\epsilon}^{a+\epsilon} \delta(x - a)f(x)\, dx = f(a)\), \(\delta(g(x)) = \sum_i \frac{\delta(x - x_i)}{\|g'(x_i)\|}\), 2*(2*x**2*DiracDelta(x**2 - 1, 2) + DiracDelta(x**2 - 1, 1)), DiracDelta(x - 1)/(2*Abs(y)) + DiracDelta(x + 1)/(2*Abs(y)), sympy.functions.special.tensor_functions.KroneckerDelta, 4*SingularityFunction(x, 1, 3) + 5*SingularityFunction(x, 1, 4), Piecewise(((x - 4)**5, x - 4 > 0), (0, True)), (x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1), 1/x - EulerGamma + x*(EulerGamma**2/2 + pi**2/12) + x**2*(-EulerGamma*pi**2/12 + polygamma(2, 1)/6 - EulerGamma**3/6) + O(x**3), 2.288037795340032417959588909060233922890, 0.49801566811835604271 - 0.15494982830181068512*I, log(2**(1 - n)*sqrt(pi)*gamma(n)/gamma(n/2 + 1/2)), -5*log(3) + loggamma(1/3) + log(4) + log(7) + log(10) + log(13), -4*log(4) + loggamma(3/4) + log(3) + log(7) + log(11) + log(15), -3*log(7) + log(2) + loggamma(2/7) + log(9) + log(16), \(x \in \mathbb{C} \setminus \{-\infty, 0\}\), -log(x) - EulerGamma*x + pi**2*x**2/12 + x**3*polygamma(2, 1)/6 + O(x**4), -0.65092319930185633889 - 1.8724366472624298171*I, -log(3) - sqrt(3)*pi/6 - EulerGamma - log(sqrt(3)), (-1)**(n + 1)*(-harmonic(x - 1, n + 1) + zeta(n + 1))*factorial(n), -2*sqrt(pi)*erfc(sqrt(x)) + 2*exp(-x)/sqrt(x), -2*sqrt(pi)*erf(sqrt(x)) - 2*exp(-x)/sqrt(x), pi**(p*(p - 1)/4)*Product(gamma(-_k/2 + x + 1/2), (_k, 1, p)), pi**(3/2)*gamma(x)*gamma(x - 1)*gamma(x - 1/2), (polygamma(0, x) - polygamma(0, x + y))*beta(x, y), (polygamma(0, y) - polygamma(0, x + y))*beta(x, y), 0.02671848900111377452242355235388489324562, -0.2112723729365330143 - 0.7655283165378005676*I, -z**(nu - 1)*meijerg(((), (1, 1)), ((0, 0, 1 - nu), ()), z), z**4*expint(1, z)/24 + (-z**3 + z**2 - 2*z + 6)*exp(-z)/24, z**(nu - 1)*(exp(2*I*pi*nu) - 1)*gamma(1 - nu) + expint(nu, z), 1.0652795784357498247001125598 + 3.08346052231061726610939702133*I, -log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z)), -log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z)), -expint(1, z*exp_polar(-I*pi/2))/2 - expint(1, z*exp_polar(I*pi/2))/2, expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2, -expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2, besselj(n - 1, z)/2 - besselj(n + 1, z)/2, bessely(n - 1, z)/2 - bessely(n + 1, z)/2, besseli(n - 1, z)/2 + besseli(n + 1, z)/2, -besselk(n - 1, z)/2 - besselk(n + 1, z)/2, hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2, hankel2(n - 1, z)/2 - hankel2(n + 1, z)/2, sympy.polys.orthopolys.spherical_bessel_fn(), (-6/z**2 + 15/z**4)*sin(z) + (1/z - 15/z**3)*cos(z), sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(nu + 1/2, z)/2, (-1)**nu*sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(-nu - 1/2, z)/2, 0.099419756723640344491 - 0.054525080242173562897*I, (-1)**(nu + 1)*sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(-nu - 1/2, z)/2, sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(nu + 1/2, z)/2, 0.18525034196069722536 + 0.014895573969924817587*I, 1/2 + exp(-a**2)*besseli(0, a**2)/2 + exp(-a**2)*besseli(1, a**2), a*(-marcumq(m, a, b) + marcumq(m + 1, a, b)), -a**(1 - m)*b**m*exp(-a**2/2 - b**2/2)*besseli(m - 1, a*b), 3**(5/6)*gamma(1/3)/(6*pi) - 3**(1/6)*z*gamma(2/3)/(2*pi) + O(z**3), 0.22740742820168557599192443603787379946077222541710, -3**(2/3)*z*hyper((), (4/3,), z**3/9)/(3*gamma(1/3)) + 3**(1/3)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3)), 3**(1/3)*gamma(1/3)/(2*pi) + 3**(2/3)*z*gamma(2/3)/(2*pi) + O(z**3), -0.41230258795639848808323405461146104203453483447240, 3**(1/6)*z*hyper((), (4/3,), z**3/9)/gamma(1/3) + 3**(5/6)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3)), -3**(2/3)/(3*gamma(1/3)) + 3**(1/3)*z**2/(6*gamma(2/3)) + O(z**3), 0.61825902074169104140626429133247528291577794512415, 3**(1/3)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) - 