[試題] 100上 王藹農 複變函數論 期中考

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課程名稱︰複變函數論
課程性質︰數學系大三必修
課程教師︰王藹農
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2011/11/10
考試時限（分鐘）：110 min
是否需發放獎勵金：是

試題 :

             Complex  Analysis  Midterm   Nov.10.2011

(一共五題，每題8分，總分為40分)


                2
     ∞        x
1. ∫    --------------  dx  = ?
    -∞    4     2
          x  + 5x  + 6


2. Ω={ |z|<1 , |z-1|<1 }   f = conformal map : Ω → { |w|<1 }

     _    _               _                    i            1
   f(z) = w  ,  f(1-z) = -w  ,  f(a+bi) = 0 + ---   ,  a = ---
                                               3            2

         __   1/3
     ? √3  (4    - 1 )
   b = ------------------  , If not , b = ?
              1/3
         2  (4    + 1 )


3. γ= { (rcosθ , 2+rsinθ) | r = a - 3sinθ , 0 ≦θ≦ 2π }

            dz
   ∮ --------------- = ?
   γ         2  1/2
       ( 1 - z  )



4. Can you find three circles C1 , C2 , C3 each orthogonal to both

   |z| = 1 and |z-1| = 4 , but the intersection C1∩C2∩C3 = empty ?

   If not , C1∩C2∩C3 = ?


5. H = upper half plane = { y > 0 } ,  p = -1 + i  ,  q = 1 + i ,

                                         |dz|
   γ= path from p to q ,  find  min ∫ ------ = ?
                                  γ  γ  y


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