最新2016年電大復(fù)變函數(shù)形成性考核冊(cè)參考答案

1、;.電大復(fù)變函數(shù)習(xí)題總匯與參考答案第1章 復(fù)數(shù)與復(fù)變函數(shù)一、單項(xiàng)選擇題1、若Z1=（a, b）,Z2=(c, d),則Z1Z2=（C）A （ac+bd, a） B (ac-bd, b)C （ac-bd, ac+bd） D (ac+bd, bc-ad)2、若R0，則N（，R）= z：（D）A |z|R B 0|z|RC R|z|R3、若z=x+iy, 則y=(D)A B C D4、若A= ，則 |A|=（C）A 3 B 0 C 1 D 2二、填空題1、若z=x+iy, w=z2=u+iv, 則v=（ 2xy ）2、復(fù)平面上滿足Rez=4的點(diǎn)集為（ z=x+iy|x=4 ）3、（ 設(shè)E為點(diǎn)集，若它

2、是開集，且是連通的，則E ）稱為區(qū)域。4、設(shè)z0=x0+iy0, zn=xn+iyn(n=1,2,)，則zn以zo為極限的充分必要條件是 xn=x0，且 yn=y0。三、計(jì)算題1、求復(fù)數(shù)-1-i的實(shí)部、虛部、模與主輻角。解：Re(-1-i)=-1 Im(-1-i)=-1|-1-i|=2、寫出復(fù)數(shù)-i的三角式。解：3、寫出復(fù)數(shù) 的代數(shù)式。解：4、求根式 的值。解：四、證明題1、證明若 ，則a2+b2=1。證明：而 3、證明：證明：第2章 解析函數(shù)一、單項(xiàng)選擇題1若f(z)= x2-y2+2xyi,則2、若f(z)=u(x, y)+iv(x,y), 則柯西黎曼條件為（D）A BC D3、若f(z)

3、=z+1, 則f(z)在復(fù)平面上（C）A 僅在點(diǎn)z=0解析 B 無(wú)處解析C 處處解析 D 在z=0不解析且在z0解析4、若f（z）在復(fù)平面解析，g(z)在復(fù)平面上連續(xù)，則f(z)+g(z)在復(fù)平面上（C）A解析 B 可導(dǎo)C連續(xù) D 不連續(xù)二、填空題1、若f(z)在點(diǎn)a不解析，則稱a為f(z)的奇點(diǎn)。2、若f(z)在點(diǎn)z=1的鄰域可導(dǎo)，則f(z)在點(diǎn)z=1解析。3、若f(z)=z2+2z+1，則 4、若 ，則 不存在。三、計(jì)算題：1、設(shè)f(z)=zRe(z), 求解： =2、設(shè)f(z)=excosy+iexsiny,求解：f(z)=excosy+iexsiny=ez,z=x+iyu=excosy

4、 v=exsinyf(z)=u+ivf(z)在復(fù)平面解析，且 =excosy+iexsiny3、設(shè)f(z)=u+iv在區(qū)域G內(nèi)為解析函數(shù)，且滿足u=x3-3xy2，f(i)=0,試求f(z)。解：依C-R條件有Vy=ux=3x2-3y2則V（x1y）=3x2y-y3+c(c為常數(shù))故f(z)=x3-3xy2+i(3x2y-y3+c)=x3-3xy2+i(cx2y-y3)+ic =z3+ic，為使f(i)=0, 當(dāng)x=0,y=1時(shí)，f(i)=0, 有f(0)=-i+ic=0c=1 f(z)=Z3+i4、設(shè)f(z)=u+iv在區(qū)域G內(nèi)為解析函數(shù)，且滿足u=2(x-1)y,f(2)=-i,試求f(z

