PHYS 216 Exam 2, Fall 2012

Physics 216,Quiz 2 December 17, 2012, 90 minutes 1. Find the Fourier Series of the function f (x) = � 0, −π < x < 0 x, 0 < x < π 2. The Fourier series of the function of period 2 which is equal to x on the interval (−1, 1) is given by f (x) = − 2 π ∞ � n=1 (−1)n 1 n sin nπx Use this expansion and Parseval theorem to show that ∞ � n=1 1 n2 = π2 6 3. Given the two recursion relations for Legendre polynomials xP ′ l (x) − P ′ l−1 (x) = lPl (x) P ′ l (x) − xP ′ l−1 (x) = lPl−1 (x) show that � 1 − x2� P ′ l (x) = lPl−1 (x) − lxPl (x) Differentiate this last equation with respect to x and eliminate P ′ l−1 (x) to show that � 1 − x2� P ′′ l (x) − 2xP ′ l (x) + l (l + 1) Pl (x) = 0 4. Expand the function f (x) = P ′ 7 (x) in terms of Legendre polynomials. Hint: Use the following recursion relation repeatedly (2l + 1) Pl (x) = P ′ l+1 (x) − P ′ l−1 (x) and P1 (x) = x. 5. Use the recursion relation for Hermite polynomials 2xHn (x) = Hn+1 (x) + 2nHn−1 (x) and orthogonality condition ∞ � −∞ e−x2Hn (x) Hm (x) dx = δnm √π2nn! to evaluate the integral ∞ � −∞ x2e−x2Hn (x) Hn (x) dx = √π2nn! � n + 1 2 � 6. (a) Using the definition P m l (x) = � 1 − x2� m 2 dm dxm Pl (x) Show that P m+1 l (x) = � 1 − x2 d dxP m l (x) + 1 √ 1 − x2 mxP m l (x) (b) Define L+ = eiϕ � ∂ ∂θ + i cot θ � to show that L+Y m l is proportional to Y m+1 l using Y m l ∼ eimϕP m l (cos θ) and the change of variable x = cos θ. 1 7. Use the relation xLk n = (2n + k + 1) Lk n − (n + k) Lk n−1 − (n + 1) Lk n+1 to show that ∞ � 0 e−xxk+1Lk n (x) Lk n (x) dx = (n + k)! n! (2n + k + 1) using the orthogonality condition ∞ � 0 e−xxkLk n (x) Lk m (x) dx = (n+k)! n! δnm.. 8. Find the Fourier transform F (ω) of the function f (t) = � 1, |t| < 1 0, |t| > 1 .Use the convolution theorem ∞ � −∞ f (t) f∗ (t) dt = ∞ � −∞ F (ω) F ∗ (ω) dω to evaluate the integral ∞ � 0 sin2 ω ω2 dω. 2