課程名稱︰工程數學
課程性質︰大二必修
課程教師︰施文彬
開課學院：工學院
開課系所︰機械系
考試日期（年月日）︰2008.06.16
考試時限（分鐘）：110 mins
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Rule: No calculators are allowed. You are allowed to bring an A4 size
information sheet. Points will not be given without providing details
of your calculation. Please carry out all integrations in your
calculation. Good luck !
1.(25%)
e^(6iz)sin(2z)
Given f(z) = ───────
(1+iz)^2
(a) Find u and v so that f(z) = u(x,y)+iv(x,y).
(b) Determine all points at which Cauchy-Riemann equations are satisfied ,and
determine all points at which f(z) is differentiable.
(c) Determine all residues of f(z).
(d) Apply Residue Theorem to find the inverse Fourier transform of the
e^(6iω)sin(2ω)
function ─────────
(1+iω)^2
2.(20%)
d^2y d^2y
Solve the wave equation ── = c^2(──) on the line for c = 2 and the given
dt^2 dx^2
initial conditions:
dy ╭ cos(πx) , -1.5≦x≦1.5
y(x,0) = xe^(-|x|) and ─(x,0) = ┤
dt ╰ 0 , |x|>1.5
3.(20%)
(a) For what value(s) of "L" does the boundary value problem ,
y" + 16y = 0 , y(0) = 0 , y(L) = 0 , L>0
have a nontrivial solution ?
(b) Write down the solution corresponding to the value(s) of "L" found in
part(a).
(c) For what values of "L" does the boundary value problem have a unique
solution ?
4.(20%)
(a) Solve (y'/x^3)' + (1+35/9x^2)y = 0 by letting u = y/x^2.
(b) Show that for any real number v ,
x^2Jv"(x) = (v^2-v-x^2)Jv(x) + xJv+1(x).
5.(15%)
d^2u du
Consider the one dimensional heat transfer problem ── = 9─ , 0≦x≦5
dx^2 dt
with boundary conditions u(0,t) = 0 ,u(5,t) = 4 ,t>0 and initial conditions
u(x,0) = 0 , 0≦x≦5.
(a) Find the long time , i.e. time independent , solution reached as t→∞.
(b) Find the time-dependent solution u(x,t) that satisfies the given boundary
and initial conditions.
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