5、)。解：依C-R條件有Vy=ux=2yV= =y2+(x) Vx=(x)=V=y2-x2+2x+c(c為常數(shù))f(z)=2(x-1)y+i(y2-x2+2x+c)為使f(z)=-i,當(dāng)x=2 y=0時(shí),f(2)=ci=-i c=-1f(z)=2(x-1)y+i(y2-x2+2x-1) =-(z-1)2i四、證明題1、試在復(fù)平面討論f(z)=iz的解析性。解：令f(z)=u+iv z=x+iy則iz=i(x+iy)=-y+ixu=-y v=x于是ux=0 uy=-1Vx=1 Vy=0ux、uy、vx在復(fù)平面內(nèi)處處連接又Ux=Vy Uy=-Vx。f(z)=iz在復(fù)平面解析。2、試證：若函數(shù)f(z)

6、在區(qū)域G內(nèi)為解析函數(shù)，且滿足條件（z）=0，zG，則f(z)在G內(nèi)為常數(shù)。證：設(shè)f(z)=u+iv,z=x+iy,zGf(z)在G內(nèi)解析，Ux=Vy, Uy=-Vx又（z）=0, （z）=Ux+iVxUx=0 Vx=0Uy=-Vx=0 Ux=Vy=0U為實(shí)常數(shù)C1，V也為實(shí)常數(shù)C2，f(z)=C1+iC2=Z0f(z)在G內(nèi)為常數(shù)。復(fù)變函數(shù)課程作業(yè)參考解答2第3章 初等函數(shù)一、單項(xiàng)選擇題1. z = ( A ) 是根式函數(shù)的支點(diǎn). (A) 0 (B) 1 (C) (D) i2. z = ( D ) 是函數(shù)的支點(diǎn). (A) i (B) 2i (C) -1 (D) 03. ei =( B ). (

7、A) e-1+e (B) cos1+isin1 (C) sin1 (D) cos14. sin1= ( A ) (A) (B) (C) (D) 二、填空題1. cosi = 2. = e(cos1+isin1)3. lni =4. ln(1+i) = k為整數(shù).三、計(jì)算題1. 設(shè)z=x+iy，計(jì)算.解: = = 2. 設(shè)z = x+iy, 計(jì)算. 解: z = x+iy 3. 求方程的解.解: lnz = 由對(duì)數(shù)函數(shù)的定義有: Z= 所給方程的解為z = i4. 求方程的解.解: =根據(jù)指數(shù)函數(shù)的定義有:z=n2+i 或z=n(1+)四、證明題1. 試證: . 證明：根據(jù)正弦函數(shù)及余弦正數(shù)定義

8、有： sin2z=2sinzcosz2. 證明: . 證明: 令A(yù)= B=sinx+sin2x+sinnx = 第4章 解析函數(shù)的積分理論一、單項(xiàng)選擇題1. ( D ) , c為起點(diǎn)在0 , 終點(diǎn)在1+i的直線段. (A) 0 (B) 1 (C) 2i (D) 2(1+i)2. . (A) 0 (B) 10 (C) i (D) 3. (A) i (B) 10 (C) 10i (D) 04. =( A ). (A) (B) (C) (D) 二、填空題1. 若與沿曲線c可積，則.2. 設(shè)L為曲線c的長(zhǎng)度, 若f(z)沿c可積, 且在c上滿足,則.3. 4. 三、計(jì)算題1.計(jì)算積分,其中c為自0到2

9、+i的直線段. 解: c的方程為: 其次由得 = =2. 計(jì)算積分. 解: = 作區(qū)域D:積分途徑在D內(nèi)被積函數(shù)的奇點(diǎn)Z=2與Z=3均不在D內(nèi),所以被積函數(shù)在D內(nèi)解析.由定理4.2得:=03. 計(jì)算積分. 解: 奇點(diǎn)z=1和z=-1不在區(qū)域D,內(nèi) 的三個(gè)根也不在D內(nèi) 由定理4.2 得 =04. 計(jì)算積分, . 解: 由定理4.6得 四、證明題1. 計(jì)算積分，并由此證明. 證明：在圓域 |z|1內(nèi)解析 = 另一方面,在圓|z|= =(實(shí)部和虛部為0) = = = = =0 而為偶函數(shù)0= = 復(fù)變函數(shù)課程作業(yè)參考解答3第5章 解析函數(shù)的冪級(jí)數(shù)表示一、單項(xiàng)選擇題1. 冪級(jí)數(shù)的收斂半徑等于( B )

