2016年广播电视大学复变函数与积分变换重点公式归纳小抄.doc

复变函数与积分变换复习提纲第一章 复变函数一、 复变数和复变函数 二、 复变函数的极限与连续极限 连续 第二章 解析函数一、 复变函数可导与解析的概念。二、 柯西黎曼方程掌握利用c-r方程判别复变函数的可导性与解析性。掌握复变函数的导数：三、 初等函数重点掌握初等函数的计算和复数方程的求解。1、幂函数与根式函数 单值函数 (k=0、1、2、n-1) n多值函数2、指数函数：性质：（1）单值.（2）复平面上处处解析，（3）以为周期3、对数函数 （k=0、1、2）性质：（1）多值函数,（2）除原点及负实轴处外解析,（3）在单值解析分枝上:。4、三角函数： 性质：（1）单值 （2）复平面上处处解析 （3）周期性 （4）无界5、反三角函数（了解）反正弦函数 反余弦函数 性质与对数函数的性质相同。6、一般幂函数： （k=0、1）四、调和函数与共轭调和函数：1) 调和函数:2) 已知解析函数的实部（虚部），求其虚部（实部）有三种方法：a）全微分法 b）利用c-r方程 c）不定积分法第三章 解析函数的积分一、复变函数的积分 存在的条件。二、复变函数积分的计算方法1、沿路径积分： 利用参数法积分，关键是写出路径的参数方程。2、闭路积分： a) 利用留数定理，柯西积分公式，高阶导数公式。b) 利用参数积分方法三、柯西积分定理： 推论1：积分与路径无关 推论2：利用原函数计算积分 推论3：二连通区域上的柯西定理推论4：复连通区域上的柯西定理四、柯西积分公式： 五、高阶导数公式：解析函数的两个重要性质：l 解析函数在任一点的值可以通过函数沿包围点的任一简单闭合回路的积分表示。l 解析函数有任意阶导数。本章重点：掌握复变函数积分的计算方法沿路径积分 1）利用参数法积分 2）利用原函数计算积分。闭路积分 利用留数定理计算积分。第四章 解析函数的级数一、幂级数及收敛半径： 1、一个收敛半径为r（0）的幂级数，在收敛圆内的和函数是解析函数，在这个收敛圆内，这个展开式可以逐项积分和逐项求导，即有： 2、收敛半径的计算方法1） 比值法：2） 根值法：二、泰勒（taylor）级数1、如函数在圆域内解析，那么在此圆域内可以展开成taylor级数1）展开式是唯一的。故将函数在解析点的邻域中展开幂级数一定是taylor级数。2 ) 收敛半径是展开点到的所有奇点的最短距离。3）展开式的系数可以微分计算： 4）解析函数可以用taylor级数表示。2、记住一些重要的泰勒级数：1） 2）3） 4）三、罗兰（laurent）级数如果函数在圆环城内解析，则= （n=0、1、2）1、展开式是唯一的，即只要把函数在圆环城内展开为幂级数即为laurent级数。2、展开式的系数是不可以利用积分计算。利用已知的幂级数，通过代数运算把函数展开成laurent级数。3、注意展开的区域，在展开点的所有解析区域展开。四、孤立奇点1、定义：若b是的孤立奇点，则在内解析。在此点可展开为罗兰级数，=2、分类：孤立奇点把函数在奇点的去心邻域中展开为罗兰级数，求解c-13、极点留数计算a) 如果b是的一阶极点，则b) 如果b是的m阶极点，则c) 如b是的一阶极点，且p(b)0，那么d) e) 若是的可去奇点，并且，关系：全平面留数之和为零。本章重点：函数展开成taylor级数，并能写出收敛半径。 函数在解析圆环城内展开成laurent级数。 孤立奇点（包含点）的判定及其留数的计算。第五章 留数定理的应用一、条件：（1）r(sinq，cosq) 为cosq与sin q 的有理函数 （2）r ( ) 在0，2p 或者 -p ，p 上连续。令，则，。 注意留数是计算单位圆中的奇点。二、条件： （1） 是x的多项式。（2） （3） 分母阶次比分子阶次至少高二次则 是在上半平面的奇点。三、 （）条件：（1），且比至少高一阶，（2），（3） ， 重点关注第一和第三种类型 第七章 fourier变换一、傅立叶变换二、 函数的傅立叶变换. 三、 一些傅立叶变换及逆变换四、性质：1、 相似性质 2、 延迟性质 位移性质3、微分性质 4、积分性质 由fourier变换的微分和积分性质，我们可以利用fourier变换求解微积分方程。四、卷积和卷积定理 五、三维fourier变换及反演 本章重点：利用定义计算fourier变换第八章 laplace变换一、拉普拉斯变换二、几个重要的拉普拉斯变换及逆变换 四、拉普拉斯变换的性质1、2、3、 4、五、卷积：六、laplace 反演七、laplace逆变换（1）部分分式法（2）卷积定理（3）laplace反演公式（留数定理）（4）利用laplace变换的性质八、利用laplace变换求解微积分方程（1）对方程取laplace变换，得到象函数的代数方程（2）解代数方程，得到像函数的表达式（3）求像函数的拉普拉斯逆变换微分方程像函数的代数方程像函数像原函数解函数 拉氏变换 解代数方程 拉氏逆变换 本章重点：利用定义和性质计算laplace变换。 计算laplace逆变换。 利用laplace变换求解微积分方程。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 carlson at pittsburgh all made significant 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 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 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 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 postseason 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. the 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 mikan 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 its 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 each year and the nits prestige began to decline. professional basketball remained a disorganized and stodgy sport up until the late 1940s, with barnstorming still central to the game and most players still using the set shot. in 1946, however, hockey owners, led by maurice podoloff, created the basketball association of america (baa) in the east to fill their arenas, but few fans came, even after joe fulks of philadelphia introduced the jump shot. the baas rival, the national basketball league, had existed since the 1930s, had better player