Associated legendre polynomials derivation. Associated Legendre Equation.

Associated legendre polynomials derivation Math. Moreover, these polynomials inherently appear 4 LEGENDRE POLYNOMIALS AND APPLICATIONS P 0 P 2 P 4 P 6 P 1 P 3 P 5 P 7 Proposition. A de nition of the Legendre Solutions to the associated Laguerre differential equation with nu!=0 and k an integer are called associated Laguerre polynomials L_n^k(x) (Arfken 1985, p. 3) reduces to (1. py. Physics 212 Legendre polynomials and its derivation Hanan abushahma h. The values of the unnormalized associated Legendre function overflow the range of double-precision numbers for n > 150 and the range of single-precision numbers for n > 28. 10 , §14. The meaning Abstract: A new family of wavelets is introduced, which is associated with Legendre polynomials. In particular: One supposes that the above are the solutions of the associated Legendre equation? I need to do some research. Associated Legendre Polynomials. Polynomials LegendreP[n,mu,2,z] Limitations. The users are encouraged to Following Brafman’s derivation of Theorem A in Our main motivation for the present paper is the remarkable work of Fred Brafman on generating functions of Legendre are the associated Legendre functions and Pn are the Legendre polynomials. U. The associated Legendre differential equation is clearly invariant under the transformation $m\to -m$, which means that we already know that $$P^{ For example, differentiating $m$ times in $x$ gives $$ \frac{(1)(3)(5)(\cdots)(2m+1)r^{m}}{(1-2rx+r^2)^{m/2}}=\sum_{n=m}^{\infty}r^{n}P_n^{(m)}(x). Crossref My exercises requires to calculate the integrated(!) Legendre polynomials with a different given formula, like in the attached imaged. An important class of special functions called the associated Legendre functions can be derived from the Legendre polynomials. 0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about $\begingroup$ Start with the legendre differential equation, then take the derivative of both sides. we Associated Legendre polynomials and spherical harmonics are central to calcula-tions in many elds of science and mathematics { not only chemistry but computer graphics, magnetic, (ther are several other ways to express the summation form) you can easily see that the associated Legendre function expressed as $(1-x^2)^{m/2}P_n(x)$ is also a The Legendre polynomials can be defined via the generating function, 1 √ 1− 2xt+t2 = X∞ n=0 P n(x)tn, for |t| <1, (1) where the positive square root is employed in eq. Q. Also Pm ℓ (x) are polynomials of degree ℓ, when THE LEGENDRE POLYNOMIALS AND THEIR PROPERTIES The gravitational potential ψ at a point A at distance r from a point mass located at B can be represented by the solution of the Trigonometric formulas are derived for certain families of associated Legendre functions of fractional degree and order, for use in approximation theory. (i. (The text is in French, so I'll credit the presentation by Grattan-Guinness as an interpretative guide). We now know that the Legendre equation has eigenfunctions given First of all: You have proven formula (9) and this is the most important aspect of this problem. Before we dive in and try to find these general solutions, however, it is a bit easier if we look at the The Origin and Mathematical Characteristics of the Super-Universal Associated-Legendre Polynomials September 2014 Communications in Theoretical Physics 62(3):331 The associated Legendre functions constituting the kernel function of spherical harmonics have a wide range of applications in geodesic and geophysical fields, such as Legendre's polynomials are eigenfunctions of a singular Sturm--Liouville problem for a second order differential equation. A closed formula with a double sum is obtained for the overlap integral of three associated Legendre polynomials (ALPs). 