COSMOLOGY AND GEGENBAUER POLYNOMIALS 73 w and X are two separation constants. For wave problems w corresponds to the angular frequency. Two linearly independent solutions of Equation (5.7) can be immediately written as T(t) = eiwt and e-iwt, (5.10) while the Second Equation (5.8) is nothing but the differential equation [Eq. (2.182)] that the spherical harmonics satisfy with X and m given as X=-1(1+1), I=O,1,2 , , and m=O,fl, , fl. (5.11) Before we try a series solution in Equation (5.9) we make the substitution X(X) = Co sin' ~~(cosx), (5.12) where z = cosx, x E [-I, 11 (5.13) and obtain the following differential equation for C(x): Substitution (5.12) is needed to ensure a two-term recursion relation with the Frobenius method. This equation has two regular singular points at the end points 2 = 3~1. We now try a series solution of the form (5.15) to get uQcr(a - l)z"-2 + Ulcu(Q + 1)Zf-l 03 f -y{uk+2(k + Q + 2)(k + Q + 1) k=O -Uk[(k f @)(k + (Y - 1) + (21 4- 3)(k fa) - A]}&" = 0. (5.16) In this equation A is defined as (5.17) 1 A = -1(1+ 2) + (w2 - -)@. % 74 GEGENBAUER AND CHEBYSHEV POLYNOMIALS Equation (5.16) cannot be satisfied for all x unless the coefficients of all the powers of x are zero, that is (5.18) (IOQ((Y - 1) = 0, a0 # 0, ala(a + 1) = 0, (5.19) a)(k + (Y - 1) + (21 + 3)(k + a) - A ak+2 = ak [@+ (k+(Y+2)(k+a+l) k=O,1,2 , ‘. (5.21) The indicia1 Equation (5.18) has two roots, 0 and 1. Starting with the smaller root, (Y = 0, we obtain the general solution as (5.22) L where a0 and a1 are two integration constants and the recursion relation for the coefficients is given as , k=0,1,2 ,.” k(k - 1) + (2l+ 3)k - A ak+2 = ak From the limit (5.23) (5.24) 1 we see that both of these series diverge at the end points, x = 3z1, as To avoid this divergence we terminate the series by restricting W& to integer values given by 1 - 22’ (5.25) Polynomial solutions obtained in this way can be expressed in terms of the Gegenbauer polynomials. Note that these frequenck mean that one can only fit integer multiples of full wavelengths around the circumference, 2~&, of the universe, that is, (IfN)XN=2T&, N=0,1,2 , . (5.26) Using the relation WN = we easily obtain the frequencies of Equation (5.25). GEGENBAUER EQUATION AND 1TS SOLUTlONS 75 5.2 GEGENBAUER EQUATION AND ITS SOLUTIONS The Gegenbauer equation is in general written as d2CX (x) dCx (x) dx2 dx (1 - x2)L - (2X + 1)xZ + n(n + 2X)C;(z) = 0. (5.27) For X = 1/2, this equation reduces to the Legendre equation. For the integer values of n, its solutions reduce to the Gegenbauer or the Legendre polyno- mials as: (5.28) 5.2.1 The orthogonality relation of the Gegenbauer polynomials is given as Orthogonality and the Generating Function The generating function of the Gegenbauer polynomials is defined as M = xC;(~)t", It1 < 1, 1x1 5 1, X > -1/2. (5.30) 1 (1 - 2xt + t2)X n=O We can now write the solution of Equation (5.14) in terms of the Gegenbauer polynomials as C:!,(x), and the complete solution for the wave Equation (5.3) becomes (5.31) +(t,x,O,4) = (cleiWNt + CZe-iuNt )(sin' x)C~~,(cosx)~m(8, 4) 5.3 CHEBYSHEV EQUATION AND POLYNOMIALS 5.3.1 Polynomials defined as Chebyshev Polynomials of the First Kind Tn(cosx) = cos(nx), n = 0,1,2 (5.32) are called the Chebyshev polynomials of first kind, and they satisfy the Cheby- shev equation where we have defined x = cosx. (5.33) (5.34) 76 GEGENBAUER AND CHEBYSHEV POLYNOMIALS 5.3.2 The Chebyshev equation after (1 + 1)-fold differentiation yields Relation of Chebyshev and Gegenbauer Polynomials d1+3 (cm nx) d'+2 (cos nx) (l-x2) - (21 + 3)" dx1+3 dx1+2 (5.35) d'+l (cos nx) + [-12 - 21 - 1 + n2] = 0, dxl+l where n = 1,2, . We now rearrange this as dL d dx { (1 - x2)s - [2(1+ 1) + I] x- + (n - 1 - 1) [(n - 1 - 1) + 2(1+ l)] (5.36) L J and compare with Equation (5.27) to obtain the following relation between the Gegenbauer and the Chebyshev polynomials of the first kind d'+lTn (X) , n=1,2 , - - dxl+I (5.37) (5.38) 5.3.3 Chebyshev polynomials of the second kind are defined as Chebyshev Polynomials of the Second Kind Un(x) = sin(nX), n = O,1,2 , (5.39) where x = cosx. Chebyshev polynomials of the first and second kinds are linearly independent, and they both satisfy the Chebyshev Equation (5.33). In terms of x the Chebyshev polynomials are written as and CHEBYSHEV EQUATION AND POLYNOMIALS 77 For some n values Chebyshev polynomials are given as Chebyshev Polynomials of the First Kind TI(.) = s 2 T2(z) = 22 - 1 T3(2) = 4z3 - 32 T~(Z) = 8x4 - 8s2 + 1 Chebyshev Polynomials of the Second Kind u, = 0 U2(z) = Jrn(2x) U3(2) = J-(4Z2 - I) U4(2) = Jrn(8s3 - 4s) (5.42) (5.43) U5(z) = Jm(16z4 - 12z2 + 1) 78 GEGENBAUER AND CHEBYSHEV POLYNOMIALS 5.3.4 Orthogonality and the Generating Function of Chebyshev Polynomials The generating functions of the Chebyshev polynomials are given as and (5.45) Their orthogonality relations are and 5.3.5 Another Definition for the Chebyshev Polynomials of the Second Kind Sometimes the polynomials defined as - U,(x) = 1 - U3(Z) = 823 - 452 - U4(x) = 16x4 - 12x2 + 1 (5.48) are also referred to as the Chebyshev polynomials of the second kind. They are related to Un(z) by = un+l(x), n = 0,1,2, . (5.49) PROBLEMS 79 - U,(x) satisfy the differential equation - dun(z) + n(n + 2)v,(x) = 0, (5.50) d2cn (x) (1 - 2)- - 3z- dx2 dx and their orthogonality relation is given as (5.51) Note that even though gm(x) are polynomials, Um(z) are not. The generating function for flm(x) is given as Tn(x) and un(x) satisfy the recursion relations (1 -2)TA(x) = -nxT,(z) +nTn-l(x) (1 - x2)u;(z) = -nxcn(x) + (n + 1)Vn-l(S). (5.53) and (5.54) Special Values of the Chebyshev Polynomials Problems 5.1 Observe that the equation 80 GEGENBAUER AND CHEBYSHEV POLYNOMIALS gives a three-term recursion relation and then drive the transformation which gives a differential equation for C(c0sx) with a two-term recursion relation. 5.2 Using the line element ds2 = c2dt2 - &(t)2[dX2 + sin2 xdB2 + sin2 Xsin2 13d4~], find the spatial volume of a closed universe. What is the circumference? 5.3 Show that the solutions of (1-z2) (21+3)xF d2C(x) [ 1 dC(x) + -1(1+ 2) + (u; - - dx2 R?i can be expressed in terms of Gegenbauer polynomials as c? 1 (x) , where 5.4 Show the orthogonality relation of the Gegenbauer polynomials: 5.5 Show that the generating function 1 = CC2(X)t", n=O (1 - 2xt + t2)X can be used to define the Gegenbauer polynomials. 5.6 equation Using the Frobenius method, find a series solution to the Chebyshev d2W dY(X) 2 ( 1 - x2) - - z- + n y (x) = 0, z E [- 1, I] . dx2 dx For finite solutions in the entire interval [-1,1] do you have to restrict n to integer values? PROBLEMS 81 5.7 Show the following special values: and 5.8 Show the relations 5.9 Using the generating function show that 5.10 Show that Tn(z) and Un(z) satisfy the recursion relations (1 -z2)T;(2) = -nzTn(z) +nTn_l(z) and (1 - 22)u;(z) = -nzU,(z) + nUn-l(.). 5.11 Using the generating function 00 1 = EC2(z)t", 111 < 1, 1x1 5 1, x > -1/2, n=O (1 - 22t + t2)X 82 GEGENBAUER AND CHEBYSHEV POLYNOMIALS show 5.12 Let x = cos x and find a series expansion of dl+’ (cos nx) d(cos x)‘+’ C(z) = in terms of z. 5.13 Using show that for X = 1/2 Gegenbauer polynomials reduce to the Legendre poly- nomials, that is CA’2(2) = Pn(z). 5.14 Prove the recursion relations Tn+l(~)-2~Tn(z) +Tn-~(z) =O and Un+1 (z) - 2zUn(z) + un- 1 (z) = 0. 5.15 Show the relations Chebyshev polynomials Tn(z) and Un(z) can be related to each other. (1 - z”)’/2Tn(z) = Un+l(z) - zUn(z) (1 - z2)”2Un(z) = zTn(z) - T,+l(S). and 5.16 Obtain the Chebyshev expansion 1 03 - 1)-1T2s(~) . [...]... 