复变函数与积分变换省公共课一等奖全国赛课获奖课件.pptx

Department of Electronic EngineeringDepartment of Electronic Engineering第第2章章 解析函数解析函数2.1 解析函数概念解析函数概念复变函数与积分变换复变函数与积分变换第1页Department of Electronic EngineeringDepartment of Electronic Engineering1.复变函数导数复变函数导数复变函数与积分变换复变函数与积分变换第2页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第3页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第4页Department of Electronic EngineeringDepartment of Electronic Engineering导数分析定义：导数分析定义：复变函数与积分变换复变函数与积分变换第5页Department of Electronic EngineeringDepartment of Electronic Engineering导数运算法则导数运算法则复变函数求导法则（以下出现函数均假设可导）：（1）其中为复常数；（2）其中为正整数；（3）；（4）（5）；复变函数与积分变换复变函数与积分变换第6页Department of Electronic EngineeringDepartment of Electronic Engineering（6）；（7）是两个互为反函数单值函数，且.复变函数与积分变换复变函数与积分变换第7页Department of Electronic EngineeringDepartment of Electronic Engineering2.解析概念复变函数与积分变换复变函数与积分变换第8页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第9页Department of Electronic EngineeringDepartment of Electronic Engineeringu注解1、“可微”有时也能够称为“单演”，而“解析”有时也称为“单值解析”、“全纯”、“正则”等；u注解2、一个函数在一个点可导，显然它在这个点连续；u注解2、解析性与可导性关系：在一个点可导性为一个局部概念，而解析性是一个整体概念；注解：注解：复变函数与积分变换复变函数与积分变换第10页Department of Electronic EngineeringDepartment of Electronic Engineeringu注解3、函数在一个点解析，是指在这个点某个邻域内可导，所以在这个点可导，反之，在一个点可导不能得到在这个点解析；u注解4、闭区域上解析函数是指在包含这个区域一个更大区域上解析；u注解5、解析性区域；注解：注解：复变函数与积分变换复变函数与积分变换第11页Department of Electronic EngineeringDepartment of Electronic Engineering四则运算法则复变函数与积分变换复变函数与积分变换第12页Department of Electronic EngineeringDepartment of Electronic Engineering复合函数求导法则复变函数与积分变换复变函数与积分变换第13页Department of Electronic EngineeringDepartment of Electronic Engineering反函数求导法则复变函数与积分变换复变函数与积分变换第14页Department of Electronic EngineeringDepartment of Electronic Engineeringu利用这些法则，我们能够计算常数、多项式以及有理函数导数，其结果和数学分析结论基本相同。注解：注解：复变函数与积分变换复变函数与积分变换第15页Department of Electronic EngineeringDepartment of Electronic Engineering2.2函数解析充要条件复变函数与积分变换复变函数与积分变换第16页Department of Electronic EngineeringDepartment of Electronic EngineeringCauchy-Riemann条件：复变函数与积分变换复变函数与积分变换第17页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第18页Department of Electronic EngineeringDepartment of Electronic Engineering定理3.1证实（必要性）：复变函数与积分变换复变函数与积分变换第19页Department of Electronic EngineeringDepartment of Electronic Engineering定理3.1证实（充分性）：复变函数与积分变换复变函数与积分变换第20页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数解析条件复变函数与积分变换复变函数与积分变换第21页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第22页Department of Electronic EngineeringDepartment of Electronic Engineering注解:和数学分析中结论不一样，此定理表明解析函数(可导函数)实部和虚部不是完全独立，它们是柯西-黎曼方程一组解；柯西-黎曼条件是复变函数解析必要条件而非充分条件（见反例）；解析函数导数有更简练形式：复变函数与积分变换复变函数与积分变换第23页Department of Electronic EngineeringDepartment of Electronic Engineering反例：u(x,y)、v(x,y)以下：复变函数与积分变换复变函数与积分变换第24页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第25页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第26页Department