10、 ( A ) 0 (B) 1 ( C ) 2 (D) 32. 點(diǎn)z=-1是f(z)=r ( B )級(jí)零點(diǎn). ( A ) 1 (B)2 (C)3 (D)53. 級(jí)數(shù)的收斂圓為( D ). (A) | z-1| 3 (B) |z|1 (D) |z| 14. 設(shè)f(z)在點(diǎn)a解析, 點(diǎn)b是f(z)的奇點(diǎn)中離點(diǎn)a最近的奇點(diǎn),于是,使f(z)=成立的收斂圓的半徑等于( C ). (A) a+b+1 (B) b-a+1(C) |a-b| (D) |a+b|二、填空題1.級(jí)數(shù)1+z+的收斂圓R=+即整個(gè)復(fù)平面2.若f(z)= (k為常數(shù)),則z=m(m=0, )為f(z)的 1 級(jí)零點(diǎn). 3.冪有數(shù)的收斂半

11、徑等于 0 . 4.z=0是f(z)=ez-1的 1 級(jí)零點(diǎn). 三、計(jì)算題 1.將函數(shù)f(z)=在點(diǎn)z=0展開冪級(jí)數(shù). 解: f(z)= =- 2.將函數(shù)f(z)=(1-z)-2在點(diǎn)z=0展開成冪級(jí)數(shù). 解：而(1-z)-1= = 3將函數(shù)f(z)=(z+2)-1在點(diǎn)z=1展開成冪級(jí)數(shù). 解：f(z)=(z+2)-1= = 4將函數(shù)f(z)=ez在點(diǎn)z=1展開成冪級(jí)數(shù). 解： f(z)=ez f(n)=ez 四、證明題 1證明:1-ei2z=-2isinzeiz 證：eiz=cosz+isinze-iz=cos-isinz eiz-e-iz=2isinz -2isinz=-( eiz-e-iz

12、) = eiz-e-iz -2isinz eiz=( e-iz- eiz) eiz =e0- e2iz=1- e2iz2試用解析函數(shù)的唯一性定理證明等式: cos2z= cos2z-sin2z 證f1(z)=cos2z,則f1(z)復(fù)平面G解析設(shè)f2(z)coszsin2，則f2(z)也在整個(gè)復(fù)平面G解析取E=K為實(shí)數(shù)軸,則E在G內(nèi)有聚點(diǎn).當(dāng)E為實(shí)數(shù)時(shí),知cos2z=cos2z-sin2z,即f1(z)= f2(z)由解析函數(shù)唯一性定理,由以上三條知f1(z)= f2(z) 成立即cos2z= cos2z-sin2z 第6章 解析函數(shù)的羅朗級(jí)數(shù)表示 一、單項(xiàng)選擇題 1函數(shù)f(z)=在點(diǎn)z=2的

13、去心鄰域( D ) 內(nèi)可展成羅朗級(jí)數(shù). (A) 0 (B) 0 (C) 1 (D) 0 2設(shè)點(diǎn)為f(z)的孤立奇點(diǎn)，若=c，則點(diǎn)為f(z)的( C ). (A) 本性奇點(diǎn) (B) 極點(diǎn) (C) 可去奇點(diǎn) (D) 解析點(diǎn) 3若點(diǎn)為函數(shù)f(z)的孤立奇點(diǎn)，則點(diǎn)為f(z)的極點(diǎn)的充分必要條件是( D ). (A) f(z)=c() (B) f(z)= (C) f(z)=c() (D) f(z)= 4若點(diǎn)為函數(shù)f(z)的孤立奇點(diǎn)，則點(diǎn)為f(z)的本性奇點(diǎn)的充要條件是（ B ）. (A) f(z)= c() (B) f(z)不存在 (C) f(z)=c() (D) f(Z)= 二、填空題 1設(shè)為函數(shù)f(