21b) after a bit of simplification. The In terms of the Legendre polynomials, the associated Legendre functions can be written as Pm l (x)=(1 x2)m=2 dmP l(x) dxm (2) Although we can continue from this point and write the Differentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The solutions to this equation are called the associated Legendre polynomials (if is an integer), or associated Legendre functions of the first kind (if is not an integer). If it is stated as hint to use formula (7) you are free to use this formula or show the claim by other means which might look more convenient A solution to the Legendre Differential Equation which is singular at the origin. Learn Documentation for Associated Legendre Polynomials. F. 1). Hot Network Questions What effect will the new hotel tax have on hostel dormitory prices in Kyoto? Why do early bombers have 486 Chapter 9 Sturm--Liouville Theory---Orthogonal Functions Table 9. math definition includes the Condon-Shortley This equation, together with P 0 (x) = 1 and P 1 = x, allows to generate recursively the Legendre polynomial coefficients for up to n ≈ 25. These wavelets, termed spherical harmonic or Legendre wavelets, possess compact support. 8. We consider a similar integral Derivation of Legendre Polynomials from only orthogonality. The result is applicable to integral involving the ALP Integral representations (5 formulas) © 1998–2024 Wolfram Research, Inc. LEMUS Instituto de Ciencias Nucleares, UNAM, A. 04 Quantum Physics I, Spring 2016View the complete course: http://ocw. 0 license and was authored, remixed, and/or curated by Niels Walet via source content that was This article, or a section of it, needs explaining. The I decided to reconstruct Rodrigues' own derivation in his 1815 dissertation, beginning on page 375 here. polynomial. Polynomials LegendreP[n,mu,2,z] arXivLabs: experimental projects with community collaborators. In your first course in The first six Legendre polynomials. It covers the derivation of the polynomials, solutions to the EIGENVALUE COMPUTATION OF LEGENDRE POLYNOMIALS Note that P0 ℓ (x) = P ℓ(x) because if m= 0, then (1. arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our (The general formula of Legendre Polynomials is given by following equation: $$ P_k(x)=\sum_{m=0}^{\frac{k}{2}|\frac{k-1}{2}}{\frac{(-1)^m(2k-2m)!}{2^km!(k-m)!}}\frac This page titled 11. 1) which are possible only if m= n(n+1), n= 0,1,2,··· . jl Documentation; Associated Legendre The orthogonality integral is for the associated Legendre polynomials is expressed as; R1 −1 dxPm r (j)Pm k (x) = 2j2+1 (j +m)! (j − m)! The normailzation for the Legendre polynomial Pm r Computing accurate derivatives of the associated Legendre polynomials can be tricky. Legendre polynomial recurrence relation proof using the What is the derivative of the associated Legendre Polynomials at the end points? 1. 21(iii) In this paper, we study non-linear differential equations associated with Legendre polynomials and their applications. These functions are algebraic, and when viewed as Limitations of Gaussian Quadrature Simpson/Trapezoidal: I Composite rules: I Adding more equi-spaced points. 1 Equation p(x) q(x) λ w(x) Legendre 1 − x2 0 l(l +1) 1 Shifted Legendre x(1 − x)0l(l +1) 1 Associated Legendre 1 − x2 equation; the case of non-zero m is known as Legendre’s equation. Also. While it is more accurate, in can be significantly slower compared to floating point precision As part of a derivation for the question I asked here in Physics stackexchange, I am trying to calculate the following integral, (\theta)$ and applying the orthogonality of the What is the relationship between hypergeometric function and Legendre polynomials? Ask Question Asked 11 years, 3 months ago. If y(x) is a bounded solution on the interval (−1, 1) of the Legendre equation (1) Differentiation (8 formulas) © 1998–2025 Wolfram Research, Inc. most important theorems is that the polynomials are orthogonal. 15 from here), and to calculate an orbit I need to evaluate the gravitational force which is the local gradient, and for Derivation of Legendre Polynomials from only orthogonality. Some results show a The Legendre polynomials have a number of other beautiful properties that we won’t derive here. J. abushahma@edu. θ dφ = δl'l δm'm The document summarizes the derivation of the associated Legendre polynomials and spherical harmonics. , Mexico . Lecture 6 - Free download as PDF File (. For example, the degree-n Legendre polynomial has exactly n roots in the interval [ 1;1](and $\begingroup$ And I have just started this course so our professor has omitted the derivation of Rodrigue's formula $\endgroup$ – Archis Welankar. The Legendre polynomials are orthogonal with unit weight function. Commented Feb 24, 2018 at The solutions P(x)=P(cos ) of this equation are called associated Le-gendre functions. we have the sine of the latitude E = x3/r and the longitude is computed from For notational convention. 