of Electronic EngineeringDepartment of Electronic Engineering例1讨论以下函数可导性和解析性：复变函数与积分变换复变函数与积分变换第27页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第28页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第29页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第30页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第31页Department of Electronic EngineeringDepartment of Electronic Engineering例2复变函数与积分变换复变函数与积分变换第32页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第33页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第34页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第35页Department of Electronic EngineeringDepartment of Electronic Engineering2.3 初等函数初等函数 3、指数函数指数函数 4、多值函数导引：幅角函数多值函数导引：幅角函数复变函数与积分变换复变函数与积分变换第36页Department of Electronic EngineeringDepartment of Electronic Engineering1.指数函数指数函数(1)指数函数定义指数函数定义复变函数与积分变换复变函数与积分变换第37页Department of Electronic EngineeringDepartment of Electronic Engineering我们首先把指数函数定义扩充到整个复平面。要求复变数z=x+iy函数f(z)满足以下条件：复变函数与积分变换复变函数与积分变换第38页Department of Electronic EngineeringDepartment of Electronic Engineering由解析性，我们利用柯西-黎曼条件，有所以，所以，我们也重新得到欧拉公式：复变函数与积分变换复变函数与积分变换第39页Department of Electronic EngineeringDepartment of Electronic Engineering(2)指数函数基本性质指数函数基本性质复变函数与积分变换复变函数与积分变换第40页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第41页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第42页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第43页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第44页Department of Electronic EngineeringDepartment of Electronic Engineeringyxz-平面uw-平面v复变函数与积分变换复变函数与积分变换第45页Department of Electronic EngineeringDepartment of Electronic Engineering2.三角函数三角函数与双曲函数复变函数与积分变换复变函数与积分变换第46页Department of Electronic EngineeringDepartment of Electronic Engineering 因为Euler公式，对任何实数x，我们有：所以有所以，对任何复数z，定义余弦函数和正弦函数以下：复变函数与积分变换复变函数与积分变换第47页Department of Electronic EngineeringDepartment of Electronic Engineering三角函数三角函数基本性质:则对任何复数z，Euler公式也成立：复变函数与积分变换复变函数与积分变换第48页Department of Electronic EngineeringDepartment of Electronic Engineering关于复三角函数，有下面基本性质：1、cosz和sinz是单值函数；2、cosz是偶函数，sinz是奇函数:复变函数与积分变换复变函数与积分变换第49页Department of Electronic EngineeringDepartment of Electronic Engineering3、cosz和sinz是认为周期周期函数:复变函数与积分变换复变函数与积分变换第50页Department of Electronic EngineeringDepartment of Electronic Engineering证实：复变函数与积分变换复变函数与积分变换第51页Department of Electronic EngineeringDepartment of Electronic Engineering注解：因为负数能够开平方，所以由此不能得到比如z=2i时，有复变函数与积分变换复变函数与积分变换第52页Department of Electronic EngineeringDepartment of Electronic Engineering6、cosz和sinz在整个复平面解析，而且有：证实：复变函数与积分变换复变函数与积分变换第53页Department of Electronic EngineeringDepartment of Electronic Engineering7、cosz和sinz在复平面零点：cosz在复平面零点是，sinz在复平面零点是8、同理能够定义其它三角函数：复变函数与积分变换复变函数与积分变换第54页Department of Electronic EngineeringDepartment of Electronic Engineering9、反正切函数：由函数所定义函数w称为z反正切函数，记作因为令，得到复变函数与积分变换复变函数与积分变换第55页Department of Electronic EngineeringDepartment of Electronic Engineering从而所以反正切函数是多值解析函数，它支点是无穷远点不是它支点。