电大复变函数形成性考核册参考答案.doc

电大复变函数习题总汇与参考答案第1章 复数与复变函数一、单项选择题1、若Z1=（a, b）,Z2=(c, d),则Z1Z2=（C）A （ac+bd, a） B (ac-bd, b)C （ac-bd, ac+bd） D (ac+bd, bc-ad)2、若R0，则N（，R）= z：（D）A |z|R B 0|z|RC R|z|R3、若z=x+iy, 则y=(D)A B C D4、若A= ，则 |A|=（C）A 3 B 0 C 1 D 2二、填空题1、若z=x+iy, w=z2=u+iv, 则v=（ 2xy ）2、复平面上满足Rez=4的点集为（ z=x+iy|x=4 ）3、（ 设E为点集，若它是开集，且是连通的，则E ）称为区域。4、设z0=x0+iy0, zn=xn+iyn(n=1,2,)，则zn以zo为极限的充分必要条件是 xn=x0，且 yn=y0。三、计算题1、求复数-1-i的实部、虚部、模与主辐角。解：Re(-1-i)=-1 Im(-1-i)=-1|-1-i|=2、写出复数-i的三角式。解：3、写出复数 的代数式。解：4、求根式 的值。解：四、证明题1、证明若 ，则a2+b2=1。证明：而 3、证明：证明：第2章 解析函数一、单项选择题1若f(z)= x2-y2+2xyi,则2、若f(z)=u(x, y)+iv(x,y), 则柯西黎曼条件为（D）A BC D3、若f(z)=z+1, 则f(z)在复平面上（C）A 仅在点z=0解析 B 无处解析C 处处解析 D 在z=0不解析且在z0解析4、若f（z）在复平面解析，g(z)在复平面上连续，则f(z)+g(z)在复平面上（C）A解析 B 可导C连续 D 不连续二、填空题1、若f(z)在点a不解析，则称a为f(z)的奇点。2、若f(z)在点z=1的邻域可导，则f(z)在点z=1解析。3、若f(z)=z2+2z+1，则 4、若 ，则 不存在。三、计算题：1、设f(z)=zRe(z), 求解： =2、设f(z)=excosy+iexsiny,求解：f(z)=excosy+iexsiny=ez,z=x+iyu=excosy v=exsinyf(z)=u+ivf(z)在复平面解析，且 =excosy+iexsiny3、设f(z)=u+iv在区域G内为解析函数，且满足u=x3-3xy2，f(i)=0,试求f(z)。解：依C-R条件有Vy=ux=3x2-3y2则V（x1y）=3x2y-y3+c(c为常数)故f(z)=x3-3xy2+i(3x2y-y3+c)=x3-3xy2+i(cx2y-y3)+ic =z3+ic，为使f(i)=0, 当x=0,y=1时，f(i)=0, 有f(0)=-i+ic=0c=1 f(z)=Z3+i4、设f(z)=u+iv在区域G内为解析函数，且满足u=2(x-1)y,f(2)=-i,试求f(z)。解：依C-R条件有Vy=ux=2yV= =y2+(x) Vx=(x)=V=y2-x2+2x+c(c为常数)f(z)=2(x-1)y+i(y2-x2+2x+c)为使f(z)=-i,当x=2 y=0时,f(2)=ci=-i c=-1f(z)=2(x-1)y+i(y2-x2+2x-1) =-(z-1)2i四、证明题1、试在复平面讨论f(z)=iz的解析性。解：令f(z)=u+iv z=x+iy则iz=i(x+iy)=-y+ixu=-y v=x于是ux=0 uy=-1Vx=1 Vy=0ux、uy、vx在复平面内处处连接又Ux=Vy Uy=-Vx。f(z)=iz在复平面解析。2、试证：若函数f(z)在区域G内为解析函数，且满足条件（z）=0，zG，则f(z)在G内为常数。证：设f(z)=u+iv,z=x+iy,zGf(z)在G内解析，Ux=Vy, Uy=-Vx又（z）=0, （z）=Ux+iVxUx=0 Vx=0Uy=-Vx=0 Ux=Vy=0U为实常数C1，V也为实常数C2，f(z)=C1+iC2=Z0f(z)在G内为常数。复变函数课程作业参考解答2第3章 初等函数一、单项选择题1. z = ( A ) 是根式函数的支点. (A) 0 (B) 1 (C) (D) i2. z = ( D ) 是函数的支点. (A) i (B) 2i (C) -1 (D) 03. ei =( B ). (A) e-1+e (B) cos1+isin1 (C) sin1 (D) cos14. sin1= ( A ) (A) (B) (C) (D) 二、填空题1. cosi = 2. = e(cos1+isin1)3. lni =4. ln(1+i) = k为整数.三、计算题1. 设z=x+iy，计算.解: = = 2. 设z = x+iy, 计算. 解: z = x+iy 3. 求方程的解.解: lnz = 由对数函数的定义有: Z= 所给方程的解为z = i4. 求方程的解.解: =根据指数函数的定义有:z=n2+i 或z=n(1+)四、证明题1. 试证: . 证明：根据正弦函数及余弦正数定义有： sin2z=2sinzcosz2. 证明: . 证明: 令A= B=sinx+sin2x+sinnx = 第4章 解析函数的积分理论一、单项选择题1. ( D ) , c为起点在0 , 终点在1+i的直线段. (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、填空题1. 若与沿曲线c可积，则.2. 设L为曲线c的长度, 若f(z)沿c可积, 且在c上满足,则.3. 4. 三、计算题1.计算积分,其中c为自0到2+i的直线段. 解: c的方程为: 其次由得 = =2. 计算积分. 解: = 作区域D:积分途径在D内被积函数的奇点Z=2与Z=3均不在D内,所以被积函数在D内解析.由定理4.2得:=03. 计算积分. 解: 奇点z=1和z=-1不在区域D,内 的三个根也不在D内 由定理4.2 得 =04. 计算积分, . 解: 由定理4.6得 四、证明题1. 计算积分，并由此证明. 证明：在圆域 |z|1内解析 = 另一方面,在圆|z|= =(实部和虚部为0) = = = = =0 而为偶函数0= = 复变函数课程作业参考解答3第5章 解析函数的幂级数表示一、单项选择题1. 幂级数的收敛半径等于( B ) ( A ) 0 (B) 1 ( C ) 2 (D) 32. 点z=-1是f(z)=r ( B )级零点. ( A ) 1 (B)2 (C)3 (D)53. 级数的收敛圆为( D ). (A) | z-1| 3 (B) |z|1 (D) |z| 14. 设f(z)在点a解析, 点b是f(z)的奇点中离点a最近的奇点,于是,使f(z)=成立的收敛圆的半径等于( C ). (A) a+b+1 (B) b-a+1(C) |a-b| (D) |a+b|二、填空题1.级数1+z+的收敛圆R=+即整个复平面2.若f(z)= (k为常数),则z=m(m=0, )为f(z)的 1 级零点. 3.幂有数的收敛半径等于 0 . 4.z=0是f(z)=ez-1的 1 级零点. 三、计算题 1.将函数f(z)=在点z=0展开幂级数. 解: f(z)= =- 2.将函数f(z)=(1-z)-2在点z=0展开成幂级数. 解：而(1-z)-1= = 3将函数f(z)=(z+2)-1在点z=1展开成幂级数. 解：f(z)=(z+2)-1= = 4将函数f(z)=ez在点z=1展开成幂级数. 解： f(z)=ez f(n)=ez 四、证明题 1证明:1-ei2z=-2isinzeiz 证：eiz=cosz+isinze-iz=cos-isinz eiz-e-iz=2isinz -2isinz=-( eiz-e-iz) = eiz-e-iz -2isinz eiz=( e-iz- eiz) eiz =e0- e2iz=1- e2iz2试用解析函数的唯一性定理证明等式: cos2z= cos2z-sin2z 证f1(z)=cos2z,则f1(z)复平面G解析设f2(z)coszsin2，则f2(z)也在整个复平面G解析取E=K为实数轴,则E在G内有聚点.