課程名稱︰複分析導論 課程性質︰數學系大三必修 課程教師︰蔡宜洵 開課學院：理學院 開課系所︰數學系 考試日期︰2020年11月12日(四) 考試時限：13:20-15:10，共計110分鐘，最後加五分鐘。 試題 : 　　　　　　　　　　　　　　　Complex Analysis 　　　　　　　　　　　　　　　 November 12, 2020 　Total points: 105 Note. 1) Your arguments and calculation details must be clear and complete; the auxiliary theorems needed for any claim of your results must be explicitly indicated. Failing to do so may result in loss of partial or full points. 1. Let D be the open disk, centered at 0, of radius r. r z 2020 n 　 i) (15 points) Use Rocuhe theorem to show that the equation e - e z = 0 has exactly n roots (counted with multiplicity) in D . 1 ii)(10 points) Replace D by D and ask for D which encloses all the above 1 r r n roots. Find the smallest r that can be obtained by your method in i). +∞ e^(ax) 2. (20 points) Compute ∫ -------- dx, 0 < a < 1. -∞ 1+2e^x 3. (20 points) Let f(z)∈Hol(C). Suppose that there exist R > 0, C > 0 and a positive integer N such that |f(z)|≦C|z|^N for all |z| > R. Show that f(z) is a polynomial of degree at most N. (hint: Cauchy's integral formula) dz 4. (20 points) Compute the integeral ∫ --------, where C is the curve with C z^4 -1 z(θ) = 2cos(2θ)e^(iθ), θ from 0 to 2π. 5. (20 points) Let f(z)∈Hol(C). Assume that f(z) is not a polynomial, Show that given any R > 0, there exists z_1 with |z_1| > R and z_2 with |z_2| < 1 such that f(z_1) = f(z_2). (hint: open mapping theorem...etc.) --

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