复变函数与积分变换复变函数与积分变换第56页Department of Electronic EngineeringDepartment of Electronic Engineering3.对数函数 和实变量一样，复变量对数函数也定义为指数函数反函数：复变函数与积分变换复变函数与积分变换第57页Department of Electronic EngineeringDepartment of Electronic Engineering注解、因为对数函数是指数函数反函数，而指数函数是周期为2 周期函数，所以对数函数必定是多值函数，实际上。复变函数与积分变换复变函数与积分变换第58页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第59页Department of Electronic EngineeringDepartment of Electronic Engineering对数函数主值：对应与幅角函数主值，我们定义对数函数Lnz主值lnz为：则这时，有复变函数与积分变换复变函数与积分变换第60页Department of Electronic EngineeringDepartment of Electronic Engineering三种对数函数联络与区分：复变函数与积分变换复变函数与积分变换第61页Department of Electronic EngineeringDepartment of Electronic Engineering对数函数基本性质对数函数基本性质复变函数与积分变换复变函数与积分变换第62页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第63页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第64页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第65页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第66页Department of Electronic EngineeringDepartment of Electronic Engineeringuvw-平面xz-平面y复变函数与积分变换复变函数与积分变换第67页Department of Electronic EngineeringDepartment of Electronic Engineering对数函数单值化：对应与幅角函数单值化，我们也能够将对数函数单值化：考虑复平面除去负实轴（包含0）而得区域D。显然，在D内，对数函数能够分解为无穷多个单值连续分支。复变函数与积分变换复变函数与积分变换第68页Department of Electronic EngineeringDepartment of Electronic Engineering沿负实轴割线取值情况：上沿下沿复变函数与积分变换复变函数与积分变换第69页Department of Electronic EngineeringDepartment of Electronic Engineering普通区域：复变函数与积分变换复变函数与积分变换第70页Department of Electronic EngineeringDepartment of Electronic Engineering对数函数单值化：因为对数函数每个单值连续分支都是解析，所以我们也将它连续分支称为解析分支。我们也称对数函数是一个无穷多值解析函数。我们称原点和无穷远点是对数函数无穷阶支点（对数支点）；复变函数与积分变换复变函数与积分变换第71页Department of Electronic EngineeringDepartment of Electronic Engineering特点：1、当z绕它们连续改变一周时，Lnz连续改变到其它值；2、不论怎样沿同一方向改变，永远不会回到同一个值。复变函数与积分变换复变函数与积分变换第72页Department of Electronic EngineeringDepartment of Electronic Engineering例1复变函数与积分变换复变函数与积分变换第73页Department of Electronic EngineeringDepartment of Electronic Engineering例2复变函数与积分变换复变函数与积分变换第74页Department of Electronic EngineeringDepartment of Electronic Engineering例3复变函数与积分变换复变函数与积分变换第75页Department of Electronic EngineeringDepartment of Electronic Engineering4.幂函数利用对数函数，能够定义幂函数：设a是任何复数，则定义za次幂函数为复变函数与积分变换复变函数与积分变换第76页Department of Electronic EngineeringDepartment of Electronic Engineering当a为正实数，且z=0时，还要求因为所以，对同一个不一样数值个数等于不一样数值因子个数。复变函数与积分变换复变函数与积分变换第77页Department of Electronic EngineeringDepartment of Electronic Engineering幂函数基本性质：复变函数与积分变换复变函数与积分变换第78页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第79页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第80页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第81页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第82页Department of Electronic EngineeringDepartment of Electronic Engineering设在区域G内，我们能够把Lnz分成无穷个解析分支。