Mathematical Methods for Physics and Engineering : A Comprehensive Guide

11.10 EXERCISES 11.12 Show that the expression below is equal to the solid angle subtended by a rectangular aperture, of sides 2 ...

LINE, SURFACE AND VOLUME INTEGRALS 11.19 Evaluate the surface integral ∫ r·dS,whereris the position vector, over that part of th ...

11.10 EXERCISES 11.24 Prove equation (11.22) and, by takingb=zx^2 i+zy^2 j+(x^2 −y^2 )k, show that the two integrals I= ∫ x^2 dV ...

LINE, SURFACE AND VOLUME INTEGRALS 11.11 Hints and answers 11.1 Show that∇×F= 0. The potentialφF(r)=x^2 z+y^2 z^2 −z. 11.3 (a)c^ ...

12 Fourier series We have already discussed, in chapter 4, how complicated functions may be expressed as power series. However, ...

FOURIER SERIES L L f(x) x Figure 12.1 An example of a function that may be represented as a Fourier series without modification. ...

12.2 THE FOURIER COEFFICIENTS we can write any function as the sum of a sine series and a cosine series. All the terms of a Four ...

FOURIER SERIES apply forr= 0 as well asr>0. The relations (12.5) and (12.6) may be derived as follows. Suppose the Fourier se ...

12.3 SYMMETRY CONSIDERATIONS 1 − 1 −T 2 0 T 2 t f(t) Figure 12.2 A square-wave function. following section). To evaluate the coe ...

FOURIER SERIES are not used as often as those above and the remainder of this section can be omitted on a first reading without ...

12.4 DISCONTINUOUS FUNCTIONS (a) (b) (c) (d) − 1 − 1 − 1 − 1 1 1 1 1 −T 2 −T 2 −T 2 −T 2 T 2 T 2 T 2 T 2 δ Figure 12.3 The conve ...

FOURIER SERIES (a) (b) (c) (d) 0 0 0 0 L L L L 2 L 2 L 2 L Figure 12.4 Possible periodic extensions of a function. 12.5 Non-peri ...

12.5 NON-PERIODIC FUNCTIONS − 220 L x f(x)=x^2 Figure 12.5 f(x)=x^2 ,0<x≤2, with the range extended to give periodicity. the ...

FOURIER SERIES converge to the correct values off(x)=±4atx=±2; it converges, instead, to zero, the average of the values at the ...

12.7 COMPLEX FOURIER SERIES where the Fourier coefficients are given by cr= 1 L ∫x 0 +L x 0 f(x)exp ( − 2 πirx L ) dx. (12.10) T ...

FOURIER SERIES 12.8 Parseval’s theorem Parseval’s theoremgives a useful way of relating the Fourier coefficients to the function ...

12.9 EXERCISES the sine and cosine form of the Fourier series, but the algebra is slightly more complicated. Parseval’s theorem ...

FOURIER SERIES be better for numerical evaluation? Relate your answer to the relevant periodic continuations. 12.7 For the conti ...

12.9 EXERCISES Deduce the value of the sumSof the series 1 − 1 33 + 1 53 − 1 73 +···. 12.15 Using the result of exercise 12.14, ...

FOURIER SERIES 12.21 Find the complex Fourier series for the periodic function of period 2πdefined in the range−π≤x≤πbyy(x)=cosh ...