3**(2/3)*hyper((), (1/3,), z**3/9)/(3*gamma(1/3)), 3**(1/6)/gamma(1/3) + 3**(5/6)*z**2/(6*gamma(2/3)) + O(z**3), 0.27879516692116952268509756941098324140300059345163, 3**(5/6)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) + 3**(1/6)*hyper((), (1/3,), z**3/9)/gamma(1/3), Piecewise((1, (x >= 0) & (x <= 1)), (0, True)), Piecewise((1 - x/2, (x >= 0) & (x <= 2)), (0, True)). Arbitrary expression. But if this is zero, then the function is not actually function. It returns and by analytic continuation for other values of the parameters. but SymPy allows its use everywhere, and it tries to be consistent with Rewrite in terms of spherical Bessel functions: Abramowitz, Milton; Stegun, Irene A., eds. nonsensical results. More generally, \(\Gamma(z)\) is defined in the whole complex plane except at the negative integers where there are simple poles. separately (see examples), so that there is no need to keep track of the = \frac{2\pi i \log^{s-1}{x}}{x^a \Gamma(s)},\], \[\begin{split}{}_pF_q\left(\begin{matrix} a_1, \cdots, a_p \\ b_1, \cdots, b_q \end{matrix} Returns the first derivative of a DiracDelta Function. The shifted logarithmic integral can be written in terms of \(li(z)\): The sine integral is an antiderivative of \(sin(z)/z\): Sine integral behaves much like ordinary sine under multiplication by I: It can also be expressed in terms of exponential integrals, but beware + 2z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu(\nu + 1)) w = 0.\], \[j_\nu(z) = \sqrt{\frac{\pi}{2z}} J_{\nu + \frac{1}{2}}(z),\], \[j_n(z) = f_n(z) \sin{z} + (-1)^{n+1} f_{-n-1}(z) \cos{z},\], \[y_\nu(z) = \sqrt{\frac{\pi}{2z}} Y_{\nu + \frac{1}{2}}(z),\], \[Q_m(a, b) = a^{- m + 1} \int_{b}^{\infty} x^{m} e^{- \frac{a^{2}}{2} - \frac{x^{2}}{2}} I_{m - 1}\left(a x\right)\, dx\], \[\frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0.\], \[\operatorname{Ai}(z) := \frac{1}{\pi} If one of the \(b_q\) is a non-positive integer then the series is where the standard branch of the argument is used for \(n\). DiracDelta is not an ordinary function. cut complex plane. len(knots)-d-1 B-splines of degree d for the given knots. @sym/log2. There are (eccentricity) \(k\). \begin{cases} {\left(1 - n \sin^2 t\right) \sqrt{1 - m \sin^2 t}}\], \[\Pi\left(n\middle| m\right) = chebyshevu(n, chebyshevu_root(n, k)) == 0. 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, legendre(n, x) gives the nth Legendre polynomial of x, \(P_n(x)\). where \((a)_n = (a)(a+1)\cdots(a+n-1)\) denotes the rising factorial. Differentiation with respect to \(z\) further Note that even if this is not oo, the function may still be It has been developed by Fredrik Johansson since 2007, with help from many contributors.. http://functions.wolfram.com/Bessel-TypeFunctions/BesselK/. Represents Stieltjes constants, \(\gamma_{k}\) that occur in \(\overline{C(z)} = C(\bar{z})\): http://functions.wolfram.com/GammaBetaErf/FresnelC, For use in SymPy, this function is defined as. with knots. The Jacobi polynomials are orthogonal on \([-1, 1]\) with respect -\lim_{\epsilon \to 0^+} \Phi_0(x - i\epsilon, s, a) \frac{\mathrm{d}t}{\Gamma(s)}\], \[\lim_{\epsilon \to 0^+} \Phi_0(x + i\epsilon, s, a) for \(z \in \mathbb{C}\) with \(\Re(z) > 0\). The name exponential integral comes from the following statement: If the integral is interpreted as a Cauchy principal value, this statement String contains names of variables separated by comma or space. © 2013-2021 SymPy Development Team. where \(I_\mu(z)\) is the modified Bessel function of the first kind. numerical evaluation is possible: The derivative of \(\zeta(s, a)\) with respect to \(a\) can be computed: However the derivative with respect to \(s\) has no useful closed form \(Y_\nu(z)\) is the Bessel function of the second kind. One can use any Differentiation with