Mathematical Methods for Physics and Engineering - Matematica.NET

23.8 EXERCISES(b) Obtain the eigenvalues and eigenfunctions over the interval [0, 2π] if∞∑ 1K(x, y) = cos nx cos ny.nn=123.8 By taking its Laplace transform, and that of x n e −ax , obtain the explicit solutionof∫ x]f(x) =e[x −x + (x − u)e u f(u) du .Verify your answer by substitution.23.9 For f(t) =exp(−t 2 /2), use the relationships of the Fourier transforms of f ′ (t) andtf(t) tothatoff(t) itself to find a simple differential equation satisfied by ˜f(ω),the Fourier transform of f(t), and hence determine ˜f(ω) to within a constant.Use this result to solve the integral equationfor h(t).23.10 Show that the equation∫ ∞−∞f(x) =x −1/3 + λ0e −t(t−2x)/2 h(t) dt = e 3x2 /8∫ ∞0f(y)exp(−xy) dyhas a solution of the form Ax α + Bx β . Determine the values of α and β, andshowthat those of A and B are11 − λ 2 Γ( 1 3 )Γ( 2 3 ) andλΓ( 2 3 )1 − λ 2 Γ( 1 3 )Γ( 2 3 ) ,where Γ(z) is the gamma function.23.11 At an international ‘peace’ conference a large number of delegates are seatedaround a circular table with each delegation sitting near its allies and diametricallyopposite the delegation most bitterly opposed to it. The position of a delegate isdenoted by θ, with0≤ θ ≤ 2π. Thefuryf(θ) felt by the delegate at θ is the sumof his own natural hostility h(θ) and the influences on him of each of the otherdelegates; a delegate at position φ contributes an amount K(θ − φ)f(φ). Thusf(θ) =h(θ)+∫ 2π0K(θ − φ)f(φ) dφ.Show that if K(ψ) takestheformK(ψ) =k 0 + k 1 cos ψ thenf(θ) =h(θ)+p + q cos θ + r sin θand evaluate p, q and r. A positive value for k 1 implies that delegates tend toplacate their opponents but upset their allies, whilst negative values imply thatthey calm their allies but infuriate their opponents. A walkout will occur if f(θ)exceeds a certain threshold value for some θ. Isthismorelikelytohappenforpositive or for negative values of k 1 ?23.12 By considering functions of the form h(x) = ∫ x(x − y)f(y) dy, show that the0solution f(x) oftheintegralequationf(x) =x + 1 2satisfies the equation f ′′ (x) =f(x).821∫ 10|x − y|f(y) dy