2(1- 21) X x >> 1, (6 .46 ) l)(2l- 3) 5 3 1 6 3 OTHER DEFINITIONS OF THE BESSEL FUNCTIONS 6 3 1 Generating Function Base1 function Jn(x) can be defined by a generating function T(x,t) as (6 .47 ) n= 03 90 BESSEL FUNCTIONS 6.3.2 Integral Definitions Bessel function J,(x) also has the following integral definitions: (6 .48 ) and Jn(IL.1 = 6 .4 (1 - t 2 ) n - 6 cosztdt, 1 ( n > ) 2 (6 .49 ) RECURSION RELATIONS... cooling of an infinitely long cylinder heated to an initial temperature f ( p ) Solve the heat transfer equation with the boundary condition and the initial condition T ( P , 0 ) = f(P) (finite) T ( p , t ) is the temperature distribution in the cylinder and the physical parameters of the problem are defined as Ic - thermal conductivity c - heat capacity po - density X - emissivity and h = X/k Hint:... Their z -+ 0 and z -+ 00 (6.36) limits are given as (real m 2 0) lim Im(x)-+ x-0 Xm 2”r(m (6.37) + 1)’ and (6.39) (6 .40 ) 6.2.3 Spherical Bessel Functions j , (z), nl(z), and (Z) Spherical Bessel functions jl(x),ni(x) ,and h1(”2)(x) defined as are (6 .41 ) 89 OTHER DEFINITIONS OF THE BESSEL FUNCTIONS Bessel functions with half integer indices, J1++(x )and N,++(x),satisfy the differential equation (6 .42 ) while... Bernoulli in 1732, however, he did not recognize the general nature of these functions As we shall see, this equation is a special case of Bessel’s equation 6 1 BESSEL’S EQUATION If we write the Laplace equation in cylindrical coordinates as 829 Id9 -+ + +-l a 2 9 ap2 p dp p2 a42 829 dz2 =o (6.13) and try a separable solution of the form W P , 4 , 4 = %)@ (4) Z(z), (6. 14) we obtain three ordinary differential... RELATIONS OF THE BESSEL FUNCTIONS Using the series definitions of the Bessel functions we can obtain the following recursion relations and First by adding and then by subtracting these equations we also obtain the relations Jm-l(X) m = -Jm(IL.) X + J:,(X) (6.52) and (6.53) Other Bessel functions, N,, Hi1’, and t ions 6.5 Hi2), satisfy the same recursion rela- ORTHOGONALITY AND THE ROOTS OF THE BESSEL FUNCTIONS... Verify the following Wronskians: n 98 BESSEL FUNCTIONS 6.6 Find the constant C in the Wronskian 6.7 Show that the stationary distribution of temperature, T ( p ,z ) , in a cylinder of length 2 and radius a with one end held a t temperature TOwhile the rest of the cylinder is held a t zero is given as Hint: Use cylindrical coordinates and solve the Laplace equation, a‘”(p, 2) = 0, by the method of separation... the noninteger values of m For the integer values of m the two solutions are related by L m ( z = (-l)mJrn(z) ) (6. 24) When m takes integer values, the second and linearly independent solution can be taken as (6.25) SOLUTIONS O f BESSEL’S EQUATION 87 which is called the Neumann function or the Bessel function of the second kind Note that N,(z) and J m ( z ) are linearly independent even for the integer... emissivity and h = X/k Hint: Use the method of separation of variables and show that the solution can be expressed as then find C, so that the initial condition T(p,O) = f ( p ) is satisfied Where , does z come from? 7 HYPERGEOMETRIC FUNCTIONS The majority of the second-order linear ordinary differential equations of science and engineering can be conveniently expressed in terms of the three parameters... among them like; (7.26) The basic integral representation of hypergeometric functions is: This integral, which can be proven by expanding (1- tz)- in binomial series and integrating term by term, transforms into an integral of the same type by Euler’s hypergeometric transformations: t t t t + + t + t, 1-t, t/(l z+tz), ( I - t ) / (I - tz) (7.28) Applications of the 4 Euler transformations t o the 6... (7. 34) Similarly, we can write the associated Legendre polynomials as P (x)= , " ( n+ m)! (1 - x y 2 ( n - m ) ! 2mm! m-n,m+n+ l , m + l ; - I-,) 2 (7.35) -n,n+2X,X+-, 1.1-x 2 2 (7.36) and the Gegenbauer polynomials as c; (x)= The main reason for our interest in hypergeometric functions is that so many of the second-order linear ordinary differential equations encountered in physics and engineering . equation in cylindrical coordinates as 829 Id9 la29 829 - + + +- =o ap2 p dp p2 a42 dz2 and try a separable solution of the form (6.13) WP, 4, 4 = %)@ (4) Z(z), (6. 14) we obtain. = J-(4Z2 - I) U4(2) = Jrn(8s3 - 4s) (5 .42 ) (5 .43 ) U5(z) = Jm(16z4 - 12z2 + 1) 78 GEGENBAUER AND CHEBYSHEV POLYNOMIALS 5.3 .4 Orthogonality and the Generating Function of Chebyshev. FUNCTIONS Using the series definitions of the Bessel functions we can obtain the following recursion relations and First by adding and then by subtracting these equations we also obtain the