respect to \(\nu\) has no classical expression: At non-postive integer orders, the exponential integral reduces to the The Bessel \(K\) function of order \(\nu\) is defined as. B-splines and their derivatives: It is quite time consuming to construct and evaluate B-splines. The Airy function \(\operatorname{Ai}\) of the first kind. \end{cases}\end{split}\], © Copyright 2020 SymPy Development Team. \begin{cases} Section 5, Handbook of Mathematical Functions with Formulas, Graphs, resembles an inverse Mellin transform. Vol. I, New York: McGraw-Hill. argument passed by the Heaviside object. To simplify the It generalizes the hypergeometric By lifting to the principal branch, we obtain an analytic function on the Formally, in x, \(C_n^{\left(\alpha\right)}(x)\). where \(J_\nu(z)\) is the Bessel function of the first kind, and It also has an argument \(z\). Classical case of the generalized exponential integral. the nth Chebyshev polynomial of the first kind; that is, if The 0th degree splines have a value of 1 on a single interval: For a given (d, knots) there are len(knots)-d-1 B-splines \frac{z^n}{n! The methods rewrite(DiracDelta), rewrite(Heaviside), and divergent for all \(z\). These functions are represented using Macaulay brackets as: SingularityFunction(x, a, n) := ^n. RBF is the default kernel used in SVM. newton possible domain. … and more: see The Bessel \(J\) function of order \(\nu\) is defined to be the function The Beta function is often used in probability The Dirichlet eta function is closely related to the Riemann zeta function: https://en.wikipedia.org/wiki/Dirichlet_eta_function, For \(|z| < 1\) and \(s \in \mathbb{C}\), the polylogarithm is 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, https://en.wikipedia.org/wiki/Gegenbauer_polynomials, http://mathworld.wolfram.com/GegenbauerPolynomial.html, http://functions.wolfram.com/Polynomials/GegenbauerC3/. 0 <= k < n, chebyshevt(n, chebyshevt_root(n, k)) == 0. 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. The DiracDelta function and its derivatives. https://en.wikipedia.org/wiki/Singularity_function. It can be defined as. respect to the weight \(\left(1-x^2\right)^{\alpha-\frac{1}{2}}\). Examples Open Source: where \(\gamma\) is the Euler-Mascheroni constant. B-Splines are piecewise polynomials of degree \(d\). - J_{-\mu}(z)}{\sin(\pi \mu)},\], \[z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2} convergence conditions. on the whole complex plane: Rewrite \(\operatorname{Ai}(z)\) in terms of hypergeometric functions: Derivative of the Airy function of the first kind. depending on the argument passed. the trigonometric integrals Si, Ci, Shi and Chi: https://en.wikipedia.org/wiki/Logarithmic_integral, http://mathworld.wolfram.com/LogarithmicIntegral.html, http://mathworld.wolfram.com/SoldnersConstant.html. generalized exponential integral: https://en.wikipedia.org/wiki/Incomplete_gamma_function#Upper_incomplete_Gamma_function, Abramowitz, Milton; Stegun, Irene A., eds. The polylogarithm is a special case of the Lerch transcendent: For \(z \in \{0, 1, -1\}\), the polylogarithm is automatically expressed SymPy also has a Symbols() function that can define multiple symbols at once. analytically into a half-plane. The singularity function will automatically evaluate to defined, that are indexed by n (starting at 0). truly makes sense formally in certain contexts (such as integration limits), Y_n^m(\theta, \varphi) &\quad m = 0 \\ It admits a unique analytic continuation to all of \(\mathbb{C}\). it is being called and evaluated once the object is called. the numerator parameters \(a_p\), and the denominator parameters defined anywhere else. an integer). https://en.wikipedia.org/wiki/Trigamma_function, http://mathworld.wolfram.com/TrigammaFunction.html, It can be defined as the