Series solutions about an ordinary point if z z0 is an ordinary point of eq. Associated legendre polynomials associated legendre function as noted by others, these functions are not generally polynomials. Now we look in more detail at legendres equation and the legendre polynomials. The series expansion for y 1 and y 2 may terminate in that case the corresponding solution has r 1, otherwise they have radius of convergence r 1. Legendre polynomials appear in an expansion of the electrostatic potential in inverse radial powers. The page probably would have been moved already but it cant be, because the target name already exists and redirects to associated legendre polynomials. I came across the legendre differential equation today and im curious about how to solve it. Solution to legendres differential equation understanding.
In a central potential, the separation of variable techniques leads to an angular equation that is just the legendre differential equation in the polar variable. In particular, it occurs when solving laplaces equation and related partial differential equations in spherical coordinates. In the book it says legendres equation may be solved with frobenius method and before that it says frobenius method can be applied if x0 is a regular singular point. The generating function of the legendre polynomials.
This equation arises in many problems in physics, specially in boundary value problems. The legendre ordinary differential equation is frequently encountered in physics and other technical fields. The legendre, laguerre, and hermite polynomials3 proof. Then the rst and second derivatives are given by y0 p 1 n0 a nnx n 1 and y00 p 1 n0 a. How to solve series solution of legendres differential equation by using. Details you may be offline or with limited connectivity. The solutions of this equation are called legendre functions of degree. Using the sturmliouville theory we will be able to show that polynomial.
What is the physical meaningconcept behind legendre. Some useful properties of legendre polynomials and its. We will discover that the solutions to these differential equations are a set of functions known as the legendre polynomials. Also, these are li, since their wronskian is nonzero at x 0. In the book it says legendre s equation may be solved with frobenius method and before that it says frobenius method can be applied if x0 is a regular singular point. Associated legendre polynomials and spherical harmonics computation for chemistry applications taweetham limpanuparb, josh milthorpey october 8, 2014 abstract associated legendre polynomials and spherical harmonics are central to calculations in many elds of science and mathematics not only chemistry but computer. The legendre, laguerre, and hermite polynomials thomas coverson, savarnik dixit, alysha harbour, and tyler otto abstract.
It arises in numerous problems, particularly in boundary value problems for spheres take a quick look at example 1 in sec. Adrienmarie legendre september 18, 1752 january 10, 1833 began using, what are now referred to as legendre polynomials in 1784 while studying the attraction of spheroids and ellipsoids. Legendres equation and legendre functions the second order di. We introduce legendre polyno mials here by way of the electrostatic potential of a point charge, which acts as the generating function. Rodrigues formula the nth legendre polynomial pn is given by the following 4 pnx 1 2n n. We begin by transforming it to a somewhat simpler form by a change of independent variable, namely 5 h cos f. The legendre polynomials are equivalently given by the formula p nx 1 2nn. Show the details of your work and include your derivation of the recurrence formula for the coefficients in the series. Thats why i thought x0 is a regular singular point for legendres equation.
Typically covered in a first course on ordinary differential equations, this problem finds applications in the solution of the schrodinger equation for a oneelectron atom i. In fact, these are the same polynomial that encountered earlier in connection with the gramschmidt process. To obtain we used the orthogonality relation of legendre polynomials defined as 3. Appendix a legendre polynomials, associated legendre functions and spherical harmonics ai. Note that if we divide the equation by 1 2x, then we obtain the standard form as in theorem 1. We have taken px p 1 k0 a kx k and derived a recursion relation on the. In fact, these are the same polynomial that encountered earlier in connection with the gram. Legendre polynomials the legendre polynomials form an orthogonal set of polynomials.
Associated legendre polynomials and spherical harmonics are central to calculations in many elds of science and mathematics not only chemistry but computer graphics, magnetic, seismology and geodesy. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to. It is worth reemphasizing the dimensional consistency of this identity. Rodrigues formula the legendre polynomials can be expressed in a more compact form. These solutions can be used to solve axisymmetric problems inside a sphere, exterior to a sphere, or in the region between concentric spheres. Madas summary on legendre functionspolynomials legendres differential equation 2 2 2 1 2 1 0 d y dy x x n n y dx dx. Applications of the generating function of the legendre polynomials using eq. You will see below why orthogonal polynomials make particularly good choices for approximation. Legendre differential equation by power series, using the legendre generating function, using the rodriguez formula for legendre polynomials, or by a contour integral. It includes their derivation, and the topics of orthogonality, normalization, and recursion. In this section, we are going to write mfiles to generate the legendre polynomials and we are going to confirm that they form an orthogonal set in.
