Chapter 6. Legendre and Bessel Functions

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1 Chapter 6 Legendre and Bessel Functions Legendre's Equation Legendre's Equation (order n): legendre d K y''k y'c n n C y = : is an important ode in applied mathematics When n is a non-negative integer, this ode has a polynomial solution of degree n, usually denoted P n, called the n th order Legendre polynomial Maple denotes it as LegendreP n, dsolve legendre = y = _C LegendreP n, C _C LegendreQ n, After loading the orthopoly pacage: with orthopoly : we can enter P n, for the n th Legendre polynomial For eample, P 3, = 3 K 3 In addition to interesting symmetry properties: plot P n, $ n =, =K = K K K K The first 6 Legendre polynomials on the interval K, the Legendre polynomials have the following orthogonality property: K P m, P n, d = n C d mn, where d mn is the Kronecer delta, d mn = if m s n and d nn = The following matri confirms this property for the first 6 Legendre polynomials Matri 6, 6, m, n / K Legendre-Fourier Approimations P m, P n, d = 3 9 The orthogonality property of the family P n, n = allows us to approimate any piecewise smooth function f on the interval K % % with linear combinations of Legendre polynomials For eample, let f d piecewise!,k, : 3 page 3

2 plot f, =K = 3 K K Define an d n C K f P n, d : and form LF, d >n an P n, = : This is called the th Legendre-Fourier Approimation for the function f For eample, the = 6 approimation is a 6th degree polynomial LF 6, = C 39 6 C K 8 3 K C C plot f, LF 6,, =K = K K And here is the = approimation, a degree polynomial (too long to show here) plot f, LF,, =K = 8 3 K K The approimations get better and better as increases because the coefficients are eactly the ones that minimize the square of the area between the graph of the function f and the graph of the approimation LF,, on the interval K, Bessel's Equation Bessel's Equation (order n): bessel d y''c y'c K n y = : is another important ode in applied mathematics Bessel's equation has independent solutions J n, R and Y n, O Maple writes these as BesselJ n, and BesselY n, Both are defined by infinite series dsolve bessel = y = _C BesselJ n, C _C BesselY n, Dividing Bessel's equation by yields y''c y'c K n y =, which loos lie a mass-spring system with damping that decreases toward and a spring constant that increases toward We can anticipate oscillating solutions, and that's eactly what we get with both J n and Y n plot BesselJ n, $ n =, =, caption = "Si Bessel functions of the first ind" page 33

3 8 6 4 K K Si Bessel functions of the first ind The functions J n are called Bessel functions of the first ind The functions Y n are called Bessel functions of the second ind Their graphs are similar ecept for the fact that each Y n is unbounded at the origin rendering it less useful in applications plot BesselY n, $ n =, =,K 6, caption = "Si Bessel functions of the second ind" K K K K Si Bessel functions of the second ind Orthogonality Bessel functions of the first ind generate families of functions that are orthogonal with respect to on the interval, For eample, consider the first zeros of J displayed below plot BesselJ,, = 33, ticmars =, caption = "The first zeros of BesselJ(,)" 6 K The first zeros of Bessel(,) The values of the zeros can be obtained using a procedure named BesselJZeros See the net entry where we assign the name z to the sequence containing the first zeros of J z d evalf BesselJZeros, 4488, 8, 86393, 93444, 4939, 8639, 63663, 4343, , Here is the third one z 3 = 86393, chec it on the graph above ow these functions BesselJ, z, =, are orthogonal on, with respect to the weight function w = The first are plotted below plot BesselJ, z $ =, =, caption = "Five functions orthogonal on [,] with respect to w() = " K4 Five functions orthogonal on [,] with respect to w() = page 34 ()

4 The following matri calculation confirms the orthogonality for these functions M d Matri,, m, n / $BesselJ, z m BesselJ, z n d : evalf %, K K K K46 - K493 - K6-368 K K36 - K8 - K44-3 K K8-3 () Observe that all of the off-diagonal entries in this matri are digit approimations to Bessel-Fourier Approimations The orthogonality property of the family J z n, with respect to on,, allows us to n = approimate any piecewise smooth function f on the interval % % with linear combinations of Bessel functions of the first ind of order For eample, consider a modified form of the function we used to illustrate Legendre-Fourier approimations f d piecewise!,, K : plot f, = = Define an d M n, n $f $ BesselJ, z n d : and BF, d >n an BesselJ, z n : = This is called the th Bessel-Fourier Approimation for the function f The = approimation, which is a linear combination of Bessel functions is shown below, followed by the = approimation 6 plot f, BF,, = = plot f, BF,, = = Once more, the approimations get better and better as increases because the coefficients are eactly the ones that minimize the square of the weighted area between the graph of the function f and the graph of the approimation BF, page 3

5 Chapter 6 Procedures Math Entry (Eample) Typical Application Calculation Legendre Polynomials The importance of Legendre polynomials derives from two facts Legendre's equation arises often in Physics and Engineering applications These polynomials can be used to easily obtain approimations to piecewise smooth functions on any finite interval a, b One only has to translate a, b to K, via T = b K a $ K m where m = a C b / is the midpoint of a, b For eample, suppose we wish to approimate this function g d piecewise!,k, plot g, = K K : 3 4 Let T d K / : and define Q n, d P n, T : This yields Legendre polynomials translated to the interval, plot Q n, $n =, = K K 3 4 Legendre polynomials on, The matri calculation on the right indicates that, while orthogonality is preserved, Q n, d = n C Matri 4, 4, m, n / Q m, Q n, d 3 The n th diagonal entry is Let an d n C 9 n C g Q n, d : and LF, d n > an Q n, : We get this 6th = degree Legendre-Fourier approimation plot g, LF 6,, = K 3 4 Bessel Functions The importance of Bessel functions also derives from two facts Bessel's equation and odes that are closely related to it come up over and over again in Physics and Engineering models Bessel functions of the first ind can be used to obtain approimations to piecewise smooth functions on any finite interval of the form, b To approimate the function g with Bessel functions of the first ind of order use the family at the top of the panel on the right J z n n = Here z n is the nth positive zero of J plot BesselJ,, = K3 z d evalf BesselJZeros, 6 : The 3rd one: z 3 = The first 4 functions in the orthogonal family are defined below and plotted on the right F n, d BesselJ, z n : plot F n, $n = 4, = ow let M d an d K3 3 4 ' $ F n, d ' $ n = 6 : M n $g F n, d : BF, d > an F n, : and approimate n = plot g, BF 6,, = K 3 4 page 36