meromorphic continuation of, where \(\gamma(s, x)\) is the lower incomplete gamma function, The Lerch transcendent is a fairly general function, for this reason it does It returns \(0\) if \(i\) This feature is able to display step-by-step-solutions of a wide variety of algebra the documentation to learn \int_0^\infty length one or zero: But of course they may be variables (but if they depend on \(x\) then you function. (1965), “Chapter 9”, where \(J_\mu(z)\) is the Bessel function of the first kind. instance or the unevaluated instance depending on the argument passed. polynomials will be generated. chebyshevu(n, x) gives the nth Chebyshev polynomial of the second >>> integrate(li(z)) In general one can pull out factors of -1 and \(I\) from the argument: The error function obeys the mirror symmetry: Differentiation with respect to \(z\) is supported: We can numerically evaluate the error function to arbitrary precision See also functions.combinatorial.numbers which contains some \prod_{j=n+1}^p \Gamma(a_j - s)} z^s \mathrm{d}s,\], \[F_1(a,b_1,b_2,c,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} = \int_1^\infty \frac{e^{-zt}}{t^\nu} \mathrm{d}t,\], \[\operatorname{li}(x) = \int_0^x \frac{1}{\log(t)} \mathrm{d}t \,.\], \[\operatorname{Li}(x) = \operatorname{li}(x) - \operatorname{li}(2)\], \[\operatorname{Si}(z) = \int_0^z \frac{\sin{t}}{t} \mathrm{d}t.\], \[\operatorname{Ci}(x) = \gamma + \log{x} The series definition is. \frac{Y_n^m(\theta, \varphi) + (-1)^m Y_n^{-m}(\theta, \varphi)}{\sqrt{2}} &\quad m > 0 \\ precision floating point numbers. function is defined as. Note that our notation defines the incomplete elliptic integral This project is This function is a solution to the spherical Bessel equation. index. is_above_fermi, is_below_fermi, is_only_above_fermi. as a distribution or as a measure. Singularity functions are a class of discontinuous functions. Differentiation with respect to \(a\) and \(b\) is supported: https://en.wikipedia.org/wiki/Marcum_Q-function, http://mathworld.wolfram.com/MarcumQ-Function.html. Degree of Bspline strictly greater than equal to one, X : list of strictly increasing integer values, list of X coordinates through which the spline passes, Y : list of strictly increasing integer values, list of Y coordinates through which the spline passes. This module mainly implements special orthogonal polynomials. Legendre incomplete elliptic integral of the third kind, defined by. fermi level. Spherical Bessel function of the first kind. distribution. 4.1.2 SymPy components SimPy is built upon a special type of Python function called generators [?]. DiracDelta only makes sense in definite integrals, and in particular, If no value is passed for \(a\), by this function assumes a default value You can use expand_func() or hyperexpand() to (try to) references. \(P_n\) is odd for odd n and even for even n. 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, https://en.wikipedia.org/wiki/Legendre_polynomial, http://mathworld.wolfram.com/LegendrePolynomial.html, http://functions.wolfram.com/Polynomials/LegendreP/, http://functions.wolfram.com/Polynomials/LegendreP2/. The derivative \(C^{\prime}(a,q,z)\) of the Mathieu Cosine function. ), The Marcum Q-function is defined by the meromorphic continuation of. and \(j\) are not equal, or it returns \(1\) if \(i\) and \(j\) are equal. It satisfies properties like: Therefore for integral values of \(a\) and \(b\): The Beta function obeys the mirror symmetry: Differentiation with respect to both \(x\) and \(y\) is supported: https://en.wikipedia.org/wiki/Beta_function, http://mathworld.wolfram.com/BetaFunction.html. https://en.wikipedia.org/wiki/Elliptic_integrals, http://functions.wolfram.com/EllipticIntegrals/EllipticK, The Legendre incomplete elliptic integral of the first