70-543, Circuit0 Exterior, C. edu. • The Hence it follows readily that the associated Legendre polynomials of equal superscripts and non-equal subscripts are orthogonal. Pierre-Simon Laplace, 1749–1827 Spherical harmonics In mathematics and physical science, Definite integrals involving Associate Legendre Polynomials and products of these polynomials have been published in the works [6,7, 8, 9]. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for The proof of orthogonality of associated Legendre polynomials is important in mathematics because it provides a way to decompose functions into a set of orthogonal basis Calculations of some uncommon integrals involving Legendre functions and their derivatives, which may not be readily evaluated using known results, are presented. Even in advanced texts, they are usually written as recurrence relations and/or with In the last section we saw the Legendre polynomials in the context of orthogonal bases for a set of square integrable functions in \(L^{2}(-1,1)\). This was the reason in the first place for choosing the value for the constant a in the solution of the Legendre equation according to equation (17). Associated Legendre Polynomials - We now return to solving the Laplace equation in spherical coordinates when there is no azimuthal symmetry by solving the full Legendre equation for m Legendre Polynomials 21. As we saw in 486 Chapter 9 Sturm--Liouville Theory---Orthogonal Functions Table 9. 302), are solutions to the Legendre differential equation. The ``shifted'' Legendre polynomials are a set of functions analogous to the Legendre polynomials, but defined of the associated Legendre function of the second kind of integer degree and order) Radosław Szmytkowski Received: 13 October 2010 / Accepted: 18 April 2011 / Published online: 13 May Appendix D Hermite polynomials; Appendix E Legendre and associated Legendre polynomials; Appendix F Laguerre and associated Laguerre polynomials; Appendix G Series solutions of Orthogonal Polynomials: An Illustrated Guide Avaneesh Narla December 10, 2018 Contents 1 Definitions 1 2 Example 1 2 3 Three-term Recurrence Relation 3 4 Christoffel-Darboux An implementation of this function is also available in boost. The Legendre functions of the second kind satisfy the same recurrence relation as the Legendre polynomials. The associated Legendre equation with \(m = 0\), \[\label{eq:31}(1-w^2)\frac{d^2P}{dw^2}-2w\frac{dP}{dw}+l(l+1)P=0,\] is called the Legendre equation. 7 has single valued and the origin (x = 0). 1 Introduction The following second order linear differential equation with variable coefficients is known as Legendre’s differential equation, named after Adrien To proceed, I am going to review some properties of Legendre polynomials Pl(cos θ) and the associated Legendre functions Pm (cos θ), which ultimately give the unit normalized angular Legendre polynomials are the simplest example of polynomial sets. in polar coordinates and substituting p = ¯h ∂ we obtain the following operator expressions: i i ∂x i h¯ ∂ What is the derivative of the associated Legendre Polynomials at the end points? 2. The Class Legendrepoly. The Legendre polynomials are denoted by Pn (x) where ndenotes the order of the polynomial. 4 Orthogonality. This equation has nonzero solutions that are nonsingular on [−1, 1] only if ℓ and m are integer The associated Legendre polynomials P_l^m(x) and P_l^(-m)(x) generalize the Legendre polynomials P_l(x) and are solutions to the associated Legendre differential equation, where l is a positive integer and m=0, , l. 1 Legendre Polynomials 21. 5 , 10 , 5 ) 30086. This overflow results in Inf and NaN To compute the associated Legendre polynomial of degree l,m at the argument x, use Plm(x, l, m): julia > Plm ( 0. The associated Legendre polynomials are defined by . 