2017-2018学年一《支西与积分换为潮路以一1 同济大学课程考核试卷(A卷) 2.Let u(x,y)=x3 -3xy2 +2x. 2017一2018学年第一 学期 (1)(6%)Show that there exists an entire function f(z)such that Re(f(z))=y) 命题教师签名： 审核教师签名： (2)(7)Find the entire function(z satisfying)=0. 课号：122144 课名：复变函数与积分变换 考试考查：考试 (3)(7%)Evaluate the integralf(z)dz,where C isthe upper half of the unit circle from I to-1. 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 6 7 8 总分 得分 (往盘：本试德共8大题3大张，满分100分，考树同为120分钟，要米可出邮圆过超否则不于计分) 1.(10%)be real constants.Show that root of the polynomial equation z”+42-1+z-2+…+司。=0，hens0sZ0
2017-2018 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） 2017 — 2018 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 总分 得分 （注意：本试卷共 8 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%) Let 1 2 , , , n a a a be real constants. Show that if 0 z is a root of the polynomial equation 1 2 1 2 0 n n n n z a z a z a − − + + + + = , then so is 0 z . 2. Let 3 2 u x y x xy x ( , ) 3 2 = − + . (1) (6%) Show that there exists an entire function f z( ) such that Re( ( )) ( , ) f z u x y = . (2) (7%) Find the entire function f z( ) satisfying f(0) 0 = . (3) (7%) Evaluate the integral ( ) C f z dz , where C is the upper half of the unit circle from 1 to -1
2017-2018学年米一学湘《支西与积分支编潮一2 3.(10%)Use the Laplace transform to solve the IVP x"(t)=-4x(t)+sint.x(0)=x'(0)=0.. 5.(1%)Use the Residue Theorem toevaluate the integral 1+sin2 0 6.()Find the Fouier uransformof)6 X 4.(10%)Finda Mobius transformation f(z)whichmaps the real axis onto the unit circle and satisfies 0=
2017-2018 学年第一学期《复变函数与积分变换》期终考试试卷--2 3. (10%) Use the Laplace transform to solve the IVP x t x t t x x ( ) 4 ( ) sin , (0) (0) 0 = − + = = . 4. (10%) Find a Möbius transformation f(z) whichmapsthe real axis onto the unit circle and satisfies 1 ( ) 2 f i = . 5. (10%) Use the Residue Theorem to evaluate the integral 2 - d 1 sin + . 6. (10%) Find the Fourier transform of 4 ( ) 16 x f x x = +
2017-2018学年一湘《支西与积分换为潮路以一3 1.LeK2）=simz 22 (1)(6%)Show that 0 and are removable singularities. (2)(7%)Find the first three nonzero terms of the Taylor series around forz.What is the radius of convergence of this series (3)(7%)Set the Laurent series for sinz in the annulus<2 to be the following 品2以 then use the result of(2)to find the coefficients ccoc and c 8.()If f(z)is analytic in D.,and isunbounded in Dfor each integer n,then what kind of singularity does f(z)have at?Prove your conclusion
2017-2018 学年第一学期《复变函数与积分变换》期终考试试卷--3 7. Let 2 2 1 1 2 ( ) sin z h z z z z = − + − . (1) (6%) Show that 0 and are removable singularities. (2) (7%) Find the first three nonzero terms of the Taylor series around 0 for h z( ) . What is the radius of convergence of this series. (3) (7%) Set the Laurent series for 1 sin z in the annulus z 2 to be the following 1 = sin n n n c z z + =− , then use the result of (2) to find the coefficients 2 1 0 1 c c c c , , , − − and 2 c .. 8. (10%) If f z( ) is analytic in D: 0 1 z , and ( ) n z f z is unbounded in D for each integer n, then what kind of singularity does f z( ) have at 0 ? Prove your conclusion