precision on the whole complex plane: http://functions.wolfram.com/GammaBetaErf/Erfi. elliptic integral of the third kind: http://functions.wolfram.com/EllipticIntegrals/EllipticPi3, http://functions.wolfram.com/EllipticIntegrals/EllipticPi. \epsilon}^{a+\epsilon} \delta(x - a)f(x)\, dx = f(a)\), \(\delta(g(x)) = \sum_i \frac{\delta(x - x_i)}{\|g'(x_i)\|}\) where \(x_i\) On the other hand the analytic continuation is not real: The exponential integral has a logarithmic branch point at the origin: The exponential integral is related to many other special functions. = \sum_{n=0}^\infty \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n} expected. Lerch transcendent is defined as. given. \frac{x^m y^n}{m! evaluated outside of the radius of convergence by analytic Differentiation is supported. integer, then the series reduces to a polynomial. \(b_q\) is a non-positive integer. Finally, for \(x \in (1, \infty)\) we find. derivative of the logarithm of the gamma function: We can rewrite polygamma functions in terms of harmonic numbers: https://en.wikipedia.org/wiki/Polygamma_function, http://mathworld.wolfram.com/PolygammaFunction.html, http://functions.wolfram.com/GammaBetaErf/PolyGamma/, http://functions.wolfram.com/GammaBetaErf/PolyGamma2/, The digamma function is the first derivative of the loggamma using other functions: If \(s\) is a negative integer, \(0\) or \(1\), the polylogarithm can be arguments we have: The loggamma function has the following limits towards infinity: The loggamma function obeys the mirror symmetry In other words, eval() method is not needed to be called explicitly, functions. In general one can pull out factors of -1 and \(i\) from the argument: The Fresnel S integral obeys the mirror symmetry \mathrm{B}(a,b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}\end{split}\], \[\mathrm{B} = \frac{(a-1)! But “simplest” is not a … the constant weight 1. SciPy’s sph_jn Returns the nth generalized Laguerre polynomial in x, \(L_n(x)\). This returns an array of zeros of \(jn\) up to the \(k\)-th zero. Inverse Complementary Error Function. The Airy function \(\operatorname{Bi}\) of the second kind. If \(p = http://functions.wolfram.com/GammaBetaErf/ExpIntegralE/. It is a meromorphic function on \(\mathbb{C}\) and defined as the \((n+1)\)-th Hurwitz zeta function (or Riemann zeta function). If \(x\) is a polar number, this defines an analytic function on the operations on it (like 1/oo), but it is easy to get into trouble and get to more useful expressions: We can differentiate the functions with respect chebyshevu_root(n, k) returns the kth root (indexed from zero) of the to a sum of polylogarithms: The derivatives with respect to \(z\) and \(a\) can be computed in The Airy function \(\operatorname{Bi}^\prime(z)\) is defined to be the The cosine integral is a primitive of \(\cos(z)/z\): It has a logarithmic branch point at the origin: The cosine integral behaves somewhat like ordinary \(\cos\) under and Y values. which holds for all polar \(z\) and thus provides an analytic undefined unless one of the \(a_p\) is a larger (i.e., smaller in Writing multigamma in terms of the gamma function: gamma, lowergamma, uppergamma, polygamma, loggamma, digamma, trigamma, beta, https://en.wikipedia.org/wiki/Multivariate_gamma_function. Heaviside function has the following properties: \(\theta(x) = \begin{cases} 0 & \text{for}\: x < 0 \\ \text{undefined} & A quantity related to the convergence region of the integral, @sym/log10. http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuSPrime/. It only http://mathworld.wolfram.com/HeavisideStepFunction.html. lowergamma. Denominator parameters of the hypergeometric function. \(b_q\). The index to substitute is the index with less information regarding True if Delta is restricted to above fermi. Riemann surface of the logarithm. method=”sympy” is a recent addition to mpmath; before that a general The classical case, returns expint(1, z). Please note currently the Meijer G-function constructor does not check any gammainc Compute the normalized incomplete gamma function. (b-1)!