当E为实数时,知cos2z=cos2z-sin2z,即f1(z)= f2(z)由解析函数唯一性定理,由以上三条知f1(z)= f2(z) 成立即cos2z= cos2z-sin2z 第6章 解析函数的罗朗级数表示 一、单项选择题 1函数f(z)=在点z=2的去心邻域( D ) 内可展成罗朗级数. (A) 0 (B) 0 (C) 1 (D) 0 2设点为f(z)的孤立奇点，若=c，则点为f(z)的( C ). (A) 本性奇点 (B) 极点 (C) 可去奇点 (D) 解析点 3若点为函数f(z)的孤立奇点，则点为f(z)的极点的充分必要条件是( D ). (A) f(z)=c() (B) f(z)= (C) f(z)=c() (D) f(z)= 4若点为函数f(z)的孤立奇点，则点为f(z)的本性奇点的充要条件是（ B ）. (A) f(z)= c() (B) f(z)不存在 (C) f(z)=c() (D) f(Z)= 二、填空题 1设为函数f(z)在点的罗朗级数,称为该级数的主要部分. 2.设点为函数f(z)的奇点,若f(z)在点的某个 某个去心邻域内解析,则称点为f(z)的孤立奇点. 3.若f(z)=，则点z=0为f(z)的 0 级极点. 不是极点,若f(z)= 则z=0为f(z)的一个极点. 4.若f(z)=(sin)-1,则点z0为f(z)非孤立 奇点. 三、计算题1将函数f(z)=(z-2)-1在点z=0的去心邻域展成罗朗级数.解: f(z)= = - = - 2将函数f(z)在点的去心邻域展成罗朗级数. 解: f(z)= 3试求函数f(z)=z-3sinz3的有限奇点,并判定奇点的类别. 解: 解析,无奇点,f(z)的有限奇点为z=0. 并且为3阶极点. 4试求函数f(z)=z-1的有限奇点，并判定奇点的类别. 解: f(z)的m阶奇点即的阶零点,而零点为z=0，z=1，z=-1，且均为1阶零点。的有限奇点为z=0，z=1,z=-1且均为1阶极点. 四、证明题 1设f(z)=,试证z=0为f(z)的6级极点. 证:要证z=0为f(z)的6级极点,只需证z=0为的6阶零点即可.而 =8z3 =8z6 令 则 为的6阶零点 z=0 为f(z)的6级极点.请您删除一下内容，O(_)O谢谢！2016年中央电大期末复习考试小抄大全，电大期末考试必备小抄，电大考试必过小抄Basketball can make a true 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 Basketball 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 space 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 (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 leather, 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. 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 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. Basketball 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 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. The 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, 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 handling 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 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 established 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 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 France 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. 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 1900s. 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, and 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-making 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 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. 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 closely 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 national, 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 first 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 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 Basketball 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 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, invented 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 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 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, the 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, depending 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 from 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 suitable 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. 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 with 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, which 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 C