MATH 224 Final, Fall 2004

AMERICAN UNIVERSITY OF BEIRUT MATHEMATICS 224, FINAL EXAMINATION FALL SEMESTER, 2004-05 Answer the following questions: 1. Let fa(z) = nz/(1+ nz) forO<2< om. (a) Find f(z) = limyeo fn (2). (2 points) (b) Does f, —+ f uniformly on (0,1)? Justify. (4 points) (c) Does f, — f uniformly on [1, oof? Justify. (4 points) 2. Use separation of variables to derive the family of solutions ux _(z,y,z) = sin(maz) cos(ny) exp(£Vm? + n? m2) of the boundary-value problem Vou = User + Uyy + Use = 9, u(O.y,z) = u(l,y.z) = u,(x,0,z) = uy(z,1,z) = 0. (14 points) 3. (a) Define a regular Sturm-Liouville problem on a closed interval fa, d]. (3 points) (b) Define an eigenvalue and eigenfunction of a regular Sturm-Liouville problem. (3 points) (c) Find the eigenvalues and the normalized eigenfunctions for the following regular Sturm-Liouville problem on {1,6}, b > 1. (a? fy +Af=0, fC) = f(b) =0. Hint: The general solution of the Cauchy-Euler differential equation az’y" + bry! + cy = 0, 2 > 0, is y = cz" + con"?. where 1,1 are the roots of the characteristic equation ar?+(b-a)r+c=O.Ifry =r =7, then the general solution is y = cx" + cox” Inz. (14 points) 4. (a) Define the Beta function and show that (5 points) {2 Biz,y) = 2 | sin**-1 6 cos’? @ dé. 0 (b) Define the Gamma function and show that (5 points) T(z) -| (in(i/t)""' dt, 2>0. 0 (c) State Stirling’s formula. . (3 points) pune. a atre JARI “NEVERSI y PAN ONPVERSIC {J Library { OF PERO? } Too aa retreat 5. (a) State the Fourier integral theorem. (3 points) (b) Conclude from the Fourier integral theorem the Fourier sine and cosine transforms. (3 points) (c) Show by use of the sine integral that 2er= | usin(eu) du, ©>0. 0 tr lt+u? Hint: f e** sin(br) dz = e**(asin(bx) — bcos(br))/(a? +b?)+c and part 5(b). (8 points) 6. (a) State Laplace’s method. (3 points) (b} Show that x [D(a + 1))/F ~ = as 2 00. (5 points) (c) Show that 4 = (2) = f 6 8! dt me? / —. 0 22 Hint: Show J(z) = e* f eWtl=cost) de. (5 points) 7. The Chebyshev polynomial T,, is defined by T,(cos @) = cos(n@), n=O,1,---. (a) Show that form =0,1,---, ! he n—2k ( 2 ke T,(£) = S- Qki(n— 2K)” Fg —1) : k<n/2 Hint: Use cos(n@) = R(cos @ + isin)" and the Binomial theorem. , (5 points) (b) Show that {7,,}§° is an orthogonal basis with respect to the function w(x) = (1 — 2?)71/? for L?(0, 7). (6 points) (c) Show that for n =0,1.---, (l= 2°)T" — aT! +n°T, = 0. Hint: cos(n@) is a solution for the differential equation y” + n?y = 0, where y° = d’y/d6?. (5 points) Good Luck and Happy Eid Al Adha