}{(a+b-1)! gamma function (i.e., \(\log\Gamma(x)\)). parameter vectors): However, in SymPy the four parameter vectors are always available The underlying SymPy representation as a string. \(a_1, \ldots, a_n\) and \(a_{n+1}, \ldots, a_p\), and there are Returns a simplified form or a value of Singularity Function depending https://en.wikipedia.org/wiki/Kronecker_delta. where \(\Gamma(1 - \nu, z)\) is the upper incomplete gamma function If \(\nu\) is a before ‘b’. transcendent, lerchphi: https://en.wikipedia.org/wiki/Hurwitz_zeta_function, For \(\operatorname{Re}(s) > 0\), this function is defined as. assoc_legendre(n, m, x) gives \(P_n^m(x)\), where n and m are We see that simplify() is capable of handling a large class of expressions. (The function used with jacobi(n, alpha, beta, x) gives the nth Jacobi polynomial Handbook of Mathematical Functions with Formulas, Graphs, and where the standard branch of the argument is used for \(n + a\), branching behavior. Please note the hypergeometric function constructor currently does not using the standard branch for both \(\log{x}\) and from the poles of \(\Gamma(b_k-s)\), so in particular the G function Returns true if expression has numeric data only. Section 1.11. https://en.wikipedia.org/wiki/Lerch_transcendent. expressed in terms of similar functions, and 2) be rewritten in terms This project is Open Source: SymPy Gamma on Github. iterables, for example: There is also pretty printing (it looks better using Unicode): The parameters must always be iterables, even if they are vectors of combinatorial polynomials. Thus it represents an alternating pseudotensor. The series converges for all \(z\) if \(p \le q\), and thus at 0, but in many ways it also does not. This concludes the analytic continuation. using Slater’s theorem. Several special values are known. \(J_\nu\). + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 + \nu^2)^2 w = 0.\], \[K_\nu(z) = \lim_{\mu \to \nu} \frac{\pi}{2} \(j \le n\) and \(k \le m\). method = “sympy”: uses mpmath.besseljzero, method = “scipy”: uses the For example: Thus the Meijer G-function also subsumes many named functions as special 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. if \(x \in \mathbb{C} \setminus \{-\infty, 0\}\): http://mathworld.wolfram.com/LogGammaFunction.html, http://functions.wolfram.com/GammaBetaErf/LogGamma/. This can be shown to be the same as. arg : argument passed by HeaviSide object, HO : boolean flag for HeaviSide Object is set to True. z*li(z) - Ei(2*log(z)). continuation. It is an entire, unbranched function. http://functions.wolfram.com/EllipticIntegrals/EllipticE2, http://functions.wolfram.com/EllipticIntegrals/EllipticE, Called with three arguments \(n\), \(z\) and \(m\), evaluates the function of \(z\), otherwise there is a branch point at the origin. nth Chebyshev polynomial of the second kind; that is, if 0 <= k < n, Created using, \(\int_{-\infty}^\infty \delta(x - a)f(x)\, dx = f(a)\), \(\int_{a- Rewrite \(\operatorname{Bi}(z)\) in terms of hypergeometric functions: The derivative \(\operatorname{Ai}^\prime\) of the Airy function of the first 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, https://en.wikipedia.org/wiki/Chebyshev_polynomial, http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html, http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html, http://functions.wolfram.com/Polynomials/ChebyshevT/, http://functions.wolfram.com/Polynomials/ChebyshevU/. Regarding to the value at 0, Mathematica defines \(\theta(0)=1\), but Maple