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(完整版)《复变函数与积分变换》习题册(2)

(完整版)《复变函数与积分变换》习题册(2)
(完整版)《复变函数与积分变换》习题册(2)

第一章 复数与复变函数

本章知识点和基本要求

掌握复数的概念和它的各种表示方法及运算; 熟悉复平面、模与辐角的概念;

熟练掌握乘积与商的模、隶莫弗公式、方根运算公式; 了解区域的概念;理解复变函数的概念; 理解复变函数的极限和连续的概念。

一、填空题

1、若等式))(()75(i y i x i i -+=-成立,则=x ______, =y _______.

2、设(12)(35)13i x i y i ++-=-,则x = ,y =

3、若1231i

z i i

+=--,则z =

4、若(3)(25)

2i i z i

+-=

,则Re z =

5、若4

21i

z i i

+=-

+,则z = 6、设(2)(2)z i i =+-+,则arg z =

7复数1z i =-的三角表示式为 ,指数表示式为 。 8、复数i z 212--=的三角表示式为 _________________,指数表示式为

_________________.

9、设i z 21=,i z -=12,则)(21z z Arg = _ _____.

10、设4

i e 2z π=,则Rez=____________. Im()z = 。z = 11、.方程0273=+z 的根为_________________________________.

12、一曲线的复数方程是2z i -=,则此曲线的直角坐标方程

为 。

13、方程3)Im(=-z i 表示的曲线是__________________________. 14、复变函数1

2

+-=

z z w 的实部=),(y x u _________,虚部=),(y x v _________. 15、不等式114z z -++<所表示的区域是曲线 的内部。

16

二、判断题(正确打√,错误打?)

1、复数7613i i +>+. ( )

2、若z 为纯虚数,则z z ≠. ( )

3、若 a 为实常数,则a a = ( )

4、复数0的辐角为0.

5、()f z u iv =+在000iy x z +=点连续的充分必要条件是(,),(,)u x y v x y 在

00(,)x y 点连续。 ( )

6、设21,z z 为复数,则2121z z z z ?=。 ( )

7、1212z z z z +=+ ( )

8、参数方程2

z t ti =+ (t 为实参数)所表示的曲线是抛物线2y x =. ( )

三、单项选择题

1、下列等式中,对任意复数z 都成立的等式是 ( ) A.z·z =Re(z·z ) B. z·z =Im(z·z ) C. z·z =arg (z·z )

D. z·z =|z|

2、方程3z =8 的复根的个数为 ( )

A. 3个

B. 1个

C. 2个

D. 0个

3、当11i

z i

+=-时,1007550z z z ++的值等于 ( )

A i

B i -

C 1

D 1-

4、方程23z i +-= ( )

A 中心为23i -的圆周

B 中心为23i -+,半径为2的圆周

C 中心为23i -+的圆周

D 中心为23i -,半径为2的圆周

四、计算题

1.求出复数4)31(i z +-=的模和辐角。

2.设iy x z +=满足,4)3Re(2

=+z 求x 与y 的关系式

3、将复数6z i =化为三角表示式和指数表示式。

4、求复数1cos sin ,(0)i j j j p -+#的三角表示式、指数表示式及幅角主值。

5.将直线方程1

+y

x化为复数形式。

2=

3

6、求以下根式的值:

(1)(2) (3)

第二章 解析函数

本章知识点和基本要求

理解复变函数的导数及复变函数解析的概念;

掌握复变函数解析的C-R 条件,并能利用C-R 条件判断复变函数的可导性和解析性;

掌握解析函数的基本性质;

了解指数函数、三角函数及对数函数的定义及它们的主要性质。

一、填空题

1、(1)Ln i +的主值为

2、()Ln i -= ,主值为

3、设i e z 43+-= , 则=)Re(iz _________________

4、=i 3_____________________________.

5、=+i i )1(________________________.

6、1i i +=

7、指数函数z e 的周期是

8、设()(1)z f z z e -=-,则()f z '=

9、设3322()f z x y ix y =++,则(1)f i '+=

10、已知函数()(21)(,)f z x y v x y i =++解析,则()f i ¢

= 11、.函数()f z u iv =+在000z x iy =+点连续是()f z 在该点解析的_________条件。

二、判断题(正确打√,错误打?)

