課程名稱︰化學數學二 課程性質︰選修 課程教師︰金必耀 開課學院：理學院 開課系所︰化學系 考試日期（年月日）︰2012/6/20 考試時限（分鐘）： 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1.Evaluate 1 x^4 ∫ ───── dx 0 √(1-x^2) (10%) 2.Given x erf(x) = 2/√pi *∫exp(-t^2) dt 0 show that ╭ 1 - i ╮ x erf│─── x│ = (1-i)*√(2/pi) ∫ [cos(u^2)+isin(u^2)] du ╰ √2 ╯ 0 (10%) 3.We have the following ODE, xy'=y a)Solve it by elementary method. b)Solve it by series method. (10%) 4.The method of order reduction is using a known solution of second order ODE to reduce it into first order ODE. We already know one solution of Legendre equation, (1-x^2)y'' - 2xy' + n(n+1)y = 0 is Legendre polynomail of the first kind Pn. Please solve a)Legendre polynomail of the second kind Qn as formal integral via the method of order reduction. b)Evaluate Q2 (20%) 5.Prove the least square approximation property of Legendre polynomial. a)Given 1 2 ∫ P_l(x)P_m(x) dx = ──── δ_lm -1 2l + 1 let p_l(x) be normalized Legendre polynomial, say 1 ∫ P_l(x)P_m(x) dx = δ_lm -1 Express p_l(x) in terms of Legendre polynomial P_l(x). b)The function to be approximated is f(x), if wewant to represent f(x) with normailzed Legendre polynomial up to n-th order, i.e. n f(x) ~= Σ c_i*p_i(x). i=0 1 Show that c_i =∫f(x)*p_i(x) dx -1 n c)Define ξ(b_i,x) = Σ b_i*p_i(x) as the model function of f(x). Use mathematical induction 1 min ∫ [f(x) - ξ(b_i,x)]^2 dx b_i -1 gives b_i = c_i, i.e. the best approximation. (15%) 6.Knowing the solution of the ODE 1 - 2a a^2 - (pc)^2 y'' + ──── y' + [(bcx^(c-1)^2 + ───────)]y = 0 x x^2 is y = x^a [AJp(bx^c) + BYp(bx^c)], and that spherical Bessel functions are j_n(x) = √(pi/2x) J(z) n+1/2 y_n(x) = √(pi/2x) Y(z) n+1/2 (1) h (x) = j_n(x) + i * y_n(x) n (2) h (x) = j_n(x) - i * y_n(x) n Show that the spherical Bessel functions are the solution of x^2 y'' + 2xy' + (x^2 - n(n+1))y = 0. (15%) 7.Prove the orthogonality of Hermite polynomial on (-∞,∞) with weight function exp(-x^2). a)Show that Hermite equation, y'' - 2xy' + 2ny = 0, can be rewritten in d exp(x^2)──(exp(-x^2)y') + 2ny = 0. dx b)Plug in Hermite polynomial Hn and Hm into the above, and show that the orthogonality of Hermite polynomial via integration by part. (10%) 8.We have raising and lowering operator for Bessel function Jp, say p d p d Rp = ─ - ── Lp = ─ - ── x dx x dx a)Dirext operator Rp and Lp on Jp to show how they are raising/lowering operators. b)Show that RLJp = Jp and LRJp = Jp both give the origin Bessel equation. x^2 y'' + xy' +(x^2 - p^2)y = 0. (10%) 9.A fullerene is any molecule composed entirely of "trivalent" carbon(sp^2), in the form of a hollow sphere, ellopsoid or closed tube. Typically, these molecules consists of only pentagons and hexagons. Show that, by using the Euler theorem F - E + V = 2, where F, E, V are numbers of faces, edges, and vertices in the polyhedron, the number of pentagons in a fullerene os always 12. (10%) --

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