[試題] 103-1 王金龍 複變函數論 期中考
課程名稱︰複變函數論
課程性質︰數學系必修
課程教師︰王金龍
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2014年11月13日
考試時限（分鐘）：170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. (15) Prove Goursat's theorem for a triangle and then deduce from it Cauchy's
theorem for arbitrary piecewise C^1 closed curves inside a disk.
2. (15) Prove that ∫(0 to ∞) (sin(x)/x) dx = π/2.
3. (15) Prove that for ξ∈R,
∫(-∞ to ∞) e^(-2πixξ)/(1+x^2)^2 dx = π/2 (1+2π|ξ|) e^(-2πξ).
4. (15) Prove that ∫(0 to 1) log(sin(πx)) dx = -log2.
5. (15) Determine the number of roots of the equation z^6 + 8z^4 + z^3 + 2z+3=0
in each quadrant of the complex plane. Determine also the number of
zeros inside each annulus k <= |z| < k+1 with k∈Z(>=0).
6. (10) Let f be an entire function such that for each a∈C at least one Taylor
coefficient at z=a is zero. Prove that f is a polynomial.
7. (10) If f is entire of growth order ρ is not integral, show that f assumes
every value w∈C infinitely many times. Give an example of such f.
8. (10) Does Cauchy's residue theorem hold for functions with not just poles
but also isolated essential singularities? Justify your answer.
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