14、z)在點(diǎn)的羅朗級(jí)數(shù),稱為該級(jí)數(shù)的主要部分. 2.設(shè)點(diǎn)為函數(shù)f(z)的奇點(diǎn),若f(z)在點(diǎn)的某個(gè) 某個(gè)去心鄰域內(nèi)解析,則稱點(diǎn)為f(z)的孤立奇點(diǎn). 3.若f(z)=，則點(diǎn)z=0為f(z)的 0 級(jí)極點(diǎn). 不是極點(diǎn),若f(z)= 則z=0為f(z)的一個(gè)極點(diǎn). 4.若f(z)=(sin)-1,則點(diǎn)z0為f(z)非孤立 奇點(diǎn). 三、計(jì)算題1將函數(shù)f(z)=(z-2)-1在點(diǎn)z=0的去心鄰域展成羅朗級(jí)數(shù).解: f(z)= = - = - 2將函數(shù)f(z)在點(diǎn)的去心鄰域展成羅朗級(jí)數(shù). 解: f(z)= 3試求函數(shù)f(z)=z-3sinz3的有限奇點(diǎn),并判定奇點(diǎn)的類別. 解: 解析,無(wú)奇點(diǎn),f(z)的有限

15、奇點(diǎn)為z=0. 并且為3階極點(diǎn). 4試求函數(shù)f(z)=z-1的有限奇點(diǎn)，并判定奇點(diǎn)的類別. 解: f(z)的m階奇點(diǎn)即的階零點(diǎn),而零點(diǎn)為z=0，z=1，z=-1，且均為1階零點(diǎn)。的有限奇點(diǎn)為z=0，z=1,z=-1且均為1階極點(diǎn). 四、證明題 1設(shè)f(z)=,試證z=0為f(z)的6級(jí)極點(diǎn). 證:要證z=0為f(z)的6級(jí)極點(diǎn),只需證z=0為的6階零點(diǎn)即可.而 =8z3 =8z6 令 則 為的6階零點(diǎn) z=0 為f(z)的6級(jí)極點(diǎn).請(qǐng)您刪除一下內(nèi)容，O(_)O謝謝！2016年中央電大期末復(fù)習(xí)考試小抄大全，電大期末考試必備小抄，電大考試必過(guò)小抄Basketball can make a true

16、 claim to being the only major sport that is an American invention. From high school to the professional level, basketball attracts a large following for live games as well as television coverage of events like the National Collegiate Athletic Association (NCAA) annual tournament and the National Ba

17、sketball Association (NBA) and Womens National Basketball Association (WNBA) playoffs. And it has also made American heroes out of its player and coach legends like Michael Jordan, Larry Bird, Earvin Magic Johnson, Sheryl Swoopes, and other great players. At the heart of the game is the playing spac

18、e and the equipment. The space is a rectangular, indoor court. The principal pieces of equipment are the two elevated baskets, one at each end (in the long direction) of the court, and the basketball itself. The ball is spherical in shape and is inflated. Basket-balls range in size from 28.5-30 in (

19、72-76 cm) in circumference, and in weight from 18-22 oz (510-624 g). For players below the high school level, a smaller ball is used, but the ball in mens games measures 29.5-30 in (75-76 cm) in circumference, and a womens ball is 28.5-29 in (72-74 cm) in circumference. The covering of the ball is l

20、eather, rubber, composition, or synthetic, although leather covers only are dictated by rules for college play, unless the teams agree otherwise. Orange is the regulation color. At all levels of play, the home team provides the ball. Inflation of the ball is based on the height of the balls bounce.

