最新電大開放教育《復(fù)變函數(shù)與積分變換》作業(yè)答案

1、復(fù)變函數(shù)與積分變換作業(yè)參考答案習(xí)題1：4、計算下列各式(1) ； (3) ；(5) ，求，； (7) 。解：(1) ；(3) ；(5) ，(7) 因為，所以，即時，；時，；時，；時，；時，；時，習(xí)題2：3、下列函數(shù)在何處可導(dǎo)？何處解析？在可導(dǎo)點求出其導(dǎo)數(shù)(2) ； (4) (6) 。解：(2) 因為，這四個一階偏導(dǎo)數(shù)都連續(xù)，故和處處可微，但柯西-黎曼方程僅在上成立，所以只在直線上可導(dǎo)，此時，但復(fù)平面上處處不解析(4) 因為，這四個一階偏導(dǎo)數(shù)都連續(xù)，故和處處可微，且滿足柯西-黎曼方程，所以在復(fù)平面內(nèi)解析，并且 (6) 所以，在除外處處解析，且4、指出下列函數(shù)的奇點(1) ； (2) 解：(1)

2、所以，的奇點為0，(2) 所以，的奇點為，10、如果在區(qū)域內(nèi)解析，并且滿足下列條件之一，試證在內(nèi)是一常數(shù)(2) 在內(nèi)解析；證明：由在區(qū)域內(nèi)解析，知、在區(qū)域內(nèi)可微，且，同理，由在內(nèi)解析，知，從而我們得到，所以、皆為常數(shù)，故在內(nèi)是一常數(shù)15、求解下列方程：(2) 解：，于是18、求，的值及主值解：，所以其主值為；，所以其主值為19、求，的值解：；20、求，的值解：；22、解方程：(1) ；解：，習(xí)題3：1、沿下列路徑計算積分：(1) 從原點至的直線段；(2) 從原點沿實軸至2，再由2鉛直向上至；(3) 從原點沿虛軸至，再由沿水平方向向右至解：(1) 從原點至的直線段的復(fù)參數(shù)方程為，參數(shù)，所以(2)

3、 從原點沿實軸至2的直線段的復(fù)參數(shù)方程為，參數(shù)，由2鉛直向上至的直線段的復(fù)參數(shù)方程為，參數(shù)，所以(3) 從原點沿虛軸至的直線段的復(fù)參數(shù)方程為，參數(shù)，由沿水平方向向右至的復(fù)參數(shù)方程為，參數(shù)，所以2、分別沿與算出積分的值解：的復(fù)參數(shù)方程為，參數(shù)所以；的復(fù)參數(shù)方程為，參數(shù)所以5、計算積分的值，其中為正向圓周：(1) 解：設(shè)是內(nèi)以被積函數(shù)的奇點為圓心的正向圓周，那么6、試用觀察法得出下列積分的值，并說明觀察時所依據(jù)的是什么？是正向圓周：(1) ； (2) ； (3) ；(4) ； (5) ； (6) 解：(1) ，根據(jù)柯西積分定理；(2) ，根據(jù)柯西積分定理；(3) ，根據(jù)柯西積分定理；(4) ，根據(jù)

4、復(fù)合閉路定理；(5) ，根據(jù)柯西積分定理；(6) ，根據(jù)柯西積分定理及復(fù)合閉路定理7、沿指定曲線的正向計算下列積分：(1) ，；(2) ，；(3) ，；(4) ，；(5) ，；(6) ，為包圍的閉曲線；(7) ，；(8) ，；(9) ，；(10) ，解：(1) ；(2) ；(3) ；(4) ；(5) ；(6) ；(7) ；(8) ；(9) ；(10) 21、證明：和都是調(diào)和函數(shù)，但是不是解析函數(shù)證明：因為，所以，且，即和都是調(diào)和函數(shù)，但是不是解析函數(shù)22、由下列各已知調(diào)和函數(shù)求解析函數(shù)，并寫出的表達(dá)式：(1) ；(2) ，；(3) ，解：(1) 因為是調(diào)和函數(shù)，所以，于是那么，則，所以，(2)

