The Chemistry Maths Book, Second Edition

430 Chapter 15Orthogonal expansions. Fourier analysis This example shows how an arbitrary wave is built up by interference of ha ...

15.4 Fourier series 431 The following example demonstrates the Fourier series for an arbitrary period, and it also prepares the ...

432 Chapter 15Orthogonal expansions. Fourier analysis and, after integration by parts, (15.54) Therefore (15.55) 0 Exercise 12, ...

15.6 Fourier transforms 433 and the coefficientsA n in (15.56) are identical to the Fourier coefficientsb n given by (15.54). In ...

434 Chapter 15Orthogonal expansions. Fourier analysis The infinite interval The Fourier series (15.48) for arbitrary interval 2 ...

15.6 Fourier transforms 435 so that, by the discussion in Section 5.4 of the Riemann integral as the limit of a sum, Equation (1 ...

436 Chapter 15Orthogonal expansions. Fourier analysis The Fourier-series representation of the function in the finite interval i ...

15.6 Fourier transforms 437 We define the function (15.78) It is shown in Example 15.9 that equation (15.69) can then be written ...

438 Chapter 15Orthogonal expansions. Fourier analysis Fourier transform pairs Equations (15.79) and (15.80) are conventionally w ...

15.6 Fourier transforms 439 EXAMPLE 15.10Find the Fourier transform of the exponential function, Figure 15.14(c), f(x) 1 = 1 e − ...

440 Chapter 15Orthogonal expansions. Fourier analysis of the results of diffraction experiments; the observed diffraction patter ...

15.7 Exercises 441 The real part of this is (see Example 15.10) (15.86) If we replace(x,y,a,b)by(t, ω, 1 2 T, ω 0 )in (15.86), w ...

442 Chapter 15Orthogonal expansions. Fourier analysis 6.For the square system of five charges in Figure 15.18, show that the lea ...

15.7 Exercises 443 12.A function with period 2 lis defined by (i)Draw the graph of the function in the interval−l 1 ≤ 1 x 1 ≤ 1 ...

16 Vectors 16.1 Concepts Physical quantities such as mass, temperature, and distance have values that are specified by single re ...

16.2 Vector algebra 445 Figures 16.2 (a) and (b) illustrate several types of vector associated with the motion of a body in spac ...

446 Chapter 16Vectors In this case aand bdefine the pairs of parallel sides of a parallelogram, and vector addition is said to o ...

16.2 Vector algebra 447 EXAMPLE 16.1Show that the diagonals of a parallelogram bisect each other. In Figure 16.6, C is the midpo ...

448 Chapter 16Vectors 16.3 Components of vectors The component of a vector in a given direction is the length of its projection ...

16.3 Components of vectors 449 The rules of vector algebra can then be formulated in terms of vector components, and it is in th ...