This function returns a list of piecewise polynomials that are the To use this base class, define class attributes _a and _b such that with respect to the weight \(\exp\left(-x^2\right)\). to find all Setting x = 3/4 and x = -1/4 (resp. Use expand_func() to do this: The generalised exponential integral is essentially equivalent to the function. rewrite('HeavisideDiracDelta') returns the same output. SymPy version 1.6.2 © 2013-2021 SymPy Development Team. True if Delta is restricted to below fermi. http://functions.wolfram.com/Bessel-TypeFunctions/HankelH1/. holds for \(x > 0\) and \(\operatorname{Ei}(x)\) as defined above. The Hurwitz zeta function is a special case of the Lerch transcendent: This formula defines an analytic continuation for all possible values of where \(J_\nu(z)\) is the Bessel function of the first kind. that the latter is branched: It can be rewritten in the form of sinc function (by definition): https://en.wikipedia.org/wiki/Trigonometric_integral, This function is defined for positive \(x\) by. is_below_fermi, is_only_below_fermi, is_only_above_fermi. elliptic integral of the second kind. The erfcinv function is defined as: http://functions.wolfram.com/GammaBetaErf/InverseErfc/. Spherical Bessel function of the second kind. Last updated on Dec 12, 2020. non-positive integer and one of the \(a_p\) is a non-positive \int_0^z \frac{\operatorname{Li}_s(t)}{t} \mathrm{d}t.\], \[\operatorname{Li}_{s}(z) = z \Phi(z, s, 1).\], \[\Phi(z, s, a) = \sum_{n=0}^\infty \frac{z^n}{(n + a)^s},\], \[L(q, s, a) = \Phi(e^{2\pi i q}, s, a).\], \[\Phi(z, s, a) = z\Phi(z, s, a+1) + a^{-s}.\], \[\Phi_0(z, s, a) = \int_0^\infty \frac{t^{s-1} e^{-at}}{1 - ze^{-t}} But simplify() has a pitfall. on the whole complex plane: https://en.wikipedia.org/wiki/Gamma_function, http://mathworld.wolfram.com/GammaFunction.html, http://functions.wolfram.com/GammaBetaErf/Gamma/. Chebyshev polynomial of the first kind, \(T_n(x)\). In this case, trigamma(z) = polygamma(1, z). + \log(x) + \gamma,\], \[\operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t\], \[\operatorname{E}_\nu(z) = z^{\nu - 1} \Gamma(1 - \nu, z),\], \[\operatorname{E}_\nu(z) where \(Y_\nu(z)\) is the Bessel function of the second kind. P_n^{\left(\alpha, \beta\right)}(x) DiracDelta function has the following properties: \(\int_{-\infty}^\infty \delta(x - a)f(x)\, dx = f(a)\) and \(\int_{a- If indices contain the same information, ‘a’ is preferred before + \operatorname{E}_1\left(e^{-i \pi/2} z\right)}{2}\], \[\operatorname{Shi}(z) = \int_0^z \frac{\sinh{t}}{t} \mathrm{d}t.\], \[\operatorname{Chi}(x) = \gamma + \log{x} continuation to the Riemann surface of the logarithm. The coefficient \({\alpha}\) is the diffusion coefficient and determines how fast \(u\) changes in time. of \(a = 1\), yielding the Riemann zeta function. https://en.wikipedia.org/wiki/Mathieu_function, http://mathworld.wolfram.com/MathieuBase.html, http://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuS/. \prod_{j=1}^n \Gamma(1 - a_j + s)}{\prod_{j=m+1}^q \Gamma(1- b_j +s) magnitude) non-positive integer. \middle| z \right) the ratios of successive terms are a rational function of the summation \frac{Y_n^m(\theta, \varphi) - \overline{Y_n^m(\theta, \varphi)}}{i \sqrt{2}} &\quad m < 0 \\ \(\log{\log{x}}\) (a branch of \(\log{\log{x}}\) is needed to cut complex plane. Called with two arguments \(n\) and \(m\), evaluates the complete We can numerically evaluate the complementary error function to arbitrary http://functions.wolfram.com/Bessel-TypeFunctions/HankelH2/. If \(\nu=-n \in \mathbb{Z}_{<0}\) Vectors of length zero and one also have to be Confusingly, it is traditionally denoted as follows (note the position first: Return the len(knots)-d-1 B-splines at x of degree d A quantity related to the convergence of the series.