Mathematical Methods for Physics and Engineering : A Comprehensive Guide

12.10 HINTS AND ANSWERS

(a) (b) (c) (d)

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Figure 12.6 Continuations of exp(−x^2 )in0≤x≤1 to give: (a) cosine terms

only; (b) sine terms only; (c) period 1; (d) period 2.

Sketch the graph of the functionf(x), where

f(x)=

{

−x(π+x)for−π≤x< 0 ,

x(x−π)for0≤x<π.

Iff(x) is to be approximated by the first three terms of a Fourier sine series, what

values should the coefficients have so as to minimiseE 3? What is the resulting

value ofE 3?

12.10 Hints and answers

12.1 Note that the only integral of a sinusoid around a complete cycle of lengthL
that is not zero is the integral of cos(2πnx/L)whenn=0.
12.3 Only (c). In terms of the Dirichlet conditions (section 12.1), the others fail as
follows: (a) (i); (b) (ii); (d) (ii); (e) (iii).
12.5 f(x)=2

∑∞

1 (−1)

n+1n− (^1) sinnx;setx=π/2.
12.7 (i) Series (a) from exercise 12.6 does not converge and cannot represent the
functiony(x)=−1. Series (b) reproduces the square-wave function of equation
(12.8).
(ii) Series (a) gives the series fory(x)=−x−^12 x^2 −^12 in the range− 1 ≤x≤ 0
and fory(x)=x−^12 x^2 −^12 in the range 0≤x≤1. Series (b) gives the series for
y(x)=x+^12 x^2 +^12 in the range− 1 ≤x≤0andfory(x)=x−^12 x^2 +^12 in the
range 0≤x≤1.
12.9 f(x)=(sinh 1)

{

1+2

∑∞

1 (−1)

n(1 +n (^2) π (^2) )− (^1) [cos(nπx)−nπsin(nπx)]}.
The series will converge to the same value as it does atx=0,i.e.f(0) = 1.
12.11 See figure 12.6. (c) (i) (1 +e−^1 )/2, (ii) (1 +e−^1 )/2; (d) (i) (1 +e−^4 )/2, (ii)e−^1.
12.13 (d) (i) The periods are both 2L; (ii)y 0 /2.
12.15 So=π^2 /8. IfSe=

∑

(2m)−^2 thenSe=^14 (Se+So), yieldingSo−Se=π^2 /12 and
Se+So=π^2 /6.
(a) (π/4)(π/ 2 −|θ|); (b) (πθ/4)(π/ 2 −|θ|/2) from integrating (a). (c) Even function;
average valueL^2 /3;y(0) = 0;y(L)=L^2 ; probablyy(x)=x^2. Compare with the
worked example in section 12.5.
12.17 coshx=(sinh1)[1+2

∑∞

n=1(−1)

n(cosnπx)/(n (^2) π (^2) + 1)] and after integrating twice
this form must be recovered. Usex^2 =^13 +4

∑

(−1)n(cosnπx)/(n^2 π^2 )] to eliminate
the quadratic term arising from the constants of integration; there is no linear
term.
12.19 C±(2m+1)=∓ 2 i/[(2m+1)π];

∑

|Cn|^2 =(4/π^2 )× 2 ×(π^2 /8); the valuesn=±1,

±3 contribute>90% of the total.