U Zʹj~@sdZddlmZddlmZmZddlmZddlm Z ddl m Z m Z m Z ddlmZddlmZdd lmZdd lmZdd lmZmZdd lmZdd lmZddlmZmZm Z m!Z!m"Z"m#Z#m$Z$m%Z%e dZ&GdddeZ'Gddde'Z(ddZ)Gddde'Z*Gddde'Z+Gddde'Z,GdddeZ-GdddeZ.Gd d!d!e'Z/Gd"d#d#eZ0Gd$d%d%e'Z1Gd&d'd'e'Z2Gd(d)d)e'Z3Gd*d+d+e'Z4d,S)-z This module mainly implements special orthogonal polynomials. See also functions.combinatorial.numbers which contains some combinatorial polynomials. )Rational)FunctionArgumentIndexError)S)Dummy)binomial factorialRisingFactorial)re)exp)floor)sqrt)cossec)gamma)hyper)chebyshevt_polychebyshevu_polygegenbauer_poly hermite_polyhermite_prob_poly jacobi_poly laguerre_poly legendre_polyxc@s$eZdZdZeddZddZdS)OrthogonalPolynomialz+Base class for orthogonal polynomials. cCs*|jr&|dkr&|t|tt|SdS)Nr) is_integer _ortho_polyint_xsubsclsnrr$b/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/functions/special/polynomials.py_eval_at_order sz#OrthogonalPolynomial._eval_at_ordercCs||jd|jdS)Nr)funcargs conjugate)selfr$r$r%_eval_conjugate%sz$OrthogonalPolynomial._eval_conjugateN)__name__ __module__ __qualname____doc__ classmethodr&r,r$r$r$r%rs rc@s>eZdZdZeddZdddZddZd d Zd d Z d S)jacobia Jacobi polynomial $P_n^{\left(\alpha, \beta\right)}(x)$. Explanation =========== ``jacobi(n, alpha, beta, x)`` gives the $n$th Jacobi polynomial in $x$, $P_n^{\left(\alpha, \beta\right)}(x)$. The Jacobi polynomials are orthogonal on $[-1, 1]$ with respect to the weight $\left(1-x\right)^\alpha \left(1+x\right)^\beta$. Examples ======== >>> from sympy import jacobi, S, conjugate, diff >>> from sympy.abc import a, b, n, 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) 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) gamma(a + n + 1)*hyper((-n, -b - n), (a + 1,), -1)/(2**n*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, hermite_prob, 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] https://mathworld.wolfram.com/JacobiPolynomial.html .. [3] https://functions.wolfram.com/Polynomials/JacobiP/ cCs||kr|tddkr4ttj|t|t||S|jrDt||S|tjkrttdtj|t|dt||St|d|td|d|t ||tj|SnV|| krt ||dt |dd||dd||dt || |S|j s| r*tj|t|||| S|jr~d| t ||dt |dt|t| || g|dgdS|tjkrt|d|t|S|tjkr|jr||d|jrtdt|||d|tjSnt||||SdS)Nr'z,Error. a + b + 2*n should not be an integer.)rr rHalfr chebyshevtis_zerolegendre chebyshevu gegenbauerrassoc_legendre is_Numbercould_extract_minus_sign NegativeOner2rOneInfinity is_positiver ValueErrorr)r"r#abrr$r$r%eval~s2  &4 J ,  z jacobi.evalc Csddlm}|dkr"t||n|dkr|j\}}}}td}d||||d}||d|dt||d||||t|||d||} ||t||||| t|||||d|dfS|dkr|j\}}}}td}d||||d}d||||d|dt||d||||t|||d||} ||t||||| t|||||d|dfS|dkr|j\}}}}tj|||dt|d|d|d|St||dS) NrSumr'r4kr5r3rG) sympy.concrete.summationsrIrr)rr r2rr6) r+argindexrIr#rDrErrJf1f2r$r$r%fdiffs. ( 4 2 4 0z jacobi.fdiffc Ksddlm}|js|jdkr$tdtd}t| |t|||d|t||d||t|d|d|}dt||||d|fS)NrrHF*Error: n should be a non-negative integer.rJr'r4)rKrI is_negativerrCrr r) r+r#rDrErkwargsrIrJkernr$r$r%_eval_rewrite_as_Sums 