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）18-19（1）试卷

2018-2019学年米一生《支西我与积分换以状一1 同济大学课程考核试卷(A卷) 2.(1%)Solve the following IVPby using of Laplace transform x(t)+2x'(t)+x(t)=e',x(0)=x(0)=1. 2018一2019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 此巷选为：期中考试（人、期终考试）、重考)试卷 年级 专业 No Name 年是数调而 题号1 2 3 4 6 7 总分 得分 (往意：本鲁共7大■，3大张，滑分100分，考树侧为120分钟，要来可出解题过握，否不于计分，) L.(I)(5%)Set C:上-=2andC生：zt)=4it-3(teR).Sketch C and C2 (2(I0%)f名C and三eC2,find the minimum of-zg 3.(10%)Find a fractional linear transformation f(z),which maps the unit cirele onto the line y =1,and satisfies f(0)=21

2018-2019 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） 2018 — 2019 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 总分 得分 （注意：本试卷共 7 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. (1) (5%) Set 1 C z : 1 2 − = and 2 C z t it t : ( ) 4 3 ( ) = −  R . Sketch 1 C and 2 C . (2) (10%) If 1 1 z C  and 2 2 z C  , find the minimum of 1 2 z z- . 2. (10%) Solve the following IVP by using of Laplace transform ( ) 2 ( ) ( ) , (0) (0) 1 t x t x t x t e x x    + + = = = . 3. (10%) Find a fractional linear transformation f(z), which maps the unit circle onto the line y = 1, and satisfies f i (0) 2 =

2013-2019年一生《文西我与积2分换》必状一2 1 4.sef2）=1-sin2 5.)Find the Fourier transform of 2 (1)(10%)Find the first four nonzero temms of the Taylor series for f(z)around 0,and determine the disc over which the Taylor series converges. (2)(1%)Find and classify all the singular points of f(z)in the extended complex plane

2018-2019 学年第一学期《复变函数与积分变换》期终考试试卷--2 4. Set 1 (z) 1 sin f z = − (1) (10%) Find the first four nonzero terms of the Taylor series for f(z) around 0, and determine the disc over which the Taylor series converges. (2) (10%) Find and classify all the singular points of f(z) in the extended complex plane. 5. (10%) Find the Fourier transform of 2 4 ( ) 1 x f x x = +

201-2019年米一学《支西我与积2分支净潮格状枣一了 6.Define log z=In.where 0 isthe value of Arg z in [0, (1)(10%)Find all the singular points of log z in the complex plane. (2)(10%)Evaluate the integral )i+7d,创g底em出 节1 ()(Does there exist a closd curve C.which may not be simple,such that? where P(x）and()are polynomial satisfying）≠0 for all x≥0and deg P(x）+1<degr）,and名，4 etherofz

2018-2019 学年第一学期《复变函数与积分变换》期终考试试卷--3 6. Define log ln z z i = +  ，where  is the value of Arg z in [0,2 )  . (1) (10%) Find all the singular points of log z in the complex plane. (2) (10%) Evaluate the integral 3 0 1 d 1 x x + +  , by using the formula 0 1 ( ) ( ) d Res log , ( ) ( ) k j j P x P z x z z Q x Q z + =   = −       , where P x( ) and Q x( ) are polynomial satisfying Q x( ) 0  for all x  0 and deg ( ) 1 deg ( ) P x Q x +  , and 1 ,..., k z z are the zeros of Q z( ). 7. (1) (8%) Does there exist a curve C, which is not closed, such that 2 d 0 C z z =  ? (2) (7%) Does there exist a closed curve C，which may not be simple, such that d 4 C z i z =   ?