5、 ，因為是調(diào)和函數(shù)，所以，從而由知，所以(3) 因為是調(diào)和函數(shù)，所以，于是那么，則，所以，由知，所以習(xí)題4：1、下列數(shù)列是否收斂？若收斂，求其極限(1) ； (2) ； (3) ； (4) 解：(1) ，當(dāng)時，實部，虛部，所以收斂于(2) ，當(dāng)時，那么，所以收斂于0(3) 當(dāng)時，實部是發(fā)散的，所以發(fā)散(4) ，實部和虛部都發(fā)散，所以發(fā)散2、判斷下列級數(shù)的收斂性與絕對收斂性：(1) ； (3) 解：(1) 記，則當(dāng)時，那么不趨近于0，所以級數(shù)發(fā)散(3) 收斂，即級數(shù)絕對收斂，所以收斂7、將下列各函數(shù)展成的冪級數(shù)，并指出它們的收斂半徑(1) ； (3) 解：(1) 因為，所以收斂半徑(3) 因為，

6、所以收斂半徑8、將下列各函數(shù)在指定點處展成泰勒級數(shù)，并指出它們的收斂半徑(3) ，； (4) ，； (6) ，解：(3) ，則因為，所以收斂半徑(4) ，則因為，所以收斂半徑(6) 因為，所以收斂半徑10、求下列各函數(shù)在指定圓環(huán)域的洛朗級數(shù)展開式：(2) ，；(5) ，在以為中心的圓環(huán)域內(nèi)；(7) ，解：(2) 在內(nèi)，由于，且，所以，從而在內(nèi)，由于，所以，從而(5) 當(dāng)時，由于，且，所以，從而當(dāng)時，由于，所以，且，從而，所以(7) 由于且，所以習(xí)題5：1、求下列函數(shù)的孤立奇點并確定它們的類別，若是極點，指出它們的級(1) ； (3) ； (4) ； (7) ； (11) 解：(1) 易見，是的

7、孤立奇點由于，所以，是極點，一級極點，二級極點(3) ，所以是極點，二級極點(4) 易見是的孤立奇點，且，所以是可去奇點；(7) ，三級極點，一級極點；(11) ，本性奇點5、求下列各函數(shù)在有限奇點處的留數(shù)(2) ； (3) ； (6) 解：(2) 記，則易見，是的孤立奇點，且他們都是一級極點由規(guī)則，(3) 記，則有二級極點由規(guī)則，(6) 記，則有本性奇點因為在的去心鄰域內(nèi)的洛朗級數(shù)為于是有其中的項的系數(shù)，所以6、利用留數(shù)定理計算下列積分(1) ，為圓周解：被積函數(shù)在圓周的內(nèi)部有一級極點和二級極點，由留數(shù)的計算規(guī)則、得，于是由留數(shù)定理得積分值(2) 解：被積函數(shù)在內(nèi)有一個二級極點，由留數(shù)的計算

8、規(guī)則得于是由留數(shù)定理得積分值(4) 解：被積函數(shù)在內(nèi)有可去奇點，則，所以由留數(shù)定理知(6) 解：被積函數(shù)在內(nèi)有一個二級極點，由留數(shù)的計算規(guī)則得于是由留數(shù)定理得積分值9、(1) 解：令，則，于是被積函數(shù)在內(nèi)有一個一級極點，其留數(shù)所以(5) 解：是偶函數(shù)，而在上半平面內(nèi)有一級極點和，且，所以(6) 解：，且在實軸上無孤立奇點，故積分存在，所求積分是它的實部函數(shù)在上半平面有兩個一級極點和，而且，從而所以習(xí)題8：4、試求的傅氏變換解：的傅里葉變化為5、試求矩形脈沖的傅氏變換解：的傅里葉變化為6、求下列函數(shù)的傅氏積分：(1) 解：是上的奇函數(shù)，則，于是7、求函數(shù)的傅氏積分，并計算解：是上的偶函數(shù)，則，于