6zjacobi._eval_rewrite_as_SumcKs|j||||f|SNrT)r+r#rDrErrRr$r$r%_eval_rewrite_as_polynomialsz"jacobi._eval_rewrite_as_polynomialcCs*|j\}}}}|||||SrUr)r(r*)r+r#rDrErr$r$r%r,szjacobi._eval_conjugateN)rG r-r.r/r0r1rFrOrTrWr,r$r$r$r%r2-sP %  r2cCsztd||dt||dt||dd|||dt|t|||d}t||||t|S)a Jacobi polynomial $P_n^{\left(\alpha, \beta\right)}(x)$. Explanation =========== ``jacobi_normalized(n, alpha, beta, x)`` gives the $n$th Jacobi polynomial in $x$, $P_n^{\left(\alpha, \beta\right)}(x)$. The Jacobi polynomials are orthogonal on $[-1, 1]$ with respect to the weight $\left(1-x\right)^\alpha \left(1+x\right)^\beta$. 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))) Parameters ========== n : integer degree of polynomial a : alpha value b : beta value x : symbol See Also ======== gegenbauer, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, hermite_prob, 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] https://mathworld.wolfram.com/JacobiPolynomial.html .. [3] https://functions.wolfram.com/Polynomials/JacobiP/ r4r')rrrr2r )r#rDrErZnfactorr$r$r%jacobi_normalizeds C2rZc@s>eZdZdZeddZdddZddZd d Zd d Z d S)r;aN Gegenbauer polynomial $C_n^{\left(\alpha\right)}(x)$. Explanation =========== ``gegenbauer(n, alpha, x)`` gives the $n$th Gegenbauer polynomial in $x$, $C_n^{\left(\alpha\right)}(x)$. The Gegenbauer polynomials are orthogonal on $[-1, 1]$ with respect to the weight $\left(1-x^2\right)^{\alpha-\frac{1}{2}}$. 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, hermite_prob, 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.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Gegenbauer_polynomials .. [2] https://mathworld.wolfram.com/GegenbauerPolynomial.html .. [3] https://functions.wolfram.com/Polynomials/GegenbauerC3/ cCs|jr tjS|tjkr t||S|tjkr4t||S|tjkrDtjS|js|tjkrt |tjkdkrntj St tj ||t tj |td||td|t|dS|rtj|t||| S|jr&d|ttj t|tj|td|dt|dt|S|tjkrZtd||td|t|dS|tjkr|jrt||tjSn t|||SdS)NTr4r')rQrZeror6r9r@r:r?r=r ZComplexInfinityrPirrr>r;r8r rArBr r)r"r#rDrr$r$r%rFgs6      ."" ( zgegenbauer.evalr5c Cs&ddlm}|dkr"t||n|dkr|j\}}}td}ddd||||||d|||}d|d|d|d|d|dd||d|}|t||||t|||} || |d|dfS|dkr|j\}}}d|t|d|d|St||dS)NrrHr'r4rJr3r5)rKrIrr)rr;) r+rLrIr#rDrrJZfactor1Zfactor2rSr$r$r%rOs,  *   zgegenbauer.fdiffcKsnddlm}td}d|t|||d||d|t|t|d|}|||dt|dfS)NrrHrJr3r4)rKrIrr rr )r+r#rDrrRrIrJrSr$r$r%rTs  (zgegenbauer._eval_rewrite_as_SumcKs|j|||f|SrUrV)r+r#rDrrRr$r$r%rWsz&gegenbauer._eval_rewrite_as_polynomialcCs"|j\}}}||||SrUrX)r+r#rDrr$r$r%r,s zgegenbauer._eval_conjugateN)r5rYr$r$r$r%r;"sD ( r;c@s>eZdZdZeeZeddZd ddZ ddZ d d Z d S) r7a Chebyshev polynomial of the first kind, $T_n(x)$. Explanation =========== ``chebyshevt(n, x)`` gives the $n$th Chebyshev polynomial (of the first kind) in $x$, $T_n(x)$. The Chebyshev polynomials of the first kind are orthogonal on $[-1, 1]$ with respect to the weight $\frac{1}{\sqrt{1-x^2}}$. Examples ======== >>> from sympy import chebyshevt, 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, hermite_prob, 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.