課程名稱︰工程數學下
課程性質︰必修
課程教師︰伍次寅
開課學院：工學院
開課系所︰機械系
考試日期（年月日）︰2010.6.21
考試時限（分鐘）：130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.(3%)Write down the real and imaginary parts of the following complex numbers
Give all distinct values if it is multi-valued.
(i) (ii) (iii) 2/3
sim(2+3i) i^(1+i) [ -1+√3i ]
[￣￣￣￣￣]
[ 1+i ]
2.(3%)Determine if each of the following complex function is analytic in the
the region |z|< ∞. If not, give the point(s) where it is not analytic.
(i) (ii) (iii)
2 z^2 + 9
f(z)=|z| f(z)= ￣￣￣￣￣￣￣ e^ln(iz)
z[z^2+(2-3i)z-6i]
3.(6%)Find all singularities of the following complex function and classify
their types. If the singularity is a pole, give the order of the pole
(i)
2 z^2 + z + 1 1-e^(iz)
f(z)= z sin(1/z-i) (ii) f(z) = ￣￣￣￣￣￣￣ (iii) f(z)= ￣￣￣￣￣￣
z^2 -iz +2 (z+3)*z^2
4.(6%)
Find the Taylor expansion(problem(i)) and Laurent expansion (problem(ii))
about the point z=a and give the range of convergence of the series in each
case.
(i) (ii) z+i
f(z)=ze^(iz),a=2 f(z)= ￣￣￣￣￣￣ , a=-1
(z+1)^2 (z-i) (<hint>: rewrite f(z) as
1 z+i z+i
￣￣￣￣ * ￣￣￣ and expand ￣￣￣ about -1)
(z+1)^2 z-i z-i
5.(4%) Determine the residue at each singular point of the following complex
function:
(i) e^(-z) (ii) cos(z)
f(z)= ￣￣￣￣￣ f(z)= ￣￣￣￣￣
z(z+i)^2 z^4 (1-z)
6.(6%) Evaluate the following comlex integrals:
(i) z+1
∮ ￣￣￣￣￣ dz Γ:|z+(1+2i)|=2
Γ z(z+2i)^2
(ii) _
e^(-iz)
∮ ￣_￣￣￣ dz Γ: |z|=1
Γ z
7.(4%)Evaluate the following real integrals by using the complex integration
technique:
(i) 2π dθ
∫ ￣￣￣￣￣￣￣ with 0<a<1
0 1-2acosθ+a^2
(ii) +∞ √x
∫ ￣￣￣￣￣￣ dx
0 x^2 +3x +2
8.(4%) 2 d:偏微
(i) du d u
solve —— = α ——— for 0<x<∞, t>0 subject to
dt dx^2
BC: u(0,t)=1, IC: u(x,0)=0；
by using the Fourier transform (or cosine, sine transforms) method (other
method are not allowed).
Write down your final result in terms of the original physical variable (x,t).
(ii)What is the expression of the solution when t → ∞ ?
2 2
9.(4%) d y 2 d y
Solve ——— = a ——— for -∞ < x < ∞ , t>0 subject to
dt^2 dx^2
dy(x.0)
IC's : y(x,0)=H(x+1)-H(x-1), ————— =0；
dt
by using the Fourier transform(or cosine,sine transforms) method (other method
are not allowable).
Write down your final result in terms of the original physical variable (x,t)
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