomial in x, :math:`P_n^{\left(\alpha, \beta\right)}(x)`. The Jacobi polynomials are orthogonal on :math:`[-1, 1]` with respect to the weight :math:`\left(1-x\right)^\alpha \left(1+x\right)^\beta`. This functions returns the polynomials normilzed: .. math:: \int_{-1}^{1} P_m^{\left(\alpha, \beta\right)}(x) P_n^{\left(\alpha, \beta\right)}(x) (1-x)^{\alpha} (1+x)^{\beta} \mathrm{d}x = \delta_{m,n} Examples ======== >>> from sympy import jacobi_normalized >>> from sympy.abc import n,a,b,x >>> jacobi_normalized(n, a, b, x) jacobi(n, a, b, x)/sqrt(2**(a + b + 1)*gamma(a + n + 1)*gamma(b + n + 1)/((a + b + 2*n + 1)*factorial(n)*gamma(a + b + n + 1))) See Also ======== gegenbauer, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly, sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Jacobi_polynomials .. [2] http://mathworld.wolfram.com/JacobiPolynomial.html .. [3] http://functions.wolfram.com/Polynomials/JacobiP/ ii(RRRR+R (R R:R;Rtnfactor((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pytjacobi_normalizedØs5pR1cB s;eZdZed„ƒZdd„Zd„Zd„ZRS(sþ Gegenbauer polynomial :math:`C_n^{\left(\alpha\right)}(x)` gegenbauer(n, alpha, x) gives the nth Gegenbauer polynomial in x, :math:`C_n^{\left(\alpha\right)}(x)`. The Gegenbauer polynomials are orthogonal on :math:`[-1, 1]` with respect to the weight :math:`\left(1-x^2\right)^{\alpha-\frac{1}{2}}`. Examples ======== >>> from sympy import gegenbauer, conjugate, diff >>> from sympy.abc import n,a,x >>> gegenbauer(0, a, x) 1 >>> gegenbauer(1, a, x) 2*a*x >>> gegenbauer(2, a, x) -a + x**2*(2*a**2 + 2*a) >>> gegenbauer(3, a, x) x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a) >>> gegenbauer(n, a, x) gegenbauer(n, a, x) >>> gegenbauer(n, a, -x) (-1)**n*gegenbauer(n, a, x) >>> gegenbauer(n, a, 0) 2**n*sqrt(pi)*gamma(a + n/2)/(gamma(a)*gamma(1/2 - n/2)*gamma(n + 1)) >>> gegenbauer(n, a, 1) gamma(2*a + n)/(gamma(2*a)*gamma(n + 1)) >>> conjugate(gegenbauer(n, a, x)) gegenbauer(n, conjugate(a), conjugate(x)) >>> diff(gegenbauer(n, a, x), x) 2*a*gegenbauer(n - 1, a + 1, x) See Also ======== jacobi, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Gegenbauer_polynomials .. [2] http://mathworld.wolfram.com/GegenbauerPolynomial.html .. [3] http://functions.wolfram.com/Polynomials/GegenbauerC3/ cC s­|jrtjS|tjkr,t||ƒS|tjkrHt||ƒS|tjkr^tjS|js™|tjkrŸt |ƒtjkt kr˜tj SdSn|j ƒrÇtj|t||| ƒS|tjkr*d|ttjƒt|tj|ƒtd|dƒt|dƒt|ƒS|tjkrgtd||ƒtd|ƒt|dƒS|tjkr©|jr–t||ƒtjSq©nt|||ƒSdS(Nii(RDRR.R,R/R6R0R5R3R tTruetComplexInfinitytNoneR4R1R tPiRR7R8R R(RR R:R((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR<Ws0     )+. ic C seddlm}|dkr.t||ƒ‚n3|dkr|j\}}}tdƒ}ddd||||||d|||}d|d|d|d|d|dd||d|}|t|||ƒ|t|||ƒ} || |d|dfƒS|dkrR|j\}}}d|t|d|d|ƒSt||ƒ‚dS(Niÿÿÿÿ(R=iiR>ii(R?R=RR#RR1( R%R@R=R R:RR>tfactor1tfactor2RG((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyRC‚s    +*  cK s‡ddlm}tdƒ}d|t|||ƒd||d|t|ƒt|d|ƒ}|||dt|dƒfƒS(Niÿÿÿÿ(R=R>ii(R?R=RR RR (R%R R:RRFR=R>RG((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyRH˜s  ,cC s1|j\}}}|j||jƒ|jƒƒS(N(R#R"R$(R%R R:R((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR&Ÿs(R'R(R)R*R<RCRHR&(((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR1s >+  R-cB s>eZdZeeƒZed„ƒZdd„Zd„Z RS(s… Chebyshev polynomial of the first kind, :math:`T_n(x)` chebyshevt(n, x) gives the nth Chebyshev polynomial (of the first