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial .. [2] https://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html .. [3] https://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html .. [4] https://functions.wolfram.com/Polynomials/ChebyshevT/ .. [5] https://functions.wolfram.com/Polynomials/ChebyshevU/ cCs|jst|r$tj|t|| S|r8t| |S|jrRttjtj|S|tj krbtj S|tj krtj Sn |j r| | |S| ||SdSrU) r=r>rr?r7r8rr6r\r@rArQr&r!r$r$r%rFs   zchebyshevt.evalr4cCsF|dkrt||n.|dkr8|j\}}|t|d|St||dSNr'r4)rr)r:r+rLr#rr$r$r%rOs   zchebyshevt.fdiffcKsZddlm}td}t|d||dd|||d|}|||dt|dfSNrrHrJr4r')rKrIrrr r+r#rrRrIrJrSr$r$r%rT%s .zchebyshevt._eval_rewrite_as_SumcKs|j||f|SrUrVr+r#rrRr$r$r%rW+sz&chebyshevt._eval_rewrite_as_polynomialN)r4) r-r.r/r0 staticmethodrrr1rFrOrTrWr$r$r$r%r7sC  r7c@s>eZdZdZeeZeddZd ddZ ddZ d d Z d S) r:a Chebyshev polynomial of the second kind, $U_n(x)$. Explanation =========== ``chebyshevu(n, x)`` gives the $n$th Chebyshev polynomial of the second kind in x, $U_n(x)$. The Chebyshev polynomials of the second kind are orthogonal on $[-1, 1]$ with respect to the weight $\sqrt{1-x^2}$. Examples ======== >>> from sympy import 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, hermite_prob, 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.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial .. [2] https://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html .. [3] https://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html .. [4] https://functions.wolfram.com/Polynomials/ChebyshevT/ .. [5] https://functions.wolfram.com/Polynomials/ChebyshevU/ cCs|js|r$tj|t|| S|r\|tjkrrr?r:r[r8rr6r\r@rArQr&r!r$r$r%rFws&     zchebyshevu.evalr4cCsd|dkrt||nL|dkrV|j\}}|dt|d||t|||ddSt||dSr])rr)r7r:r^r$r$r%rOs   0zchebyshevu.fdiffcKsnddlm}td}tj|t||d||d|t|t|d|}|||dt|dfSNrrHrJr4rKrIrrr?rr r`r$r$r%rTs  zchebyshevu._eval_rewrite_as_SumcKs|j||f|SrUrVrar$r$r%rWsz&chebyshevu._eval_rewrite_as_polynomialN)r4) r-r.r/r0rbrrr1rFrOrTrWr$r$r$r%r:1sC  r:c@seZdZdZeddZdS)chebyshevt_roota ``chebyshev_root(n, k)`` returns the $k$th root (indexed from zero) of the $n$th Chebyshev polynomial of the first kind; that is, if $0 \le k < n$, ``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, hermite_prob, 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.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly cCs>d|kr||ks td||fttjd|dd|S)Nr+must have 0 <= k < n, got k = %s and n = %sr4r'rCrrr\r"r#rJr$r$r%rFs zchebyshevt_root.evalNr-r.r/r0r1rFr$r$r$r%rfs rfc@seZdZdZeddZdS)chebyshevu_rootaw ``chebyshevu_root(n, k)`` returns the $k$th root (indexed from zero) of the $n$th Chebyshev polynomial of the second kind; that is, if $0 \le k < n$, ``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, hermite_prob, 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.