The Chemistry Maths Book, Second Edition

410 Chapter 14Partial differential equations 14.7 The vibrating string We consider an elastic string, such as a guitar string, o ...

14.7 The vibrating string 411 and the initial value problem (14.92) with initial conditions (14.88), wheref(x)andg(x)are given f ...

412 Chapter 14Partial differential equations in space of the first few of these normal modes are illustrated in Figure 14.7 (see ...

14.8 Exercises 413 The general solution of the wave equation that satisfies the boundary conditions is therefore a superposition ...

414 Chapter 14Partial differential equations Section 14.3 Find solutions of the following equations by the method of separation ...

14.8 Exercises 415 (i)Show that the equation is separable in spherical polar coordinates, with the same angular wave functions, ...

15 Orthogonal expansions. Fourier analysis 15.1 Concepts We saw in Section 7.6 that many functions can be expanded as power seri ...

15.2 Orthogonal expansions 417 and that, when the value of the variable is restricted to the interval− 11 ≤ 1 x 1 ≤ 1 + 1 ,the o ...

418 Chapter 15Orthogonal expansions. Fourier analysis and this can be used to obtain a general formula for the coefficientsc l i ...

15.2 Orthogonal expansions 419 Therefore and this is in agreement with the coefficient ofP 0 shown in (15.6). 0 Exercises 1, 2 T ...

420 Chapter 15Orthogonal expansions. Fourier analysis The norm of a function is sometimes interpreted as the ‘magnitude’ or ‘len ...

15.3 Two expansions in Legendre polynomials 421 (1) By everyfunction is meant not only continuous functions but the more general ...

422 Chapter 15Orthogonal expansions. Fourier analysis EXAMPLE 15.2Derive the expansion (15.22). We treat the function to be expa ...

15.3 Two expansions in Legendre polynomials 423 whereR q is the distance of point Pfrom the charge (if a chargeq′is placed at P, ...

424 Chapter 15Orthogonal expansions. Fourier analysis This analysis of the potential is important for the description of the ele ...

15.4 Fourier series 425 The potential at point P is then When Ris large enough (R 1 >> 1 r), the expansion of the potentia ...

426 Chapter 15Orthogonal expansions. Fourier analysis The (trigonometric) Fourier series is usually written in the form 2 (15.37 ...

15.4 Fourier series 427 Figure 15.4 Figure 15.5 Form 1 = 1 n 1 > 10 , 0 Exercise 7 Periodicity The trigonometric functions (1 ...

428 Chapter 15Orthogonal expansions. Fourier analysis EXAMPLE 15.6The Fourier series of the function (15.42) The graph of the (e ...

15.4 Fourier series 429 The coefficientsb n . By equation (15.39), We havecos 1 nπ 1 = 1 + 1 when nis an even integer, andcos 1 ...