21、Inside the covering or casing, a rubber bladder holds air. The ball must be inflated to a pressure sufficient to make it rebound to a height (measured to the top of the ball) of 49-54 in (1.2-1.4 m) when it is dropped on a solid wooden floor from a starting height of 6 ft (1.80 m) measured from the

22、bottom of the ball. The factory must test the balls, and the air pressure that makes the ball legal in keeping with the bounce test is stamped on the ball. During the intensity of high school and college tourneys and the professional playoffs, this inflated sphere commands considerable attention. Ba

23、sketball is one of few sports with a known date of birth. On December 1, 1891, in Springfield, Massachusetts, James Naismith hung two half-bushel peach baskets at the opposite ends of a gymnasium and out-lined 13 rules based on five principles to his students at the International Training School of

24、the Young Mens Christian Association (YMCA), which later became Springfield College. Naismith (1861-1939) was a physical education teacher who was seeking a team sport with limited physical contact but a lot of running, jumping, shooting, and the hand-eye coordination required in handling a ball. Th

25、e peach baskets he hung as goals gave the sport the name of basketball. His students were excited about the game, and Christmas vacation gave them the chance to tell their friends and people at their local YMCAs about the game. The association leaders wrote to Naismith asking for copies of the rules

26、, and they were published in the Triangle, the school newspaper, on January 15,1892. Naismiths five basic principles center on the ball, which was described as large, light, and handled with the hands. Players could not move the ball by running alone, and none of the players was restricted against h

27、andling the ball. The playing area was also open to all players, but there was to be no physical contact between players; the ball was the objective. To score, the ball had to be shot through a horizontal, elevated goal. The team with the most points at the end of an allotted time period wins. Early

28、 in the history of basketball, the local YMCAs provided the gymnasiums, and membership in the organization grew rapidly. The size of the local gym dictated the number of players; smaller gyms used five players on a side, and the larger gyms allowed seven to nine. The team size became generally estab

29、lished as five in 1895, and, in 1897, this was made formal in the rules. The YMCA lost interest in supporting the game because 10-20 basketball players monopolized a gymnasium previously used by many more in a variety of activities. YMCA membership dropped, and basketball enthusiasts played in local

30、 halls. This led to the building of basketball gymnasiums at schools and colleges and also to the formation of professional leagues. Although basketball was born in the United States, five of Naismiths original players were Canadians, and the game spread to Canada immediately. It was played in Franc

31、e by 1893; England in 1894; Australia, China, and India between 1895 and 1900; and Japan in 1900. From 1891 through 1893, a soccer ball was used to play basketball. The first basketball was manufactured in 1894. It was 32 in (81 cm) in circumference, or about 4 in (10 cm) larger than a soccer ball.

32、The dedicated basketball was made of laced leather and weighed less than 20 oz (567 g). The first molded ball that eliminated the need for laces was introduced in 1948; its construction and size of 30 in (76 cm) were ruled official in 1949. The rule-setters came from several groups early in the 1900

33、s. Colleges and universities established their rules committees in 1905, the YMCA and the Amateur Athletic Union (AAU) created a set of rules jointly, state militia groups abided by a shared set of rules, and there were two professional sets of rules. A Joint Rules Committee for colleges, the AAU, a

34、nd the YMCA was created in 1915, and, under the name the National Basketball Committee (NBC) made rules for amateur play until 1979. In that year, the National Federation of State High School Associations began governing the sport at the high school level, and the NCAA Rules Committee assumed rule-m

35、aking responsibilities for junior colleges, colleges, and the Armed Forces, with a similar committee holding jurisdiction over womens basketball. Until World War II, basketball became increasingly popular in the United States especially at the high school and college levels. After World War II, its

36、popularity grew around the world. In the 1980s, interest in the game truly exploded because of television exposure. Broadcast of the NCAA Championship Games began in 1963, and, by the 1980s, cable television was carrying regular season college games and even high school championships in some states.