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly cCs:d|kr||ks td||fttj|d|dS)Nrrgr'rhrir$r$r%rFs zchebyshevu_root.evalNrjr$r$r$r%rks rkc@s>eZdZdZeeZeddZd ddZ ddZ d d Z d S) r9a ``legendre(n, x)`` gives the $n$th Legendre polynomial of $x$, $P_n(x)$ Explanation =========== The Legendre polynomials are orthogonal on $[-1, 1]$ with respect to the constant weight 1. They satisfy $P_n(1) = 1$ for all $n$; further, $P_n$ 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, hermite_prob, 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.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Legendre_polynomial .. [2] https://mathworld.wolfram.com/LegendrePolynomial.html .. [3] https://functions.wolfram.com/Polynomials/LegendreP/ .. [4] https://functions.wolfram.com/Polynomials/LegendreP2/ cCs|js|r$tj|t|| S|rL| dsLt| tj|S|jrttjt tj |dt tj|dS|tjkrtjS|tj krtj Sn|j r| tj}| ||SdSr])r=r>rr?r9r@r8r r\rr6rArQr&r!r$r$r%rF=s.   z legendre.evalr4cCs`|dkrt||nH|dkrR|j\}}||dd|t||t|d|St||dSr])rr)r9r^r$r$r%rOUs   ,zlegendre.fdiffcKs`ddlm}td}tj|t||dd|d||d|d|}|||d|fSr_)rKrIrrr?rr`r$r$r%rTms <zlegendre._eval_rewrite_as_SumcKs|j||f|SrUrVrar$r$r%rWssz$legendre._eval_rewrite_as_polynomialN)r4) r-r.r/r0rbrrr1rFrOrTrWr$r$r$r%r9s5  r9c@sJeZdZdZeddZeddZdddZd d Zd d Z d dZ dS)r<a ``assoc_legendre(n, m, x)`` gives $P_n^m(x)$, where $n$ and $m$ are the degree and order or an expression which is related to the nth order Legendre polynomial, $P_n(x)$ 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} Explanation =========== Associated Legendre polynomials are orthogonal on $[-1, 1]$ with: - weight $= 1$ for the same $m$ and different $n$. - weight $= \frac{1}{1-x^2}$ 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, hermite_prob, 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.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Associated_Legendre_polynomials .. [2] https://mathworld.wolfram.com/LegendrePolynomial.html .. [3] https://functions.wolfram.com/Polynomials/LegendreP/ .. [4] https://functions.wolfram.com/Polynomials/LegendreP2/ cCs@t|tddt|f}tj|dtdt|d|S)NT)Zpolysr'r4)rrdiffrr?rZas_expr)r"r#mPr$r$r%r&szassoc_legendre._eval_at_ordercCs|r:tj| t||t||t|| |S|dkrLt||S|dkrd|ttjtd||dtd||dS|j r|j r|j r|j r|j rt d||ft ||krt d|||f|t|t t|t|SdS)Nrr4r'z3%s : 1st index must be nonnegative integer (got %r)z9%s : abs('2nd index') must be <= '1st index' (got %r, %r))r>rr?rr<r9r r\rr=rrQrCabsr&rr r)r"r#rmrr$r$r%rFs2 : zassoc_legendre.evalr5cCs|dkrt||nn|dkr(t||nZ|dkrx|j\}}}d|dd||t|||||t|d||St||dS)Nr'r4r5)rr)r<)r+rLr#rmrr$r$r%rOs   <zassoc_legendre.fdiffcKsddlm}td}td|d|d|t||t|t|d||tj||||d|}d|d|d|||dt||tjfSr_)rKrIrrrr?r r6)r+r#rmrrRrIrJrSr$r$r%rTs &z#assoc_legendre._eval_rewrite_as_SumcKs|j|||f|SrUrV)r+r#rmrrRr$r$r%rWsz*assoc_legendre._eval_rewrite_as_polynomialcCs"|j\}}}||||SrUrX)r+r#rmrr$r$r%r,s zassoc_legendre._eval_conjugateN)r5) r-r.r/r0r1r&rFrOrTrWr,r$r$r$r%r<ys:   r<c@sFeZdZdZeeZeddZdddZ ddZ d d Z d d Z d S)hermitea. ``hermite(n, x)`` gives the $n$th Hermite polynomial in $x$, $H_n(x)$. Explanation =========== The Hermite polynomials are orthogonal on $(-\infty, \infty)$ with respect to the weight $\exp\left(-x^2\right)$. 