The Chemistry Maths Book, Second Edition

370 Chapter 13Second-order differential equations. Some special functions For this to be equal to zero for all values of x(withi ...

13.3 The Frobenius method 371 Then and this is zero if the coefficient of each power of xis separately zero: a m 1 + 1 (m 1 + 1 ...

372 Chapter 13Second-order differential equations. Some special functions in whichb(x)andc(x)are polynomials or can be expanded ...

13.3 The Frobenius method 373 EXAMPLE 13.3Indicial equations (i) We haveb 0 1 = 1 − 122 ,c 0 1 = 1122 , and the indicial equatio ...

374 Chapter 13Second-order differential equations. Some special functions 3 Roots differ by an integer One solution has the form ...

13.4 The Legendre equation 375 This is zero if the coefficient of each power of xis separately zero. The coefficient of x 1 is 2 ...

376 Chapter 13Second-order differential equations. Some special functions Similarly for the odd values of m, A power-series solu ...

13.4 The Legendre equation 377 interest in the physical sciences. 3 By convention, the nonzero arbitrary constant in (13.16) is ...

378 Chapter 13Second-order differential equations. Some special functions The Legendre polynomials satisfy the recurrence relati ...

13.4 The Legendre equation 379 We have Then P 3 2 (x) 1 = 1 15(1 1 − 1 x 2 )xP 3 2 (cos 1 θ) 1 = 1151 sin 2 1 θ 1 cos 1 θ P 3 3 ...

380 Chapter 13Second-order differential equations. Some special functions (b) In this case, bothP 1 (x)andP 3 (x)are odd functio ...

13.5 The Hermite equation 381 The integrand is an even function of xin the interval− 11 ≤ 1 x 1 ≤ 11. Therefore (Section 5.3, eq ...

382 Chapter 13Second-order differential equations. Some special functions The Hermite polynomials satisfy the recurrence relatio ...

13.5 The Hermite equation 383 The expression in square brackets is the left side of the Hermite equation (13.30), so that partic ...

384 Chapter 13Second-order differential equations. Some special functions By dividing throughout by−A 2 α 22 mand noting thatα 1 ...

13.7 Bessel functions 385 The associated Laguerre polynomials arise in the solution of the radial part of the Schrödinger equati ...

386 Chapter 13Second-order differential equations. Some special functions The solutions of the indicial equation (13.8) are r 1 ...

13.7 Bessel functions 387 Both the expansions and the graphs show that the Bessel functions have properties similar to the trigo ...

388 Chapter 13Second-order differential equations. Some special functions so that Similarly, withn 1 = 1 − 122 in (13.56), 0 Exe ...

13.8 Exercises 389 The functionsj l (x)are called spherical Bessel functions of order l; the functions η l are the spherical Neu ...