37、 Players like Bill Russell, Wilt Chamberlain, and Lew Alcindor (Kareem Abdul-Jabbar) became nationally famous at the college level and carried their fans along in their professional basketball careers. The womens game changed radically in 1971 when separate rules for women were modified to more clos

38、ely resemble the mens game. Television interest followed the women as well with broadcast of NCAA championship tourneys beginning in the early 1980s and the formation of the WNBA in 1997. Internationally, Italy has probably become the leading basketball nation outside of the United States, with nati

39、onal, corporate, and professional teams. The Olympics boosts basketball internationally and has also spurred the womens game by recognizing it as an Olympic event in 1976. Again, television coverage of the Olympics has been exceptionally important in drawing attention to international teams. The fir

40、st professional mens basketball league in the United States was the National Basketball League (NBL), which debuted in 1898. Players were paid on a per-game basis, and this league and others were hurt by the poor quality of games and the ever-changing players on a team. After the Great Depression, a

41、 new NBL was organized in 1937, and the Basketball Association of America was organized in 1946. The two leagues came to agree that players had to be assigned to teams on a contract basis and that high standards had to govern the game; under these premises, the two joined to form the National Basket

42、ball Association (NBA) in 1949. A rival American Basketball Association (ABA) was inaugurated in 1967 and challenged the NBA for college talent and market share for almost ten years. In 1976, this league disbanded, but four of its teams remained as NBA teams. Unification came just in time for major

43、television support. Several womens professional leagues were attempted and failed, including the Womens Professional Basketball League (WBL) and the Womens World Basketball Association, before the WNBA debuted in 1997 with the support of the NBA. James Naismith, originally from Al-monte, Ontario, in

44、vented basketball at the International YMCA Training School in Springfield, Massachusetts, in 1891. The game was first played with peach baskets (hence the name) and a soccer ball and was intended to provide indoor exercise for football players. As a result, it was originally a rough sport. Although

45、 ten of Naismiths original thirteen rules remain, the game soon changed considerably, and the founder had little to do with its evolution. The first intercollegiate game was played in Minnesota in 1895, with nine players to a side and a final score of nine to three. A year later, the first five-man

46、teams played at the University of Chicago. Baskets were now constructed of twine nets but it was not until 1906 that the bottom of the nets were open. In 1897, the dribble was first used, field goals became two points, foul shots one point, and the first professional game was played. A year later, t

47、he first professional league was started, in the East, while in 1900, the first intercollegiate league began. In 1910, in order to limit rough play, it was agreed that four fouls would disqualify players, and glass backboards were used for the first time. Nonetheless, many rules still differed, depe

48、nding upon where the games were played and whether professionals, collegians, or YMCA players were involved. College basketball was played from Texas to Wisconsin and throughout the East through the 1920s, but most teams played only in their own regions, which prevented a national game or audience f

49、rom developing. Professional basketball was played almost exclusively in the East before the 1920s, except when a team would barnstorm into the Midwest to play local teams, often after a league had folded. Before the 1930s very few games, either professional or amateur, were played in facilities sui

50、table for basketball or with a perfectly round ball. Some were played in arenas with chicken wire separating the players from fans, thus the word cagers, others with posts in the middle of the floor and often with balconies overhanging the corners, limiting the areas from which shots could be taken.

51、 Until the late 1930s, all players used the two-hand set shot, and scores remained low. Basketball in the 1920s and 1930s became both more organized and more popular, although it still lagged far behind both baseball and college football. In the pros, five urban, ethnic teams excelled and played wit

52、h almost no college graduates. They were the New York Original Celtics; the Cleveland Rosenblums, owned by Max Rosenblum; Eddie Gottliebs Philadelphia SPHAs (South Philadelphia Hebrew Association); and two great black teams, the New York Renaissance Five and Abe Sapersteins Harlem Globetrotters, whi

53、ch was actually from Chicago. While these teams had some notable players, no superstars, such as Babe Ruth, Jack Dempsey, or Red Grange, emerged to capture the publics attention as they did in other sports of the period. The same was true in college basketball up until the late 1930s, with coaches dominating the game and its development. Walter Doc Meanwell at Wisconsin, Forrest Phog Allen at Kansas, Ward Piggy Lambert at Purdue, and Henry Doc