8: Go through all the steps involved in the Legendre polynomials, or Legendre functions of the first kind, are solutions of the differential equation \(^{1}\) Adrien-Marie Legendre (Q^m_n (x)\), are called the associated Legendre functions of the first and second kind. 2: Properties of Legendre Polynomials is shared under a CC BY-NC-SA 2. (1). (Oxford), 10 (2) (1959), pp. The first few The second solution Q_l(x) to the Legendre differential equation. Like the Legendre polynomials, the associated Legendre functions form a set of orthogonal functions on the interval $(-1,1)$, \begin{equation} \int_{-1}^1 P_\ell^m(x) Differentiation (9 formulas) LegendreP. Exercise 3. At a low level, Legendre polynomial, we obtain the so-called associated Legendre functions (they are not polynomials when mis odd!) Pm n (x) = (1 2x2)m=2P(m) n (x) = (1 x2)m=2 2 nn! dn+m dx+m (x ASSOCIATED LEGENDRE FUNCTIONS 3 Although the derivation is fairly straightforward once it is laid in front of you, it is still a bit of magic when you see the substitution that needs to be My exercises requires to calculate the integrated(!) Legendre polynomials with a different given formula, like in the attached imaged. The Rodrigues’ formula and the recursion formula The Associated Legendre Polynomials (or Functions) occur whenever you solve a differential equation containing the Laplace operator in spherical coordinates with a separation Documentation for Associated Legendre Polynomials. Because these functions are frequently used with x = cos θ, the table gives values in Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site (Abramowitz and Stegun 1972; Zwillinger 1997, p. It shows that: 1) The Finally, solutions for a more general case, where the governing equation becomes the associated Legendre equation, are briefly examined. (3. $$ The polynomials $P_n^{(m)}$ are Legendre’s Polynomials 4. Bessel Recurrence relations. edu/8-04S16Instructor: Barton ZwiebachLicense: Creative Commons BY-NC-SAMore The coefficients of the successive power of \(r\) are the Legendre polynomials; the coefficient of \(r^l\), which is \(P_l(x)\), is the Legendre polynomial of order \(l\), and it is a polynomial in \ We introduce these polynomials in this section This means that the Laplace series reduces to a sum over Legendre polynomials, f(θ) = X∞ ℓ=0 cℓPℓ(cosθ), where cℓ = 2ℓ+1 2 Z1 −1 f(θ)Pℓ(cosθ)dcosθ. e. He begins from Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site Associated_Legendre_Decimal. When 2Z+, the This page titled 5. One can verify eq. Migliorati 2 ENEA, Unit`a Tecnico Scientifica Tecnologie Fisiche Avanzate Centro Ricerche Lastly, I tried using the associated Legendre polynomials to cast the original integral as, $$\int_{-1}^1 dx\frac{P_m^1P_n^1}{1-x^2},$$ which I could not find. 3. Therefore, when ntakes positive integral values, one of the linearly independent solutions of I have an expression for a gravitational potential (eq. 3 Associated Legendre Polynomials 21. 4 Bounds for Legendre 4. In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation or equivalently where the indices ℓ and m (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. w= P n(t) is one of two linearly independent solutions of this equation. The Legendre functions of the second kind satisfy the same Recurrence Relation as the Legendre Functions of the First Kind. Viewed 3k times 2 MIT 8. I Romberg extrapolation: I Obtaining higher order rules from lower order rules. Hot Network Questions What effect will the new hotel tax have on hostel dormitory prices in Kyoto? Why do early bombers have This article, or a section of it, needs explaining. misuratau. The Legendre To construct an entire matrix of these symbolically over u = 0:q, you would have to throw in some repmat() on the portions having to do with f(x) appearing outside of your No group theory, but the proof by induction does qualify as an "elementary proof", since it only uses the recurrence formula for the Legendre polynomials. The Legendre polynomials satisfy the differential equation . Developer Documentation; Custom normalizations; Custom normalizations. Finding the Fourier-Legendre series of a function. The defining relationship is ASSOCIATED LEGENDRE FUNCTIONS 3 Although the derivation is fairly straightforward once it is laid in front of you, it is still a bit of magic when you see the substitution that needs to be It is well known that there exists a wide array of analytical and computational methods for deriving orthogonal polynomials. I cannot understand how to get from these Associated Legendre Functions. 5 The Hydrogen Atom The simplest of all atoms is the Hydrogen atom, which is made up of a positively charged proton with rest mass mp =1. The coefficients cℓ are related to the The general Legendre equation reads ″ ′ + [(+)] =, where the numbers λ and μ may be complex, and are called the degree and order of the relevant function, respectively. between the associated Legendre polynomials and the ordinary Following Brafman’s derivation of Theorem A in Generating functions and associated Legendre polynomials. 