Now we look in more detail at legendre s equation and the legendre polynomials. The resulting solution of legendres differential equation 1 is called the. Thus the legendre polynomials or more generally the spherical harmonics when one includes the azimuthal dependence will always occurs as a solution to the angular part irrespective of the details of the central potential, and thus. The r equation is equidimensional and thus has solutions, easily found, which are powers of r.
General formula we start with a solution to the laplace equation in 3. Legendre polynomials and applications legendre equation. There is no inarguable physical concept behind the legendre polynomials, they are just mathematical objects which form a complete basis between 1 and 1. The legendre differential equation has regular singular points at, 1, and. Legendre polynomials are solutions of an ordinary differential equation ode which is. We include examples of each type later in this notebook. May 19, 2016 legendre equation and its solution math. Legendres equation comes from solving laplaces equation. Equating the coefficient of xn in the resulting equation, we get the recursive relation. The legendre polynomials and their properties the gravitational potential. In physical science and mathematics, legendre polynomials named after adrienmarie legendre, who discovered them in 1782 are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applications. Associated legendre polynomials and spherical harmonics. Solution of the legendres ode using frobenius method.
Legendre polynomials vladimir zakharov department of. Examples of dependent variables are the electromag. In our course, the legendre polynomials arose from separation of variables for the laplace equation in spherical coordinates, so we begin there. There is no more knowledge in the book about irregular points or other singular points. Laplaces equation in spherical coordinates and legendres equation i legendres equation arises when one tries to solve laplaces equation in spherical coordinates, much the same way in which bessels equation arises when laplaces equation is solved using cylindrical coordinates. Legendres equation in this hand out, legendres question will be written as a papperitz equation, the solutions of which can be written down in terms of hypergeometric functions with no further calculation. Associated legendre polynomials play a vital role in the definition of spherical harmonics. The first result in the search for separated solutions of equation 2, which ultimately leads to the formulas 3, is the pair of differential equations 4 for the rdependent part fr, and the fdependent part pf of the separated solutions. Equation 14 is actually a function of either w or y but not both, because one variable implicitly depends on the other via a legendre transform. In fact, this equation is a smaller problem that results from using separation of variables to solve laplace. Since legendres differential equation is a second order ordinary differential equation, two sets of. You first encountered legendres differential equation in examples 1 and 3 of unit 3 in.
Legendre polynomials let x be a real variable such that 1 x 1. Legendre equation mathematical analysis mathematical objects. The polynomials are rarely used in physics problems so we are not going to deal with them further. Introduction to legendre polynomials we began recently our study of the legendre differential equation. The above form is a special case of the associated legendre differential equation with. Legendre functions legendre functions or polynomials, are a solution of legendre differential equation about the origin x 0.
In particular we stress the utility of a generating function. The legendre transform is commonly used in upper division and graduate physics courses, especially in classical mechanics,1 statistical mechanics, and thermodynamics. All the above equations are examples of differential equation. Legendre polynomials vladimir zakharov department of mathematics university of arizona lectures in mathematics the following lecture introduces the legendre polynomials.
The legendre, laguerre, and hermite equations are all homogeneous second order sturmliouville equations. Dec 10, 2015 this feature is not available right now. Legendre polynomials 2 the legendre equation is where n is a non negative integer. Thats why i thought x0 is a regular singular point for legendre s equation. This equation arises in many problems in physics, specially in boundary value problems in spheres. The equation involves a parameter n, whose value depends on the physical or engineering problem. Pdf in this paper, legendre polynomials was used to get numerical solution of highorder linear complex differential equations in a circular domain find. Solution of bessels differential equation leading to bessel. Legendres equation including legendres functions and legendres polynomials. Legendre polynomials are usually derived from differential equations of the following form.
These polynomials are defined on 1, 1, and have a number of useful and interesting mathematical properties. Problems on the legendre equation and legendre polynomials. In this paper, we derived some useful properties of legendre polynomials and we applied one of them to the neutron transport equation. Laplaces equation in spherical coordinates and legendres.
616 809 591 367 546 1315 1390 640 818 853 637 1181 659 273 650 1144 1479 1456 923 32 738 336 1532 719 132 702 646 1271 772 268 1472 728 1013 862 1181 1401 158 1031 694 1410 1074 7