電大復(fù)變函數(shù)與積分變換(修訂版復(fù)旦大學(xué))課后的習(xí)題答案小抄參考

1、習(xí)題 七1.證明：如果f(t)滿足傅里葉變換的條件，當(dāng)f(t)為奇函數(shù)時，則有其中當(dāng)f(t)為偶函數(shù)時，則有其中證明：因為其中為f(t)的傅里葉變換當(dāng)f(t)為奇函數(shù)時，為奇函數(shù)，從而為偶函數(shù)，從而故 有為奇數(shù)。 =所以，當(dāng)f(t)為奇函數(shù)時，有同理，當(dāng)f(t)為偶函數(shù)時，有.其中 2.在上一題中，設(shè).計算的值.解:3.計算函數(shù).解:4.求下列函數(shù)的傅里葉變換 解：(2)解：因為所以根據(jù)傅里葉變換的微分性質(zhì)可得(3)解：(4)解:令，則在上半平面有兩個一級極點.故.(5) 解:同(4).利用留數(shù)在積分中的應(yīng)用,令則.5.設(shè)函數(shù)f(t)是解析函數(shù)，而且在帶形區(qū)域內(nèi)有界.定義函數(shù)為證明當(dāng)時，有 對

2、所有的實數(shù)t成立.(書上有推理過程)6.求符號函數(shù) 的傅里葉變換.解:因為把函數(shù).不難看出 故：7.已知函數(shù)的傅里葉變換求解:8.設(shè)函數(shù)f(t)的傅里葉變換，a為一常數(shù). 證明f當(dāng)a0時,令u=at.則當(dāng)a0時,令u=at,則.故原命題成立.9.設(shè)證明.證明：10.設(shè)，證明：以及證明：同理：11.設(shè)計算.解：當(dāng)時，若則故=0.若則若則故12.設(shè)為單位階躍函數(shù)，求下列函數(shù)的傅里葉變換.習(xí)題八1.求下列函數(shù)的拉普拉斯變換.(1)， (2)， (3)(4)， (5) 解: (1) (2) (3) (4) (5) 2.求下列函數(shù)的拉普拉斯變換.（1） (2)解: (1) (2) 3.設(shè)函數(shù),其中函數(shù)為

3、階躍函數(shù), 求的拉普拉斯變換.解: 4.求圖8.5所表示的周期函數(shù)的拉普拉斯變換解：5. 求下列函數(shù)的拉普拉斯變換.(1) (2) (3) (4) (5 (6 (7) (8) 解:(1) (2)(4)(5) (6)(7)(8)6.記，對常數(shù)，若，證明證明: 7 記，證明:證明：當(dāng)n=1時,所以，當(dāng)n=1時, 顯然成立。假設(shè)，當(dāng)n=k-1時, 有現(xiàn)證當(dāng)n=k時8. 記，如果a為常數(shù),證明:證明：設(shè)，由定義9. 記，證明:,即證明：10.計算下列函數(shù)的卷積(1) (2) (3) (4) (5) (6 解：(1) (2) (3) (4)(5) (6)11.設(shè)函數(shù)f, g, h均滿足當(dāng)t0時恒為零，證

4、明以及證明：12.利用卷積定理證明證明：設(shè)，則，則，所以13. 求下列函數(shù)的拉普拉斯逆變換.(1) (2) (3)(4) (5) (6 解：(1)(2)(3故(4)因為所以(5)其中所以(6)所以14.利用卷積定理證明證明：又因為所以，根據(jù)卷積定理15.利用卷積定理證明證明：因為所以，根據(jù)卷積定理有16. 求下列函數(shù)的拉普拉斯逆變換.(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)設(shè)方程兩邊取拉氏變換,得為y(s)的三個一級極點,則待添加的隱藏文字內(nèi)容1(2) 方程兩邊同時取拉氏變換,得(3)方

5、程兩邊取拉氏變換,得因為由拉氏變換的微分性質(zhì)知，若lf(t)=f(s),則即因為所以故有(4)方程兩邊取拉氏變換,設(shè)ly(t)=y(s),得故(5)設(shè)ly(t)=y(s),則方程兩邊取拉氏變換,得故18.求下列微分方程組的解(1) (2) 解:(1) 設(shè)微分方程組兩式的兩邊同時取拉氏變換,得得(2)代入(1)，得(3)代入(1)，得(2)設(shè) 方程兩邊取拉氏變換,得(3)代入(1)：所以故19.求下列方程的解(1) (2) 解：(1)設(shè)lx(t)=x(s), 方程兩邊取拉氏變換,得(2)設(shè)ly(t)=y(s), 方程兩邊取拉氏變換,得if we dont do that it will go o

6、n and go on. 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

7、has been exaggerated 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 i

8、t is against 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.

9、the (lack of) 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 m

10、eet yaya toure, 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 gov

11、erning body, 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 supersoni

12、c descent could 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 le

13、vel of the 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

14、 boundary layer (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

15、pool that he 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

16、- 690 mph (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

17、deployed to 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