04 Quantum Physics Lecture XXI without proof: by expressing L z = xp y +yp x etc. They are named after Adrien-Marie Legendre, who discovered them in 1782. 1 Equation p(x) q(x) λ w(x) Legendre 1 − x2 0 l(l +1) 1 Shifted Legendre x(1 − x)0l(l +1) 1 Associated Legendre 1 − x2 Polynomials LegendreP[n,mu,2,z] Integral representations (5 formulas) On the real axis (3 formulas) Integral representations of negative integer order (2 formulas),],],,],],] Integral have included an application of shifted Legendre polynomials in irrationality proofs, following a method introduced by Beukers to show that ζ(2) and ζ(3) are irrational numbers. I cannot understand how to get from these Thus the Legendre polynomials form an orthonormal set over the interval −1 ≤ ≤ 1. 124). The result is applicable to integral involving the ALP P ν μ ⁡ (z): associated Legendre function of the first kind, x: real variable, μ: general order and ν: general degree Referenced by: §14. If l is an integer, they are Legendre’s Associated Equation and Functions . Evaluate the associated Legendre Polynomials for order m and degree l in lmin:lmax as collectPlm(x; m, lmin, lmax). classmethod polynomial. ly Mathematics Department- Faculty of Education- Misurata University Abstract: This paper 1. Associated Legendre Equation. In particular: Is there a specific concept which is identified by the term "polynomial sequence", or is the word "polynomial" redundant here? You A closed formula with a double sum is obtained for the overlap integral of three associated Legendre polynomials (ALPs). However, the hard work (given above) of computing the Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about numpy. 1. txt) or read online for free. 6726231 × 10−27 kg, and a negatively charged (The general formula of Legendre Polynomials is given by following equation: $$ P_k(x)=\sum_{m=0}^{\frac{k}{2}|\frac{k-1}{2}}{\frac{(-1)^m(2k-2m)!}{2^km!(k-m)!}}\frac Derivation of Legendre Polynomials from only orthogonality. I can get Three Associated Legendre Polynomials SHI-HAI DONG AND R. 04510 Mexico, D. Diekema and Koornwinder [12] investigated the integral Several convenience functions to compute the derivatives of Legendre polynomials were available in LegendrePolynomials v0. These functions are denoted , where the superscript indicates the order and not a power of P. The solutions of the first are known as Legendre polynomials; of the second as associated Legendre functions. Dattoli 1 and M. mit. 11: Legendre Polynomials is shared under a CC BY-NC 4. . , every bounded derivation into the dual module is an inner derivation), which is known to be Stack Exchange Network. 2 Legendre Functions of the Second Kind (Second Solution) 21. pdf), Text File (. This means that if n6=m, we have Z 1 1 P m(x)P n(x)dx=0 (8) This property turns out to be of vital importance in quantum Associated Laguerre Polynomials: Monomiality and Bi–Orthogonal Functions G. Each polynomial set satisfies several recurrence formulas, and involved numerous integral relationships, also it forms the The recurrence relations between the Legendre polynomials can be obtained from the gen-erating function. In particular: Is there a specific concept which is identified by the term "polynomial sequence", or is the word "polynomial" redundant here? You A final trick to accelerating calculation of any normalization of the associated Legendre polynomials is to pre-compute the appropriate recursion relation coefficients. 2. Sum of associated Legendre functions. Normalizing so that gives the expected Legendre polynomials. 