Mathematical Methods for Physics and Engineering - Matematica.NET

23.9 HINTS AND ANSWERS23.9 Hints and answers23.1 Define y(−x) =y(x) and use the cosine Fourier transform inversion theorem;y(x) =(2/π) 1/2 exp(−x 2 /2).23.3 f ′′ (x) − f(x) =expx; α =3/4, β =1/2, γ =1/4.23.5 (a) φ(x) =f(x) − (1+2n)F n x n − (1 − 2n)F −n x −n . (b) There are no solutions forλ =[1± (1 − 4n 2 ) −1/2 ] −1 unless F ±n =0orF n /F −n = ∓[(1 − 2n)/(1 + 2n)] 1/2 .23.7 (a) a (i)n= ∫ ba h n(x)ψ (i) (x) dx; (b) use (1/ √ π)cosnx and (1/ √ π)sinnx; M is diagonal;eigenvalues λ k = k/π with eigenfunctions ψ (k) (x) =(1/ √ π)coskx.23.9 d˜f/dω = −ω˜f, leading to ˜f(ω) =Ae −ω2 /2 . Rearrange the integral as a convolutionand deduce that ˜h(ω) = Be −3ω2 /2 ; h(t) = Ce −t2 /6 , where resubstitution andGaussian normalisation show that C = √ 2/(3π).23.11 p = k 0 H/(1 − 2πk 0 ), q = k 1 H c /(1 − πk 1 )andr = k 1 H s /(1 − πk 1 ),where H = ∫ 2πh(z) dz, H0 c = ∫ 2πh(z)coszdz,andH0 s = ∫ 2πh(z)sinzdz. Positive0values of k 1 (≈ π −1 ) are most likely to cause a conference breakdown.23.13 For eigenvalue 0 : f(x) =0for|x|