课程名称︰应用数学三 课程性质︰物理系大二必修 课程教师︰何小刚 开课学院：理学院 开课系所︰物理系 考试日期（年月日）︰21/06/2011 考试时限（分钟）：180分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : Problem 1. (25 points) a) Write down the expression for the Fourier transformed function f (x) and f Laplace transformed function f (s) of a function F(t). l b) Write down the expression for the inverse Fourier transform of the function f (x), and inverse Laplace transform of the function f (s) in the Bromwich f l integral form. ∞ c) For a fourier series expansion f(x)=a_0/2+Σ(a_ncos(nπx/L)+b_nsin(nπx/L)) n=1 in the interval [-L,L], express a_n and b_n in terms of integration of product of f(x), and cos(nπx/L) and sin(nπx/L). ∞ inπx/L d) Show that f(x) can be written as f(x) = Σ c_n e and express c_n in n=1 terms of a_n and b_n. e) Write down the convolution formula which relates the functions F1(t) and F2(t) and their Laplace transformed functions f1(s) and f2(s). Problem 2. (25 points) a) Obtain the Fourier series expansion for the function F(x) = x for -π< x <π b) Obtain the Fourier transformed function f(t) of the function F(x) = x for -π< x <π and 0 for |x| > π. c) Obtain the inverse of Laplace tranform for the function f(s) = 1/(s-a)(s-b) (s-c). Problem 3. (25 points) a) Obtain the Fourier transformed function f(ω) for the function F(x) = 1 for |x| < 1 and F(x) = 0 for |x| > 1. ∞ b) Show that F(x) can be expressed as F(x) = (2/π)∫ dωsin(ω)cos(ωx)/ω. 0 c) Evaluate the integral in b) explicitly and show that for x = 1, F(x) = 1/2. Problem 4. (25 points) a) Obtain the Laplace transformed function x(s) for X(t) satisfying the driven oscillator with damping equation: mX''(t) + bX'(t) + kX(t) = F(t) with initial conditions X(0) = A (with A being a constant) and X'(0) = 0. (X'(t) = dX(t)/dt 2 2 and X''(t) = d X(t)/dt ) b) Using convolution formula to show that one can write the solution of X(t) -bt/2m t -bz/2m as: X(t) = Ae (cos(ω1t)+(b/2mω1)sin(ω1t))+(1/mω1)∫F(t-z)e sin(ω1z)dz 2 2 0 Here ω1 = (k/m)-(b/2m) . c) Assuming that F(t) is equal to a for 0 < t < t0 and 0 else where. Obtain the expression for X(t) by explicitly carrying out the integral in b). 2 2 ct (Useful equations: the inverse of f(s) = d/((s-c) + d ) is equal to e sin(dt) 2 2 ct and f(s) = (s-c)/((s-c) + d ) is equal ot e cos(dt). The Laplace transform (n) (n) n n-1 for the nth derivative F of F is: L(F (t)) = s L(F(t)) - s F(0) - ... (n-1) - F (0)). --

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