9、是10、求符號函數(shù)的傅氏變換（提示：）解：方法一：方法二：11、求函數(shù)的傅氏變換解：，則15、利用位移性質(zhì)計算下列函數(shù)的傅氏變換：(1) ；(2) 解：(1) ；(2) 23、求下列函數(shù)的傅氏變換：(2) ；(3) ；(4) 解：(2) 記，由卷積定理有(3) 記，由卷積定理有(4) 記，由卷積定理有習(xí)題9：2、求下列函數(shù)的拉氏變換：(1) (3) .解：(1) (3) 3、求下列周期函數(shù)的拉氏變換：(1) 以為周期且在一個周期內(nèi)的表達(dá)式為解：4、求下列函數(shù)的拉氏變換：(1) ；(2) ；(3) ；(6) （為實常數(shù)）；(9) ；(10) ；(11) 解：(1) (2) (3) ；(6) ，則

10、由位移性質(zhì)有；(9) ，則；(10) ，則，從而；(11) ，則請您刪除一下內(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

11、 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

12、 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 t

13、he 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 (75-7

14、6 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 l

15、evels 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

16、 (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 s

17、chool 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

18、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 physi

19、cal 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 pe

20、ople 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

21、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 th

22、rough 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

23、 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

24、 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, fi

25、ve 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

26、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

27、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 sh

28、ared 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 s

29、chool 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 in

30、creasingly 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 19

31、80s, 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 bask

32、etball 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

33、. 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, televis

34、ion 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 ot

35、hers 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 basi

36、s 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

37、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,

38、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 wa

39、s 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 mi

40、nnesota 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

41、, 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 d

42、isqualify 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

43、 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

44、 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

45、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 l

46、agged 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 as

47、sociation); 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

48、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 carlson at pittsburgh all made signifi

49、cant contributions to the games development: zone defenses, the weave, the passing game, and the fast break. in the decade preceding world war ii, five events changed college basketball and allowed it to become a major spectator sport. in 1929, the rules committee reversed a decision that would have

50、 outlawed dribbling and slowed the game considerably. five years later, promoter edward ned irish staged the first intersectional twin bill in madison square garden in new york city and attracted more than 16,000 fans. he demonstrated the appeal of major college ball and made new york its center. in

51、 december 1936, hank luisetti of stanford revealed the virtues of the one-handed shot to an amazed garden audience and became the first major collegiate star. soon thereafter, luisetti scored an incredible fifty points against duquesne, thus ending the easts devotion to the set shot and encouraging

52、a more open game. in consecutive years the center jump was eliminated after free throws and then after field goals, thus speeding up the game and allowing for more scoring. in 1938, irish created the national invitation tournament (nit) in the garden to determine a national champion. although postse

53、ason tournaments had occurred before, the nit was the first with major colleges from different regions and proved to be a great financial success. the national collegiate athletic association (ncaa) created its own postseason tournament in 1939 but did not rival the nit in prestige for some time. th

54、e 1940s saw significant changes for college basketball. players began using the jump shot after kenny sailors of wyoming wowed the east with it in 1943. the behind-the-back dribble and pass also appeared, as did exceptional big men. bob kurland at oklahoma a&m was almost seven feet tall and george m

55、ikan at depaul was six feet ten inches. while kurland had perhaps the better college career and played in two olympics, he chose not to play professional ball, whereas mikan became the first dominant star in the pros. their defensive play inspired the rule against goal tending (blocking a shot on it

56、s downward flight). adolph rupp, who played under phog allen, also coached the first of his many talented teams at kentucky in that decade. however, in 1951, rupp and six other coaches suffered through a point-shaving scandal that involved thirty-two players at seven colleges and seriously injured college basketball, particularly in new york, where four of the seven schools were located. while the game survived, the ncaa moved its tournament away from madison square garden to different cities ea