PHYS 216 Exam 2, Fall 2014

Physics 216 Mathematical Physics Quiz 2, December 4, 2014, Time: 90 minutes 1. Expand Fe) x?, 0 < x < 2n, if the period is 27. Use the result for f (7) to prove that vs =" = . (17points) 2. The Fourier series for the function f (x) = sinz, 0 < x < 7 is given by 2 4 \cos2nz f@)= 2 - geod n=1 co Use this result and Parseval’s identity to show that } > a—heyr = Fg — 2: (14) 3. Use the Fourier sine integral to solve the equation oo F, (w) = [(@)sinweds = { bby; noes H , to determine the function f (x). (14p). co 4. Expand the function f (r) = { 0 oss) in a Legendre series Yar (x). Hint: , l=1 Use the recurrence relation (21 +1) P, = Pi,, (x) — Pi_y(x) , (1) = 1 and P, (0) = 0 for l=odd. Express your answer in terms of P; (0) for | =even. (14p) oo 5. The generating function for Hermie Polynomials is g (x,t) = errt-? _ Son (x) f. Do 90 oo the integral feve (x,t) dx to deduce that [eee (x) Hp (xz) dx = 0, n # m. Use —0o eo / e-?"dz = Vr. (14p) 6. Consider the Sturm-Liouville equation Ly = a (a (x) “u(2)) +7(xr)y(x) = 0 with weight function w (x) = 1. Let y; (x) and y2 (x) be two independent solutions of this equation. Define the Wronskian yi(z) y2(z) a) 2) Show that W (x) = wey where C is constant. Hint: Differentiate a(x) W (x) with respect to x. (15p) 7. Derive the spherical harmonics closure relations W (x) = det oo l >> YS Hi" (1, 41) 17" (02, G2) = 5 (cos 1 — cos 62) 5 (1 — 2) l=0m=-l (12p) tad oe ious oF 6) tye dryer ey, PEG whe yyy 1. wo Area roe weuyt es CeCe ION = CE HEBD GE GBT IO G = mete a + Yoga = fe)