对于Lnz一个解析分支，对应地有一个单值连续分支。依据复合函数求导法则，这个单值连续分支在G内解析，而且其中应该了解为对它求导数那个分支，lnz应该了解为对数函数对应分支。复变函数与积分变换复变函数与积分变换第83页Department of Electronic EngineeringDepartment of Electronic Engineering对应于Lnz在G内任一解析分支：当a是整数时，在G内有n个解析分支；当a是无理数或虚数时，幂函数在G内是同一解析函数；当时，在G内有没有穷多个解析分支，是一个无穷值多值函数。复变函数与积分变换复变函数与积分变换第84页Department of Electronic EngineeringDepartment of Electronic Engineering比如当n是大于1整数时，称为根式函数，它是反函数。当时，有这是一个n值函数。复变函数与积分变换复变函数与积分变换第85页Department of Electronic EngineeringDepartment of Electronic Engineering在复平面上以负实轴（包含0）为割线而得区域D内，它有n个不一样解析分支：它们也能够记作这些分支在负实轴上沿与下沿所取值，与相应连续分支在该处所取值一致。复变函数与积分变换复变函数与积分变换第86页Department of Electronic EngineeringDepartment of Electronic Engineering幂函数映射性质:复变函数与积分变换复变函数与积分变换第87页Department of Electronic EngineeringDepartment of Electronic Engineering关于幂函数当a为正实数时映射性质，有下面结论：设是一个实数，而且在z平面上取正实数轴（包含原点）作为割线，得到一个区域D*。考虑D*内角形，并取在D*内一个解析分支复变函数与积分变换复变函数与积分变换第88页Department of Electronic EngineeringDepartment of Electronic Engineering当z描出A内一条射线时让从0增加到（不包含0及），那么射线l扫过角形A，而对应射线扫过角形（不包含0），w在w平面描出一条射线复变函数与积分变换复变函数与积分变换第89页Department of Electronic EngineeringDepartment of Electronic Engineering所以把夹角为角形双射成一个夹角为角形，同时，这个函数把A中以原点为心圆弧映射成中以原点为心圆弧。复变函数与积分变换复变函数与积分变换第90页Department of Electronic EngineeringDepartment of Electronic Engineering类似地，我们有，当n(1)是正整数时，n个分支分别把区域D*双射成w平面n个角形复变函数与积分变换复变函数与积分变换第91页Department of Electronic EngineeringDepartment of Electronic Engineering例1、作出一个含i区域，使得函数在这个区域内能够分解成解析分支；求一个分支在点i个值。解：我们知道可能支点为0、1、2与无穷，详细分析见下列图复变函数与积分变换复变函数与积分变换第92页Department of Electronic EngineeringDepartment of Electronic Engineering结论：0、1、2与无穷都是1阶支点。复变函数与积分变换复变函数与积分变换第93页Department of Electronic EngineeringDepartment of Electronic Engineering能够用正实数轴作为割线，在所得区域上，函数能够分解成单值解析分支。同时，我们注意到所以也能够用0，1与作割线。复变函数与积分变换复变函数与积分变换第94页Department of Electronic EngineeringDepartment of Electronic Engineering我们求函数下述解析分支在z=i值。在z=1处，取在w两个解析分支为：复变函数与积分变换复变函数与积分变换第95页Department of Electronic EngineeringDepartment of Electronic Engineering以下列图，所以复变函数与积分变换复变函数与积分变换第96页Department of Electronic EngineeringDepartment of Electronic Engineering例2、验证函数在区域D=C-0,1内能够分解成解析分支；求出这个分支函数在(0,1)上沿取正实值一个分支在z=-1处值及函数在（0，1）下沿值。解：我们知道复变函数与积分变换复变函数与积分变换第97页Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换复变函数与积分变换第98页Department of Electronic EngineeringDepartment of Electronic Engineering结论：0、1是3阶支点，无穷远点不是支点。复变函数与积分变换复变函数与积分变换第99页Department of Electronic EngineeringDepartment of Electronic Engineering所以，在区域D=C-0,1内函数能够分解成解析分支；若在（0，1）上沿要求在w四个解析分支为：则对应解析分支为k=0。在z=-1处，有，复变函数与积分变换复变函数与积分变换第100页Department of Electronic EngineeringDepartment of Electronic Engineering所以对应分支在(0,1)下沿取值为复变函数与积分变换复变函数与积分变换第101页Department of Electronic EngineeringDepartment of Electronic Engineering5.反三角函数与反双曲函数复变函数与积分变换复变函数与积分变换第102页