kind) in x, :math:`T_n(x)`. The Chebyshev polynomials of the first kind are orthogonal on :math:`[-1, 1]` with respect to the weight :math:`\frac{1}{\sqrt{1-x^2}}`. Examples ======== >>> from sympy import chebyshevt, chebyshevu, diff >>> from sympy.abc import n,x >>> chebyshevt(0, x) 1 >>> chebyshevt(1, x) x >>> chebyshevt(2, x) 2*x**2 - 1 >>> chebyshevt(n, x) chebyshevt(n, x) >>> chebyshevt(n, -x) (-1)**n*chebyshevt(n, x) >>> chebyshevt(-n, x) chebyshevt(n, x) >>> chebyshevt(n, 0) cos(pi*n/2) >>> chebyshevt(n, -1) (-1)**n >>> diff(chebyshevt(n, x), x) n*chebyshevu(n - 1, x) See Also ======== jacobi, gegenbauer, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial .. [2] http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html .. [3] http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html .. [4] http://functions.wolfram.com/Polynomials/ChebyshevT/ .. [5] http://functions.wolfram.com/Polynomials/ChebyshevU/ cC sÌ|jsž|jƒr.tj|t|| ƒS|jƒrHt| |ƒS|tjkrottjtj|ƒS|tj kr…tj S|tj krÈtj Sn*|j r¸|j | |ƒS|j ||ƒSdS(N( R3R4RR5R-R.RR,RNR6R7RDR!(RR R((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR<és     icC sa|dkrt||ƒ‚n?|dkrN|j\}}|t|d|ƒSt||ƒ‚dS(Nii(RR#R0(R%R@R R((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyRCs   cK soddlm}tdƒ}t|d|ƒ|dd|||d|}|||dt|dƒfƒS(Niÿÿÿÿ(R=R>iii(R?R=RRR (R%R RRFR=R>RG((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyRH s 3( R'R(R)t staticmethodRRR*R<RCRH(((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR-¨s =  R0cB s>eZdZeeƒZed„ƒZdd„Zd„Z RS(s© Chebyshev polynomial of the second kind, :math:`U_n(x)` chebyshevu(n, x) gives the nth Chebyshev polynomial of the second kind in x, :math:`U_n(x)`. The Chebyshev polynomials of the second kind are orthogonal on :math:`[-1, 1]` with respect to the weight :math:`\sqrt{1-x^2}`. Examples ======== >>> from sympy import chebyshevt, chebyshevu, diff >>> from sympy.abc import n,x >>> chebyshevu(0, x) 1 >>> chebyshevu(1, x) 2*x >>> chebyshevu(2, x) 4*x**2 - 1 >>> chebyshevu(n, x) chebyshevu(n, x) >>> chebyshevu(n, -x) (-1)**n*chebyshevu(n, x) >>> chebyshevu(-n, x) -chebyshevu(n - 2, x) >>> chebyshevu(n, 0) cos(pi*n/2) >>> chebyshevu(n, 1) n + 1 >>> diff(chebyshevu(n, x), x) (-x*chebyshevu(n, x) + (n + 1)*chebyshevt(n + 1, x))/(x**2 - 1) See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial .. [2] http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html .. [3] http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html .. [4] http://functions.wolfram.com/Polynomials/ChebyshevT/ .. [5] http://functions.wolfram.com/Polynomials/ChebyshevU/ cC s |jsÀ|jƒr.tj|t|| ƒS|jƒrf|tjkrPtjSt| d|ƒ Sn|tjkrttjtj|ƒS|tj kr§tj |S|tj krtj SnH|j rø|tjkrßtjS|j | d|ƒ Sn|j ||ƒSdS(Ni( R3R4RR5R0R.RR,RNR6R7RDR!(RR R((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR<Us$      icC s‚|dkrt||ƒ‚n`|dkro|j\}}|dt|d|ƒ|t||ƒ|ddSt||ƒ‚dS(Nii(RR#R-R0(R%R@R R((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyRCts   6cK s‡ddlm}tdƒ}tj|t||ƒd||d|t|ƒt|d|ƒ}|||dt|dƒfƒS(Niÿÿÿÿ(R=R>ii(R?R=RRR5RR (R%R RRFR=R>RG((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyRHs K( R'R(R)RQRRR*R<RCRH(((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR0s =  tchebyshevt_rootcB seZdZed„ƒZRS(sC chebyshev_root(n, k) returns the kth root (indexed from zero) of the nth Chebyshev polynomial of the first kind; that is, if 0 <= k < n, chebyshevt(n, chebyshevt_root(n, k)) == 0. Examples ======== >>> from sympy import chebyshevt, chebyshevt_root >>> chebyshevt_root(3, 2) -sqrt(3)/2 >>> chebyshevt(3, chebyshevt_root(3, 2)) 0 See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly cC sRd|ko||ks1td||fƒ‚nttjd|dd|ƒS(Nis+must have 0 <= k < n, got k = %s and n = %sii(R9RRRN(RR R>((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR<§s(R'R(R)R*R<(((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyRR‡stchebyshevu_rootcB seZdZed„ƒZRS(s- chebyshevu_root(n, k) returns the kth root (indexed from zero) of the nth Chebyshev polynomial of the second kind; that is, if 0 <= k < n, chebyshevu(n, chebyshevu_root(n, k)) == 0. Examples ======== >>> from sympy import chebyshevu, chebyshevu_root >>> chebyshevu_root(3, 2) -sqrt(2)/2 >>> chebyshevu(3, chebyshevu_root(3, 2)) 0 See Also ======== chebyshevt, chebyshevt_root, chebyshevu, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly cC sNd|ko||ks1td||fƒ‚nttj|d|dƒS(Nis+must have 0 <= k < n, got k = %s and n = %si(R9RRRN(RR R>((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR<Ïs(R'R(R)R*R<(((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyRS¯sR/cB s>eZdZeeƒZed„ƒZdd„Zd„Z RS(sY legendre(n, x) gives the nth Legendre polynomial of x, :math:`P_n(x)` The Legendre polynomials are orthogonal on [-1, 1] with respect to the constant weight 1. They satisfy :math:`P_n(1) = 1` for all n; further, :math:`P_n` is odd for odd n and even for even n. Examples ======== >>> from sympy import legendre, diff >>> from sympy.abc import x, n >>> legendre(0, x) 1 >>> legendre(1, x) x >>> legendre(2, x) 3*x**2/2 - 1/2 >>> legendre(n, x) legendre(n, x) >>> diff(legendre(n,x), x) n*(x*legendre(n, x) - legendre(n - 1, x))/(x**2 - 1) See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, assoc_legendre, hermite, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Legendre_polynomial .. [2] http://mathworld.wolfram.com/LegendrePolynomial.html .. [3] http://functions.wolfram.com/Polynomials/LegendreP/ .. [4] http://functions.wolfram.com/Polynomials/LegendreP2/ cC sò|jsÄ|jƒr.tj|t|| ƒS|jƒrOt| tj|ƒS|tjkr•ttjƒt tj |dƒt tj|dƒS|tjkr«tjS|tj krîtj Sn*|j rÞ| tj}n|j ||ƒSdS(Ni(R3R4RR5R/R6R.R RNRR,R7RDR!(RR R((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR<s   7  icC s~|dkrt||ƒ‚n\|dkrk|j\}}||dd|t||ƒt|d|ƒSt||ƒ‚dS(Nii(RR#R/(R%R@R R((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyRC&s   2cK sqddlm}tdƒ}d|t||ƒdd|d||d|d|}|||d|fƒS(Niÿÿÿÿ(R=R>iii(R?R=RR(R%R RRFR=R>RG((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyRH2s ?( R'R(R)RQRRR*R<RCRH(((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR/Ûs /  R2cB sJeZdZed„ƒZed„ƒZdd„Zd„Zd„ZRS(s, assoc_legendre(n,m, x) gives :math:`P_n^m(x)`, where n and m are the degree and order or an expression which is related to the nth order Legendre polynomial, :math:`P_n(x)` in the following manner: .. math:: P_n^m(x) = (-1)^m (1 - x^2)^{\frac{m}{2}} \frac{\mathrm{d}^m P_n(x)}{\mathrm{d} x^m} Associated Legendre polynomial are orthogonal on [-1, 1] with: - weight = 1 for the same m, and different n. - weight = 1/(1-x**2) for the same n, and different m. Examples ======== >>> from sympy import assoc_legendre >>> from sympy.abc import x, m, n >>> assoc_legendre(0,0, x) 1 >>> assoc_legendre(1,0, x) x >>> assoc_legendre(1,1, x) -sqrt(1 - x**2) >>> assoc_legendre(n,m,x) assoc_legendre(n, m, x) See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, hermite, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Associated_Legendre_polynomials .. [2] http://mathworld.wolfram.com/LegendrePolynomial.html .. [3] http://functions.wolfram.com/Polynomials/LegendreP/ .. [4] http://functions.wolfram.com/Polynomials/LegendreP2/ cC sOt|tdtƒjt|fƒ}d|dtdt|dƒ|jƒS(Ntpolysiÿÿÿÿii(RRRKtdiffRtas_expr(RR tmtP((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR!os$cC sQ|jƒrEtj| t||ƒt||ƒt|| |ƒS|dkr^t||ƒS|dkr«d|ttjƒtd||dƒtd||dƒS|j rM|j rM|j rM|j rM|j rñt d||fƒ‚nt |ƒ|krt d|||fƒ‚n|jt|ƒt t|ƒƒƒjt|ƒSdS(Niiis3%s : 1st index must be nonnegative integer (got %r)s9%s : abs('2nd index') must be <= '1st index' (got %r, %r)(R4RR5RR2R/R RNRR3RRDR9tabsR!RRR(RR RWR((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR<ts 9   A$ icC s±|dkrt||ƒ‚n|dkr<t||ƒ‚nq|dkrž|j\}}}d|dd||t|||ƒ||t|d||ƒSt||ƒ‚dS(Niii(RR#R2(R%R@R RWR((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyRC…s   DcK sÁddlm}tdƒ}td|d|ƒd|t||ƒt|ƒt|d||ƒd||||d|}d|d|d|||dt||tjƒfƒS(Niÿÿÿÿ(R=R>iii(R?R=RRR RR,(R%R RWRRFR=R>RG((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyRH”s jcC s1|j\}}}|j||jƒ|jƒƒS(N(R#R"R$(R%R RWR((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR&›s( R'R(R)R*R!R<RCRHR&(((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR29s 4  thermitecB s>eZdZeeƒZed„ƒZdd„Zd„Z RS(sÔ hermite(n, x) gives the nth Hermite polynomial in x, :math:`H_n(x)` The Hermite polynomials are orthogonal on :math:`(-\infty, \infty)` with respect to the weight :math:`\exp\left(-x^2\right)`. Examples ======== >>> from sympy import hermite, diff >>> from sympy.abc import x, n >>> hermite(0, x) 1 >>> hermite(1, x) 2*x >>> hermite(2, x) 4*x**2 - 2 >>> hermite(n, x) hermite(n, x) >>> diff(hermite(n,x), x) 2*n*hermite(n - 1, x) >>> hermite(n, -x) (-1)**n*hermite(n, x) See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Hermite_polynomial .. [2] http://mathworld.wolfram.com/HermitePolynomial.html .. [3] http://functions.wolfram.com/Polynomials/HermiteH/ cC s°|js€|jƒr.tj|t|| ƒS|tjkrgd|ttjƒttj |dƒS|tj kr¬tj Sn,|j rœt d|ƒ‚n|j ||ƒSdS(Nis0The index n must be nonnegative integer (got %r)(R3R4RR5RZR.R RNRR6R7RDR9R!(RR R((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR<Ös  *  icC se|dkrt||ƒ‚nC|dkrR|j\}}d|t|d|ƒSt||ƒ‚dS(Nii(RR#RZ(R%R@R R((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyRCês   cK s€ddlm}tdƒ}d|t|ƒt|d|ƒd||d|}t|ƒ|||dt|dƒfƒS(Niÿÿÿÿ(R=R>ii(R?R=RRR (R%R RRFR=R>RG((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyRHõs :( R'R(R)RQRRR*R<RCRH(((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyRZ¤s .  tlaguerrecB s>eZdZeeƒZed„ƒZdd„Zd„Z RS(s Returns the nth Laguerre polynomial in x, :math:`L_n(x)`. Parameters ========== n : int Degree of Laguerre polynomial. Must be ``n >= 0``. Examples ======== >>> from sympy import laguerre, diff >>> from sympy.abc import x, n >>> laguerre(0, x) 1 >>> laguerre(1, x) 1 - x >>> laguerre(2, x) x**2/2 - 2*x + 1 >>> laguerre(3, x) -x**3/6 + 3*x**2/2 - 3*x + 1 >>> laguerre(n, x) laguerre(n, x) >>> diff(laguerre(n, x), x) -assoc_laguerre(n - 1, 1, x) See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial .. [2] http://mathworld.wolfram.com/LaguerrePolynomial.html .. [3] http://functions.wolfram.com/Polynomials/LaguerreL/ .. [4] http://functions.wolfram.com/Polynomials/LaguerreL3/ cC s±|js|jƒr1t|ƒt|d| ƒS|tjkrGtjS|tjkr]tjS|tjkr­tj |tjSn,|j rt d|ƒ‚n|j ||ƒSdS(Nis0The index n must be nonnegative integer (got %r)( R3R4R R[RR.R6tNegativeInfinityR7R5RDR9R!