课程名称︰古典椭圆函数论一 课程性质︰数学系选修 课程教师︰蔡宜洵 开课学院：理学院 开课系所︰数学系 考试日期︰2011年11月11日 考试时限：170分钟，3:30-5:20 是否需发放奖励金：是 试题 : Total points: 100 Read the following carefully. Choose two problems in Topci-Part, and two prob -lems in Exercise-part. But if you have chosen Problem 1 or 5, then you may eit -her i) choose one problem in Topic-Part and only one problem in Exercise-Part, or ii) two problems in Exercise-Part. See more restrictions stated in A), B), C) and D). Do not work more than the total points 100. 　Topic-Part: A) and B) A) For the following two problems, choose only one (or neither), but not both. Note: If you choose Problem 1, then you should not choose Problem 8 in Exercise Part, netier Problem 5 in B). 1 (50 pts). a) As the inversion of lemniscatic integral, construct the lemnisca -te sine function sl(s) and show how to extend its domain of defintion to all r -eal numbers. b) State (without proof) basic properties of sl(s) similar to tri -gonomertic sine function. c) Extend the domain of defintion of sl(s) to the co -mplex plane. Show, by checking hte pair of Cauchy-Riemann equations, that it i -s complex analytic where it is defined. 2 (25 pts). a) Give the statement of periods, zeros and poles of sl(z), where z denotes the complex variable. b)Assume basic properties of sl(z). Prove the tw -o key results of sl(z) needed for the proof,i.e. one result for the relationsh -ip between sl(z+(m+inω) and sl(z), and the other result for the relationship between sl(z) and sl(z+(1+i)ω/2) (resp. sl(z+(1-i)ω/2)). c) Complete the proo -f for a), basing on the result of b). B) For the following, you may choose one (or two if you skip A) above) of the t -hree problems. But if you have already cgiseb Problem 1 in A), then you should not choose Problem 5. If you did not choose Problem 1 in A), you may choose Pr -oblem 5 or not, and if you do choose Problem 5 in this case, then you should n -ot choose Problem 9 in Exercise Part. 3 (25 pts). a) By following Euler's argument, explain the construction of a pol -ynomial relating sin(nz) and sin(z). Can you write down the general form? b) G -iven sin(nz) and fixed throughout, give explicitly the roots of the polynomial in a), in terms of sin(z) and its variants. c) State and prove the infinite se -ries expansion of 1/sin(z). d) State and prove the infinite product expansion of sin(z). 4 (25 pts). a)By following Euler's 1st treatment if the infinite product expans -ion of sin(z), write down the expected form for the intinite product expansion of sl(z). b) Simplify the formula given in a) to the effect that it is ready f -or the further simplification by using (without proof) the infinite product ex -pansion of sin(z) and cos(z). c) Simplify further and prove the final result o -f the infinite product expansion of sl(z) in terms of suitable trigonometric s -ine function. 5 (50 pts). a) By starting with Euler's (or Abel's) addition formula, give a de -rivation of the recursive relation for computing sl(nz). Then state (without p -roof) the general form of the relationship between sl(nz) and sl(z) for odd n and even n respectively. b) For the case n=2, turn the equation of a) into a po -lynomial, give the roots of this polynomial when sl(sz) is given and fixed, an -d show that there roots are distinct. (you need not prove the theorems used fo -r the construction of the roots, but you need to give carfully the statement o -f the theorems you are using.) c) Examine the case for n=3 in the same way as b). Exercise-Part: C) and D) C) If you have chosen Problem 1 in Topic-Part, then you can choose Problem 6 or 7 (or neither) but not both. 6 (25 pts). State and give a proof of, the addition theorem according to Abel's formulation. 7 (25 pts). State and prove Gauss' addition formula for sl(a+b) involving sl(a) ,sl(b),cl(a) and cl(b). (You may freely use the relationship between sl(s) and cl(s)). 8 (25 pts). Firts give (without proof) sl(x+iy) in terms of sl(x) and sl(y) whe -n x,y are real. Then by verifying the two Cauchy-Riemann equations, show that the above function sl(z) where z=x+iy, is complex analytic where it is defined. D) If you have chosen Problem 5 of B), then you should not chose Problem 9. 9 (25 pts). Show explicitly the relationship between sl(3z) and sl(z) and turn it into a polynomial in terms of x=sl(z) for sl(3z) fixed. Then give the roots of that polynomial and show that these roots are all distinct. --

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