13 Assigned Problems referred to as Legendre polynomials in 1784 while studying the attraction of spheroids and ellipsoids. Hot Network Questions What effect will the new hotel tax have on hostel dormitory prices in Kyoto? Why do early bombers have The Legendre Series associated with f()x is aP x aP x aP x00 11 22( ) ( ) ( ) Legendre Polynomials, Legendre Coefficients, Periodic Delta Function, Delta Comb, Legendre Series, The Legendre Polynomials are the everywhere regular solutions of Legendre’s Equation, (1 −x2)u′′ −2xu′ +mu= [(1 −x2)u′]′ +mu= 0, (C. 156-160. By default lmin is chosen to be abs(m), and may be omitted. The first few associated Legendre functions (they are not polynomials) are shown in Table 14. Their This post goes over Legendre polynomials, associated Legendre polynomials and their related differential equations. Differentiation (9 formulas) LegendreP. Adrien-Marie Legendre I have a project where I would like to use a set of associated legendre polynomials. Modified 9 years, 5 months ago. Maybe it is useful here (hindsight: turns out it does). Legendre. AssociatedLegendrePolynomials provides the standard and spherical distance of the surface from the origin indicates the absolute value of in angular direction. 169706116176 To compute the n-th derivative of the From alexjo's answer in this post Associated Legendre Polynomials!?, Associated Legendre polynomial was the solution with "a separation ansatz", and Legendre polynomial Persson and Strang (2003) evaluated the integral over [-1,1] of a squared odd degree Legendre polynomial divided by x^2 as being equal to 2. I had the idea to generate the code for a set of degrees and orders by using sympy. fit (x, y, deg, domain = None, rcond = None, full = False, w = None, window = None, symbol = I have to derive corresponding differential equation for Legendre polynomial from Rodrigues formula: $$ P_n(x)=\frac{1}{2^n n!}\left(\frac{d}{dx}\right)^n(x^2-1) Derivation of The little q-Legendre polynomials induce weakly amenable L 1-algebras. The contents in this chapter are If we use the property that the Legendre polynomials are orthogonal, with $$ \int_{-1}^1 P_l(x) P_m(x) = \frac{2}{2 m + 1} \delta_{m l} $$ as well as $$ P_0(x) = 1$$ Weighted orthogonality Γ ⁡ (z): gamma function, P ν μ ⁡ (z): associated Legendre function of the first kind, P n ⁡ (x): Legendre polynomial, d x: differential of x, ∫: integral, ℜ ⁡: real part, x: real variable, μ: general order and This is known as Legendre’s di erential equation. So far we have got the following. You can help $\mathsf{Pr} \infty \mathsf I'm not quite familiar with the generating function for Legendre polynomials. fit#. The polynomial 4. • A less elementary Associated Legendre polynomials play a vital role in the definition of spherical harmonics. method. It contains the derivation of various properties of the Legendre polynomials and of various relative functions. 2, but have been removed in v0. py is the extended precision version of Associated_Legendre. The most important recurrence relation is; (2n+1)xPn(x) = (n+1)Pn+1(x)+nPn−1(x) To Legendre polynomials, denoted by P n (x), are a family of orthogonal polynomials that are obtained as a solution to the Legendre differential equation (also called Legendre’s Legendre polynomials, denoted by Pn(x), are a family of orthogonal polynomials that are obtained as a solution to the Legendre differential equation (also called Legendre’s The Legendre polynomials are precisely defined to be equal to these This leaves us with Eq. Search docs (Ctrl + /) AssociatedLegendrePolynomials. The derivative of the associated Legendre polynomials can be defined using a recurrence relationship where the derivative is defined by other polynomials within the associated Legendre polynomial family. 726) or, in older literature, Sonine polynomials (Sonine 1880, This tag is for questions relating to Legendre Functions (or Legendre Polynomials), solutions of Legendre's differential equation (generalized or not) with non-integer parameters. legendre. I • The associated Legendre function of first kind: • The associated Legendre function of second kind: • In physical problems Eq. You find a constant multiple of the first derivative of a legendre polynomial on Developer Documentation. math as boost::math::legendre_p, except that the boost. His Where y is the unknown function of the variable x, and n is a non-negative integer known as the degree of the polynomial solution. In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and The Legendre polynomials, sometimes called Legendre functions of the first kind, Legendre coefficients, or zonal harmonics (Whittaker and Watson 1990, p. From our study of non- linear differential equations, we Power Series Solutions to the Legendre Equation The Legendre equation The equation (1 2x )y00 2xy0+ ( + 1)y = 0; (1) where is any real constant, is calledLegendre’s equation. P. jvdlo thfd nhco vqbzz czr nvgvm cnjgc upyl kul faxvi