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, hermite_prob, 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.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Hermite_polynomial .. [2] https://mathworld.wolfram.com/HermitePolynomial.html .. [3] https://functions.wolfram.com/Polynomials/HermiteH/ cCs|js`|r$tj|t|| S|jrNd|ttjttj |dS|tj krtj Sn |j rtt d|n | ||SdS)Nr40The index n must be nonnegative integer (got %r))r=r>rr?rpr8r r\rr@rArQrCr&r!r$r$r%rF&s$ z hermite.evalr4cCsJ|dkrt||n2|dkr<|j\}}d|t|d|St||dSr])rr)rpr^r$r$r%rO:s   z hermite.fdiffcKsjddlm}td}tj|t|t|d|d||d|}t||||dt|dfSrdrer`r$r$r%rTEs 6zhermite._eval_rewrite_as_SumcKs|j||f|SrUrVrar$r$r%rWKsz#hermite._eval_rewrite_as_polynomialcKstd|t||tdSrc)r hermite_probrar$r$r%_eval_rewrite_as_hermite_probPsz%hermite._eval_rewrite_as_hermite_probN)r4) r-r.r/r0rbrrr1rFrOrTrWrsr$r$r$r%rps5  rpc@sFeZdZdZeeZeddZdddZ ddZ d d Z d d Z d S)rra ``hermite_prob(n, x)`` gives the $n$th probabilist's Hermite polynomial in $x$, $He_n(x)$. Explanation =========== The probabilist's Hermite polynomials are orthogonal on $(-\infty, \infty)$ with respect to the weight $\exp\left(-\frac{x^2}{2}\right)$. They are monic polynomials, related to the plain Hermite polynomials (:py:class:`~.hermite`) by .. math :: He_n(x) = 2^{-n/2} H_n(x/\sqrt{2}) Examples ======== >>> from sympy import hermite_prob, diff, I >>> from sympy.abc import x, n >>> hermite_prob(1, x) x >>> hermite_prob(5, x) x**5 - 10*x**3 + 15*x >>> diff(hermite_prob(n,x), x) n*hermite_prob(n - 1, x) >>> hermite_prob(n, -x) (-1)**n*hermite_prob(n, x) The sum of absolute values of coefficients of $He_n(x)$ is the number of matchings in the complete graph $K_n$ or telephone number, A000085 in the OEIS: >>> [hermite_prob(n,I) / I**n for n in range(11)] [1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496] See Also ======== jacobi, gegenbauer, chebyshevt, 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.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Hermite_polynomial .. [2] https://mathworld.wolfram.com/HermitePolynomial.html cCs||jsX|r$tj|t|| S|jrFttjttj |dS|tj krxtj Sn |j rlt d|n | ||SdS)Nr4z'n must be a nonnegative integer, not %r)r=r>rr?rrr8r r\rr@rArQrCr&r!r$r$r%rFs zhermite_prob.evalr4cCs2|dkr$|j\}}|t|d|St||dS)Nr4r')r)rrrr^r$r$r%rOs zhermite_prob.fdiffcKshddlm}td}tj |||d|t|t|d|}t||||dt|dfSrd)rKrIrrr6rr r`r$r$r%rTs 4z!hermite_prob._eval_rewrite_as_SumcKs|j||f|SrUrVrar$r$r%rWsz(hermite_prob._eval_rewrite_as_polynomialcKs td| t||tdSrc)r rprar$r$r%_eval_rewrite_as_hermitesz%hermite_prob._eval_rewrite_as_hermiteN)r4) r-r.r/r0rbrrr1rFrOrTrWrtr$r$r$r%rrTs9  rrc@s>eZdZdZeeZeddZd ddZ ddZ d d Z d S) laguerreaL Returns the $n$th Laguerre polynomial in $x$, $L_n(x)$. 