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看板NTU-Exam
标题[试题] 106-2 萧浩明 工程数学下 期末考
时间Mon Jul 23 06:27:43 2018
课程名称︰工程数学下
课程性质︰机械系大二必修
课程教师︰萧浩明
开课学院：工学院
开课系所︰机械系
考试日期（年月日）︰107/6/6
考试时限（分钟）：60 min
试题 :
1. Derive the Cauchy-Riemann equation in the polar coordinate (r,Θ). (20%)
Given that r = √(x^2+y^2), Θ= tan^-1(y/x), ∂Θ/∂x = -sinΘ/r,
∂Θ/∂y = cosΘ/r.
2. Calculate the residue of f(z) = cot(z) at z = ±nπ, n = 0,1,2,... (20%)
╭ -e^z
3. │ -------dz, c:|z| = 1 (20%)
╯c cosπz
╭π 1
4. │ ----------- dΘ, given that cos2Θ = 1 - 2sin^2Θ (20%)
╯0 1 + sin^2Θ
Hint: Try to change the upper limit from πto 2πso the Cauchy integral of
a full unit circle can be applied.
5. The Laurent series is the generalization of the Taylor series and can be
obtained by using the known Taylor series. Find the Laurent series for the
following complex function. Identify its singularity. (20%)
sinz u^3 u^5 u^7
f(x) = -------, given that sinu = u - --- + --- - --- + ...
z - π 3! 5! 7!
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