Mathematical Methods for Physics and Engineering - Matematica.NET

FOURIER SERIES12.21 Find the complex Fourier series for the periodic function of period 2π defined inthe range −π ≤ x ≤ π by y(x) =coshx. By setting x = 0 prove that∞∑ (−1) nn 2 +1 = 1 ( π)2 sinh π − 1 .n=112.22 The repeating output from an electronic oscillator takes the form of a sine wavef(t) =sint for 0 ≤ t ≤ π/2; it then drops instantaneously to zero and startsagain. The output is to be represented by a complex Fourier series of the form∞∑c n e 4nti .n=−∞Sketch the function and find an expression for c n .Verifythatc −n = c ∗ n.Demonstratethat setting t =0andt = π/2 produces differing values for the sum∞∑ 116n 2 − 1 .n=1Determine the correct value and check it using the result of exercise 12.20.12.23 Apply Parseval’s theorem to the series found in the previous exercise and soderive a value for the sum of the series17(15) + 652 (63) + 1452 (143) + ···+ 16n2 +12 (16n 2 − 1) + ··· .212.24 A string, anchored at x = ±L/2, has a fundamental vibration frequency of 2L/c,where c is the speed of transverse waves on the string. It is pulled aside at itscentre point by a distance y 0 and released at time t = 0. Its subsequent motioncan be described by the series∞∑y(x, t) = a n cos nπx nπctcosL L .n=1Find a general expression for a n and show that only the odd harmonics of thefundamental frequency are present in the sound generated by the released string.By applying Parseval’s theorem, find the sum S of the series ∑ ∞0 (2m +1)−4 .12.25 Show that Parseval’s theorem for two real functions whose Fourier expansionshave cosine and sine coefficients a n , b n and α n , β n takes the form∫1 Lf(x)g ∗ (x) dx = 1 L 04 a 0α 0 + 1 ∞∑(a n α n + b n β n ).2n=1(a) Demonstrate that for g(x) =sinmx or cos mx this reduces to the definitionof the Fourier coefficients.(b) Explicitly verify the above result for the case in which f(x) =x and g(x) isthe square-wave function, both in the interval −1 ≤ x ≤ 1.12.26 An odd function f(x) ofperiod2π is to be approximated by a Fourier sine serieshaving only m terms. The error in this approximation is measured by the squaredeviation∫ [2 πm∑E m = f(x) − b n sin nx]dx.−πn=1By differentiating E m with respect to the coefficients b n , find the values of b n thatminimise E m .430