(RR R((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR<9s   icC sa|dkrt||ƒ‚n?|dkrN|j\}}t|dd|ƒ St||ƒ‚dS(Nii(RR#tassoc_laguerre(R%R@R R((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyRCPs   cK sddlm}|js(|jtkr7tdƒ‚ntdƒ}t| |ƒt|ƒd||}|||d|fƒS(Niÿÿÿÿ(R=s*Error: n should be a non-negative integer.R>ii( R?R=RDRRER9RR R(R%R RRFR=R>RG((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyRH[s  &( R'R(R)RQRRR*R<RCRH(((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR[s 5  R]cB s;eZdZed„ƒZdd„Zd„Zd„ZRS(s2 Returns the nth generalized Laguerre polynomial in x, :math:`L_n(x)`. Parameters ========== n : int Degree of Laguerre polynomial. Must be ``n >= 0``. alpha : Expr Arbitrary expression. For ``alpha=0`` regular Laguerre polynomials will be generated. Examples ======== >>> from sympy import laguerre, assoc_laguerre, diff >>> from sympy.abc import x, n, a >>> assoc_laguerre(0, a, x) 1 >>> assoc_laguerre(1, a, x) a - x + 1 >>> assoc_laguerre(2, a, x) a**2/2 + 3*a/2 + x**2/2 + x*(-a - 2) + 1 >>> assoc_laguerre(3, a, x) a**3/6 + a**2 + 11*a/6 - x**3/6 + x**2*(a/2 + 3/2) + x*(-a**2/2 - 5*a/2 - 3) + 1 >>> assoc_laguerre(n, a, 0) binomial(a + n, a) >>> assoc_laguerre(n, a, x) assoc_laguerre(n, a, x) >>> assoc_laguerre(n, 0, x) laguerre(n, x) >>> diff(assoc_laguerre(n, a, x), x) -assoc_laguerre(n - 1, a + 1, x) >>> diff(assoc_laguerre(n, a, x), a) Sum(assoc_laguerre(_k, a, x)/(-a + n), (_k, 0, n - 1)) See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial#Assoc_laguerre_polynomials .. [2] http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html .. [3] http://functions.wolfram.com/Polynomials/LaguerreL/ .. [4] http://functions.wolfram.com/Polynomials/LaguerreL3/ cC sÍ|tjkrt||ƒS|js|tjkrEt|||ƒS|tjkru|tjkrutj|tjS|tjkrÉ|tjkrÉtjSn,|jr¹t d|ƒ‚nt |||ƒSdS(Ns0The index n must be nonnegative integer (got %r)( RR.R[R3RR7R5R\RDR9R(RR talphaR((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR<ªs    icC sÐddlm}|dkr.t||ƒ‚nž|dkr†|j\}}}tdƒ}|t|||ƒ|||d|dfƒS|dkr½|j\}}}t|d|d|ƒ St||ƒ‚dS(Niÿÿÿÿ(R=iiR>ii(R?R=RR#RR](R%R@R=R R^RR>((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyRCÀs   . cK s©ddlm}|js(|jtkr7tdƒ‚ntdƒ}t| |ƒt|t dƒt |ƒ||}t|t dƒt |ƒ|||d|fƒS(Niÿÿÿÿ(R=s*Error: n should be a non-negative integer.R>ii( R?R=RDRRER9RR RR^R(R%R RRFR=R>RG((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyRHÑs  4cC s1|j\}}}|j||jƒ|jƒƒS(N(R#R"R$(R%R R^R((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR&Ûs(R'R(R)R*R<RCRHR&(((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyR]es C  N(5R)t __future__RRt sympy.coreRtsympy.core.functionRRtsympy.core.singletonRtsympy.core.symbolRt(sympy.functions.combinatorial.factorialsRRR t$sympy.functions.elementary.complexesR t&sympy.functions.elementary.exponentialR t#sympy.functions.elementary.integersR t(sympy.functions.elementary.miscellaneousR t(sympy.functions.elementary.trigonometricRt'sympy.functions.special.gamma_functionsRtsympy.functions.special.hyperRtsympy.polys.orthopolysRRRRRRRRRR+RJR1R-R0RRRSR/R2RZR[R](((sBlib/python2.7/site-packages/sympy/functions/special/polynomials.pyts84 ¢ ?‘ls(,^k\e