1、.若)(z f '在区域D 内处处为零,则)(z f 在D 内必恒为常数。 ( )

2、.若()f z 在0z 点不解析,则()f z 在0z 点必不可导。 ( )

3、函数()(,)(,)f z u x y iv x y =+在点000z x iy =+可微等价于(,)(,)u x y v x y 和在点

00(,)x y 可微。 ( )

4、sin 1z ≤.. ( )

5、函数z e 是周期函数。 ( )

6、设函数()f z 在点0z 处可导,则()f z 在点0z 处解析。 ( )

7、对于任意的复数12,z z ,等式1212(.)Ln z z Lnz Lnz =+恒成立。 ( )

8、不等式Re()2z ≤ 表示的是有界闭区域。 ( )

9、对于任意的复数z ,整数n ,等式n Lnz nLnz =恒成立 ( )

三、单项选择题

1、下列点集是单连域的是 ( )

A .Re()2z > B.13z

C.1z £

D.2arg 2Z p #+

2、下列所示区域中是多连域的为 ( ) A.Im 0z > B.Re 0z < C.01z << D.

arg 4

3

z π

π

<<

3、函数()f z 在点z 可导是()f z 在点z 解析的 ( )

A .充分不必要条件 B.必要不充分条件

C. 充分必要条件

D.既不充分又不必要条件 4、下列说法正确的是 ( )

A 、()f z 在0z 可导的充要条件是()f z 在0z 处解析。

B 、()f z 在0z 可导的充要条件是 ,u v 在0z 处偏导数连续且满足

C R -条件。 C 、()f z 在0z 可导的充要条件是()f z 在0z 处连续。

D 、()f z 在0z 可导的充要条件是,u v 在0z 处可微且满足C R -条件

5、在复平面上,下列关于正弦函数sinz 的命题中,错误的是( ) A.sinz 是周期函数 B.sinz 是解析函数 C.|sinz|1≤

D.(sin )cos z z '=

6、以下说法中,错误的是 ( )

A .复指数函数z e 具有周期 B.幂函数a z (a 为非零的复常数)是多值函数 C .对数函数Lnz 为多值函数 D.在复数域内sin z 和cos z 都是有界函数 7、设()sin f z z =,则下列命题中错误的是(

)。

A .()f z 在复平面内处处解析

B .()f z 以2π为周期

C .()2

iz iz e e f z --= D .()f z 是无界的

四、计算题

判断下列函数在何处可导,在何处解析? (1)33()23f z x y i =+

(2)2()()2()f z x y x y i =-++

(3) 22()f z xy ix y =+

第三章 复变函数的积分

本章知识点和基本要求

了解复变函数积分的定义及性质; 会求复变函数的积分;

理解柯西积分定理,掌握柯西积分公式;0 掌握解析函数的高阶导数公式;

了解解析函数无限次可导的性质;会综合利用各定理计算闭路积分。

一、填空题

1、设曲线C 是正向圆周2z =,则1

1C

dz z =-?? ,21(1)C dz z =-?? ,2

(1)

z

C

e dz z =-?? 。

2、设C 为从点1z i =-到点20z =的直线段,则C

zdz =?_______.

3、若C 为正向圆周2z =,则1

C dz z =??________.

4、若22

21

()z z z f dz z ξξ=++=-??,2ξ≠,则(35)f i +=__ ___,(1)f = . (1)f '=

5、(:4)3

z c e dz c z z =-??的值是________ 二、单项选择题

1、若f(z)在D 内解析,()z Φ为f(z)的一个原函数,则( ) A.()()f z z '=Φ B. ()()f z z ''=Φ C.()()z f z 'Φ=

D. ()()z f z ''Φ=

2、下列积分中,积分值不为0的是 ( )

A.3(2)C

z z dz +?? ,12z -= B.z

c

e dz ?? ,2z = C.sin c z dz z ??,1z = D.cos 1c

z

dz z -??,2z = 三、计算题

1、沿下列路径计算积分C

zdz ?

(1) 从原点到3i +的直线段

(2) 从原点沿实轴到3,再从3垂直向上到3i +。

2、沿下列路径计算积分2C

z dz ?

(1)从原点到1i +的直线段

(2)从原点沿实轴到1,再从1垂直向上到1i +。

3、计算0

cos i zdz ?。 4、计算积分30

(23).i

z dz +-?

