电大复变函数与积分变换(修订版复旦大学)课后的习题答案小抄参考

1、习题 七1.证明：如果f(t)满足傅里叶变换的条件，当f(t)为奇函数时，则有其中当f(t)为偶函数时，则有其中证明：因为其中为f(t)的傅里叶变换当f(t)为奇函数时，为奇函数，从而为偶函数，从而故 有为奇数。 =所以，当f(t)为奇函数时，有同理，当f(t)为偶函数时，有.其中 2.在上一题中，设.计算的值.解:3.计算函数.解:4.求下列函数的傅里叶变换 解：(2)解：因为所以根据傅里叶变换的微分性质可得(3)解：(4)解:令，则在上半平面有两个一级极点.故.(5) 解:同(4).利用留数在积分中的应用,令则.5.设函数f(t)是解析函数，而且在带形区域内有界.定义函数为证明当时，有 对

2、所有的实数t成立.(书上有推理过程)6.求符号函数 的傅里叶变换.解:因为把函数.不难看出 故：7.已知函数的傅里叶变换求解:8.设函数f(t)的傅里叶变换，a为一常数. 证明f当a0时,令u=at.则当a0时,令u=at,则.故原命题成立.9.设证明.证明：10.设，证明：以及证明：同理：11.设计算.解：当时，若则故=0.若则若则故12.设为单位阶跃函数，求下列函数的傅里叶变换.习题八1.求下列函数的拉普拉斯变换.(1)， (2)， (3)(4)， (5) 解: (1) (2) (3) (4) (5) 2.求下列函数的拉普拉斯变换.（1） (2)解: (1) (2) 3.设函数,其中函数为

3、阶跃函数, 求的拉普拉斯变换.解: 4.求图8.5所表示的周期函数的拉普拉斯变换解：5. 求下列函数的拉普拉斯变换.(1) (2) (3) (4) (5 (6 (7) (8) 解:(1) (2)(4)(5) (6)(7)(8)6.记，对常数，若，证明证明: 7 记，证明:证明：当n=1时,所以，当n=1时, 显然成立。假设，当n=k-1时, 有现证当n=k时8. 记，如果a为常数,证明:证明：设，由定义9. 记，证明:,即证明：10.计算下列函数的卷积(1) (2) (3) (4) (5) (6 解：(1) (2) (3) (4)(5) (6)11.设函数f, g, h均满足当t0时恒为零，证

4、明以及证明：12.利用卷积定理证明证明：设，则，则，所以13. 求下列函数的拉普拉斯逆变换.(1) (2) (3)(4) (5) (6 解：(1)(2)(3故(4)因为所以(5)其中所以(6)所以14.利用卷积定理证明证明：又因为所以，根据卷积定理15.利用卷积定理证明证明：因为所以，根据卷积定理有16. 求下列函数的拉普拉斯逆变换.(1) (2) (3)(4)解:(1) 故(2)：(3)故(4)故且所以17.求下列微分方程的解(1) (2) (3) (4) (5) 解: (1)设方程两边取拉氏变换,得为y(s)的三个一级极点,则(2) 方程两边同时取拉氏变换,得(3)方程两边取拉氏变换,得因

5、为由拉氏变换的微分性质知，若lf(t)=f(s),则即因为所以故有(4)方程两边取拉氏变换,设ly(t)=y(s),得故(5)设ly(t)=y(s),则方程两边取拉氏变换,得故18.求下列微分方程组的解(1) (2) 解:(1) 设微分方程组两式的两边同时取拉氏变换,得得(2)代入(1)，得(3)代入(1)，得(2)设 方程两边取拉氏变换,得(3)代入(1)：所以故19.求下列方程的解(1) (2) 解：(1)设lx(t)=x(s), 方程两边取拉氏变换,得(2)设ly(t)=y(s), 方程两边取拉氏变换,得if we dont do that it will go on and go on

6、. we have to stop it; we need the courage to do it.his comments came hours after fifa vice-president jeffrey webb - also in london for the fas celebrations - said he wanted to meet ivory coast international toure to discuss his complaint.cska general director roman babaev says the matter has been ex

7、aggerated by the ivorian and the british media.blatter, 77, said: it has been decided by the fifa congress that it is a nonsense for racism to be dealt with with fines. you can always find money from somebody to pay them.it is a nonsense to have matches played without spectators because it is agains

8、t the spirit of football and against the visiting team. it is all nonsense.we can do something better to fight racism and discrimination.this is one of the villains we have today in our game. but it is only with harsh sanctions that racism and discrimination can be washed out of football.the (lack o

9、f) air up there watch mcayman islands-based webb, the head of fifas anti-racism taskforce, is in london for the football associations 150th anniversary celebrations and will attend citys premier league match at chelsea on sunday.i am going to be at the match tomorrow and i have asked to meet yaya to

10、ure, he told bbc sport.for me its about how he felt and i would like to speak to him first to find out what his experience was.uefa hasopened disciplinary proceedings against cskafor the racist behaviour of their fans duringcitys 2-1 win.michel platini, president of european footballs governing body

11、, has also ordered an immediate investigation into the referees actions.cska said they were surprised and disappointed by toures complaint. in a statement the russian side added: we found no racist insults from fans of cska.baumgartner the disappointing news: mission aborted.the supersonic descent c

12、ould happen as early as sunda.the weather plays an important role in this mission. starting at the ground, conditions have to be very calm - winds less than 2 mph, with no precipitation or humidity and limited cloud cover. the balloon, with capsule attached, will move through the lower level of the

13、atmosphere (the troposphere) where our day-to-day weather lives. it will climb higher than the tip of mount everest (5.5 miles/8.85 kilometers), drifting even higher than the cruising altitude of commercial airliners (5.6 miles/9.17 kilometers) and into the stratosphere. as he crosses the boundary l

14、ayer (called the tropopause),e can expect a lot of turbulence.the balloon will slowly drift to the edge of space at 120,000 feet ( then, i would assume, he will slowly step out onto something resembling an olympic diving platform.below, the earth becomes the concrete bottom of a swimming pool that h

15、e wants to land on, but not too hard. still, hell be traveling fast, so despite the distance, it will not be like diving into the deep end of a pool. it will be like he is diving into the shallow end.skydiver preps for the big jumpwhen he jumps, he is expected to reach the speed of sound - 690 mph (

16、1,110 kph) - in less than 40 seconds. like hitting the top of the water, he will begin to slow as he approaches the more dense air closer to earth. but this will not be enough to stop him completely.if he goes too fast or spins out of control, he has a stabilization parachute that can be deployed to

17、 slow him down. his team hopes its not needed. instead, he plans to deploy his 270-square-foot (25-square-meter) main chute at an altitude of around 5,000 feet (1,524 meters).in order to deploy this chute successfully, he will have to slow to 172 mph (277 kph). he will have a reserve parachute that will open autom