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) Parameters ========== n : int Degree of Laguerre polynomial. Must be `n \ge 0`. See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, hermite_prob, 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.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial .. [2] https://mathworld.wolfram.com/LaguerrePolynomial.html .. [3] https://functions.wolfram.com/Polynomials/LaguerreL/ .. [4] https://functions.wolfram.com/Polynomials/LaguerreL3/ cCs|jdkrtd|js|rH| dsHt|t| d| S|jrTtjS|tj krdtj S|tj krtj |tj Sn,|j rt|t| d| S| ||SdS)NFError: n should be an integer.r')rrCr=r>r rur8rr@NegativeInfinityrAr?rQr&r!r$r$r%rFs   z laguerre.evalr4cCsF|dkrt||n.|dkr8|j\}}t|dd| St||dSr])rr)assoc_laguerrer^r$r$r%rO s   zlaguerre.fdiffcKs|ddlm}|jr2t||j| d| f|S|jdkrDtdtd}t| |t |d||}|||d|fS)NrrHr'FrvrJr4) rKrIrQr rTrrCrr rr`r$r$r%rTs    zlaguerre._eval_rewrite_as_SumcKs|j||f|SrUrVrar$r$r%rW"sz$laguerre._eval_rewrite_as_polynomialN)r4) r-r.r/r0rbrrr1rFrOrTrWr$r$r$r%rus8   ruc@s>eZdZdZeddZdddZddZd d Zd d Z d S)rxaj Returns the $n$th generalized Laguerre polynomial in $x$, $L_n(x)$. Examples ======== >>> from sympy import 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)) Parameters ========== n : int Degree of Laguerre polynomial. Must be `n \ge 0`. alpha : Expr Arbitrary expression. For ``alpha=0`` regular Laguerre polynomials will be generated. See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, hermite_prob, 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.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial#Generalized_Laguerre_polynomials .. [2] https://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html .. [3] https://functions.wolfram.com/Polynomials/LaguerreL/ .. [4] https://functions.wolfram.com/Polynomials/LaguerreL3/ cCs|jrt||S|jsf|jr*t|||S|tjkrL|dkrLtj|tjS|tjkr|dkrtjSn |jrzt d|n t |||SdS)Nrrq) r8rur=rrrAr?rwrQrCr)r"r#alpharr$r$r%rFos zassoc_laguerre.evalr5cCsddlm}|dkr t||nt|dkr`|j\}}}td}|t||||||d|dfS|dkr|j\}}}t|d|d| St||dS)NrrHr'r4rJr5)rKrIrr)rrx)r+rLrIr#ryrrJr$r$r%rOs   $ zassoc_laguerre.fdiffcKsddlm}|js|jdkr$tdtd}t| |t||dt|||}t||dt||||d|fS)NrrHFrPrJr') rKrIrQrrCrr rr)r+r#ryrrRrIrJrSr$r$r%rTs z#assoc_laguerre._eval_rewrite_as_SumcKs|j|||f|SrUrV)r+r#ryrrRr$r$r%rWsz*assoc_laguerre._eval_rewrite_as_polynomialcCs"|j\}}}||||SrUrX)r+r#ryrr$r$r%r,s zassoc_laguerre._eval_conjugateN)r5rYr$r$r$r%rx(sF   rxN)5r0Z sympy.corerZsympy.core.functionrrZsympy.core.singletonrZsympy.core.symbolrZ(sympy.functions.combinatorial.factorialsrrr Z$sympy.functions.elementary.complexesr Z&sympy.functions.elementary.exponentialr Z#sympy.functions.elementary.integersr Z(sympy.functions.elementary.miscellaneousr Z(sympy.functions.elementary.trigonometricrrZ'sympy.functions.special.gamma_functionsrZsympy.functions.special.hyperrZsympy.polys.orthopolysrrrrrrrrrrr2rZr;r7r:rfrkr9r<rprrrurxr$r$r$r%s<         ((Nv~)-tuffn