5、2()C

x y ix dz -+?,其中C 是从点0到1i +的直线段。

6、设C 为从-2到2的上半圆周,计算积分23

C z dz z

-?的值。

7、21

1

C

dz z -?

?,C 为正向圆周2z =

8、计算积分()(4)

C dz

z i z -+??,其中C 为圆周3Z =,且取正向。

9、计算212(1)(2)

C z i

dz z z i ++++?

?,其中C 为正向圆周3z =.

10、求下列积分之值(积分沿闭曲线的正向)

(1) 1

(2)c z dz z z --??,3z = (2) ()(2)

2

c

dz

i

z z -+?

?,1z =

(3) 3cos c z

dz z ??,1z = (4) 3()iz

c e dz z i -??,1z i -=

第七章 傅里叶变换

本章知识点和基本要求

掌握傅氏积分定理、理解傅氏积分公式; 理解傅立叶变换及傅立叶逆变换的概念; 了解δ函数的概念、性质及其傅氏变换, 了解傅氏变换的物理意义;

掌握傅氏变换的性质,熟悉常用傅氏变换对。

一、填空题

1、设50 ,0(),0

t t f t e t -ì

(), t 0t t f t e β-

3、[1]_______F =

4、设1

[()]F f t i αω

=

+,则()f t = ; 5、设2()sin f t t =,则[()]F f t = ;

6、设[()]()F f t F w =,则[(5)()]F t f t += ;

7、设[()]()F f t F ω=,0t 为实常数,则0[()]F f t t -= ; 8、0[()]F t t δ-= ;

9、设[()]()F f t F w =,则(1)f t -的傅氏变换[(1)]F f t -= ; 10、[()]()F f t F ω=,则[()]_______t F f d ττ-∞

=?

11、已知()f t t =,且2

2

[()]F f t ω=-

,则12

2

[](2)

F ω--

=- 二、单项选择题

1、下列变换中,正确的是 ( )

A.[()]1F t δ=

B. [1]()F δω=

C. 1[()]1F δω-=

D. 1[1]()F u t -= 2、设[()]()F f t F ω=,则[(1)()]F t f t -为 ( )

A. ()()iF F ωω'+

B. ()()iF F ωω'-

C. ()()iF F ωω'-+

D. ()()iF F ωω'-- 3、()0t t δ-的傅里叶变换[]0()F t t δ-为 ( )

A .1

B 。0t

C 。0

i t e

ω- D 。0

i t e

ω

4、设[()]()F f t F ω=,则[(23)()]F t f t -= ( )

A.2()3()iF F ωω'-

B. 2()3()iF F ωω'+

C. 2()3()iF F ωω'-+

D. 2()3()iF F ωω'-- 5、设[()]()F f t F ω=,则[(2)()]F t f t -= ( )

A.()2()F F ωω'-

B. ()2()F F ωω'--

C. ()2()iF F ωω'-

D. ()2()iF F ωω'-- 6、设0()cos f t t ω=,则[()]F f t = ( )

A.00[()()]πδωωδωω++-

B. 00[()()]πδωωδωω+--

C.00[()()]i πδωωδωω+--

D. 00[()()]i πδωωδωω++- 7、设0()(2)i t f t t e ωδ=-+,则[()]F f t = ( )

A 202()i e ωπδωω-+-

B 202()i e ωπδωω+-

C 202()i e ωπδωω-++

D 202()i e ωπδωω++ 8、设0()sin f t t ω=,则其傅氏变换[()]F f t = ( )

A.00[()()]δωωδωω+--

B. 00[()()]i πδωωδωω+--

C.00[()()]πδωωδωω+--

D. 00[()()]i πδωωδωω++-

三、计算题

1、已知函数0,12,10()1,020,2t t f t t t -∞<<-??-≤

=?≤

?,求它的傅里叶变换。

2、求函数2,10() 2, 01 0, t f t t --<<=<<ì???

í?????其他的傅里叶变换;

3、求函数0,0

()0t t f t e t β- )的傅氏变换及其积分表达式。

4、求函数 sin ()0t t f t t π

π? ≤?=? ,>??

的傅氏变换,

并证明2

sin ,sin sin 210,t t t d t π

π

ωπωωωπ+∞

?≤?=?-?≥?

