同济大学：《复变函数和积分变换》课程教学资源（试卷习题）复变函数与积分变换（全英语）期末试卷（A卷）16-17（1）试卷

2016-2017华年常一学湘《复支西我与积2分支》以一 同济大学课程考核试卷(A卷) 2.(1%)Use the Laplace transform to solve the IVP x"t)+9x(t)=e‘,x(0)=0,x(0)=2. 2016一2017学年第一 学期 命题教师签名： 审核教师签名： 课号：122144 课名：复变函数与积分变换 考试考查：考试 此卷选为：期中考试(、期终考试（√）、重考)试卷 年级 专业 No. Name 任课教师 题号12345 6 89 总分 得分 (往盘：本试德共9大题3大张，满分100分.考树为120分钟，要球可出圆过提，否则不于计分) 1.(10%)Using Euler's equation el=os+isin together with the binomial formula to derive the identity:sin30 =3cos28 sine-sin30,where 0 is real. 3.(10%)Find a Mobius transformation mapping the unit disc [l<1]onto the right half-plane and taking-i to the origin

2016-2017 学年第一学期《复变函数与积分变换》期终考试试卷--1 同济大学课程考核试卷（A 卷） 2016 — 2017 学年第 一 学期 命题教师签名： 审核教师签名： 课号：122144 课名：复变函数与积分变换 考试考查：考试 此卷选为：期中考试( )、期终考试( √)、重考( )试卷 年级 专业 No. Name 任课教师 ___ _ 题号 1 2 3 4 5 6 7 8 9 总分 得分 （注意：本试卷共 9 大题 3 大张，满分 100 分．考试时间为 120 分钟。要求写出解题过程，否则不予计分） 1. (10%) Using Euler’s equation e iθ = cos 𝜃 + 𝑖 sin 𝜃 together with the binomial formula to derive the identity: sin 3𝜃 = 3 cos2 𝜃 sin 𝜃 − sin3 𝜃, where θ is real. 2. (10%) Use the Laplace transform to solve the IVP 𝑥 ′′(𝑡) + 9𝑥(𝑡) = 𝑒 𝑡 , 𝑥(0) = 0, 𝑥 ′(0) = 2. 3. (10%) Find a Möbius transformation mapping the unit disc {|𝑧| < 1} onto the right half-plane and taking −𝑖 to the origin

206-2017学一学《支西与积分支换南一2 6.(1%)Evaluatesind annulus of convergence. 5.(Find and casify the isolated singularity ofin the extnded complexplane. 7.()Fnthe Fourerf

2016-2017 学年第一学期《复变函数与积分变换》期终考试试卷--2 4. (10%) Find the Laurent series for z+1 z(z−4)3 at the punctured neighborhood of z=4, and determine the annulus of convergence. 5. (10%) Find and classify the isolated singularity of sin 3𝑧 z 2 − 3 z in the extended complex plane. 6. (10%) Evaluate ∫ 𝑒 1 𝑧 sin 1 𝑧 𝑑𝑧 |𝑧|=1 7. (10%) Find the Fourier transform of 𝑡 𝑡 4+1

2016-2017年第一生湘《文西与积2分换终以状枣一3 8.(10%)Find an analytic functionf(z)satisfying Re f()=x+y-xy and f(0)=1. 9.Set the function =introducing the change of variables 6塔 (1)(10%)Using the chain rule formally,showthat 器=装+瑞 (2)()Show tha,f isanalytic function,)isanalytic functionimplie is independent of2

2016-2017 学年第一学期《复变函数与积分变换》期终考试试卷--3 8. (10%) Find an analytic function f(z) satisfying Re 𝑓(z) = x + y − xy and 𝑓(0) = 1. 9. Set the function 𝑢̃ (𝑧, 𝑧̅) = 𝑢(x(z, 𝑧̅), 𝑦(z, 𝑧̅)) by introducing the change of variables { x = z+𝑧̅ 2 y = z−𝑧̅ 2i . (1) (10%) Using the chain rule formally, show that { 𝜕𝑢̃ 𝜕𝑧 = 1 2 𝜕𝑢 𝜕𝑥 − 𝑖 2 𝜕𝑢 𝜕𝑦 𝜕𝑢̃ 𝜕𝑧̅ = 1 2 𝜕𝑢 𝜕𝑥 + 𝑖 2 𝜕𝑢 𝜕𝑦 . (2) (10%) Show that 𝜕𝑓 𝜕𝑧̅ = 𝜕𝑓̅ 𝜕𝑧=0, if f(z) is analytic function, i.e. “f(z) is analytic function” implies “f is independent of 𝑧̅