作者TptPTM (zennen)
看板NTU-Exam
標題[試題] 106-2 蕭浩明 工程數學下 期末考
時間Mon Jul 23 06:27:43 2018
課程名稱︰工程數學下
課程性質︰機械系大二必修
課程教師︰蕭浩明
開課學院：工學院
開課系所︰機械系
考試日期（年月日）︰107/6/6
考試時限（分鐘）：60 min
試題 :
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!
--
※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 123.192.34.163
※ 文章網址: https://webptt.com/m.aspx?n=bbs/NTU-Exam/M.1532298465.A.F42.html