Given the Rodriguez formula P₁(x) = 2"n! dx" (x²-1)", show that Pm (x) = (1-x²)m/2 mm Pn(x) dm dam is a solution of the associated Legendre eqn. (1-x²)y" - 2xy' + [n(n + 1) — [m²]y = 0. Hint: show that the equation. can be obtained by differentiating the Legendre Equation (1-x²)z" - 2xz' + n(n+1)z = 0 m times w.r.t. x, followed by the substitution z = y(1-x²)-m/2.

Given that L* = L,±iL, and L+Ym (0, 0) = NËYm÷1 (0, 0), where N = h/(e+1+m) (l = m), %3D show that, L+ + L- L+ - L- (a) L, = and L, 2 2i (b)L,Y10 (0, 6) = cY1 (0,6) + c„Y1-1 (0, 6) and find the constants e and c2. (c)Consider the function (0,6), which has the expansion 1 v (0, 6) = E amYım (0,9), where ao = 2v2, a1 = a-1 = 2. %3D m=-1 Show that (0, 0) is an eigenfunction of L, and find the corresponding eigenvalue.

7letA-Cam)e+ Com)-Cam)Rand B (4m) Î + Camdî (2m) R. A)(8m)î+ Ciamdî- Com) i B) (1am)î-Ciy m) ŷ-Caom) R C23 %3D Then À.B equals: Dwone of these

S Can 6 PAR к Торс K Unit K In x K Moti = Cop K Unit S Spee S Topo S Math Micr eb.kamihq.com/web/viewer.html?source-filepicker&document_identifier=137VZR5BZOVSAIMOA55WU555_CvJ9NacO + 100 P e Interpreting Graphs Answer the questions following the graphs on each side Dietance va. Time 2 7 10 11 12 13 14 Time in soconds 1. From 1 second to 2 seconds, how fast is the object traveling. (Take the difference in distance and divide it by the time in between the 2 distances) 2. Is the object going as fast between 9 and 12 seconds as it is between 1 and 4 seconds? How can you tell? 3. What is the motion of the object between 4 and 6 seconds? acer