?;

5、利用定义或查表求下列函数的傅里叶逆变换

(1)00()[()()]555i F p w w

w d w d w =+--

(2)00()[()()]555

F p w w

w d w d w =++-

6、用傅里叶变换求解下面的微分方程

()()(),x t x t t t δ'+=-∞<<+∞

7、设[()]()F f t F ω=,列表给出下列函数的付里叶变换:

2

00'(),"(),(),(),(),()f t f t tf t t f t f t t f t t -+,(),()t

f d f at ττ-∞

?

001,(),(),(),t t t t t δδδ-+ 0,0

(),0t t f t e t β-

并证明付里叶变换的微分性质和位移性质。

第八章 拉普拉斯变换

本章知识点和基本要求

理解拉普拉斯变换及拉普拉斯逆变换的概念; 了解拉普拉斯变换存在定理; 掌握拉普拉斯变换的性质; 掌握用留数求拉氏逆变换的方法; 了解拉氏变换卷积概念及卷积定理;

应用拉氏变换求解常微分方程及常微分方程组。

一、填空题

1、设2

1()F S S

=

,则[()]S

L e F S --1= 2、[(sin 3)]L t ¢

= 3、[sin ]t L e t =

4、设()(35)f t u t =-,3[()]t L e f t -=

5、[cos ]t L e t =

6、设2

2[()],4

L f t s =

+ 则 3[()]t

L e f t -= 7、设2()(1)t f t t e =-,[()]L f t = 8、设 22

1()(1)

F s s =

+,则1

[()]L F s -= 9、设11[()](), L f t F S =22[()]()F f t F S =,则12[()*()]L f t f t = 10、设 2

2()16

s F s s +=

+,则1

[()]L F s -= 二、单项选择题

1、下列变换中,不正确的是 ( )

A.[()]1F t δ=

B.[()]1L t δ=

C. [1]()L t δ=

D. [1]2()F πδω= 2、设[()]()L f t F s =,其中正确的是( )

A .[()]()L f t sF s '=

B 。[()]()at L e f t F s a =+

C .1

[()]()L f at F s a

=

D 。[()]()at L e f t F s a =- 3、()(0)at f t te a => 的拉氏变换为 ( ) A.

1S a - B.1

s a

-- C.21()s a - D.2

1()s a -- 4、若2

1()1

S

F S e S -=

+,则1[()]L F S -= ( ) A.sin(1)t - B.(1)sin u t t - C. ()sin(1)u t t - D.(1)sin(1)u t t -- 5、设2()cos3t f t e t -= 则[()]L f t = ( ) A.

23(2)9S ++ B.2

2

(2)9S S +++ C.

23(2)9S S ++ D. 23(2)

(2)9

S S +++

6、函数2

2()1

s F s s =+ 的拉氏逆变换为 ( )

A.()cos t t δ

B.()cos t t δ-

C.()(1sin )t t δ-

D.()sin t t δ-

7、设()(2)

S

e F s S S -=+,则1[()]L F S -= ( )

A 2(1)(1)t e u t ---

B 2(1)(1)(1)t u t e u t ----- C

2(1)1[1](1)2t e u t ---- D 2(1)1

[()(1)]2

t u t e u t ---- 三、计算题

1、利用定义或查表求下列函数的拉普拉斯变换。

(1) 2()sin at f t e t = (2)2()cos 5

t

f t =

(3)()2sin sin f t at t at =- (4) 25()3t t f t e e =+

(5)2()5()t f t e t δ=+ (6)2()t t f t e te =+

(7)()sin at f t e t = (8)2()sin 2f t t =

(9) ()cos f t t at = (10)2()(2)t f t t e =-

(11)()sin (2)f t t u t =?- (12)()sin(2) f t t =-

(13)()sin 2t f t te t -= (14)()sin(2)(2)f t t u t =-?-

2、已知10 ,0

()sin ,0

t f t t t ì

?3??,20 ,0()cos ,0

t f t t t ì

?3??,求1()f t 与2()f t 的卷积12()()f t f t *.

3、用定义或查表求下列函数的拉普拉斯逆变换。

(1) 1

()(1)

F S S S =+ (2) 52()1S e F S S -=+

(3) ()()()b F S S a S b =-- (4)1

()(1)

F S S S =-

(5)3

()(2)(3)S F S S S +=+- (6)22()1

S F S S =+

(7)223

()9

s F s s +=+ (8)4()(4)(2)F s s s =++

(9)22

21

()(1)s s F s s s +-=-

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