[数学]复变函数与积分变换-第二章课件.ppt

Department of Electronic EngineeringDepartment of Electronic Engineering第2章 解析函数2.1 解析函数的概念Department of Electronic EngineeringDepartment of Electronic Engineering1.复变函数的导数Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering导数的分析定义：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering 导数运算法则 复变函数的求导法则（以下出现的函数均假设可导）：（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Department of Electronic EngineeringDepartment of Electronic Engineering（6）；（7）是两个互为反函数的单值函数，且.复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering2.解析的概念复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineeringu 注解1、“可微”有时也可以称为“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、解析性与可导性的关系：在一个点的可导性为一个局部概念，而解析性是一个整体概念；注解：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineeringu 注解3、函数在一个点解析，是指在这个点的某个邻域内可导，因此在这个点可导，反之，在一个点的可导不能得到在这个点解析；u 注解4、闭区域上的解析函数是指在包含这个区域的一个更大的区域上解析；u 注解5、解析性区域；注解：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering四则运算法则复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复合函数求导法则复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering反函数求导法则复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineeringu 利用这些法则，我们可以计算常数、多项式以及有理函数的导数，其结果和数学分析的结论基本相同。注解：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering2.2函数解析的充要条件复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic EngineeringCauchy-Riemann 条件：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering定理3.1的证明（必要性）：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering定理3.1的证明（充分性）：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数的解析条件复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering注解:和数学分析中的结论不同，此定理表明解析函数(可导函数)的实部和虚部不是完全独立的，它们是柯西-黎曼方程的一组解；柯西-黎曼条件是复变函数解析的必要条件而非充分条件（见反例）；解析函数的导数有更简洁的形式：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering反例：u(x,y)、v(x,y)如下：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering例1讨论下列函数的可导性和解析性：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering例2复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering2.3 初等函数 3、指数函数 4、多值函数导引：幅角函数Department of Electronic EngineeringDepartment of Electronic Engineering1.指数函数(1)指数函数的定义Department of Electronic EngineeringDepartment of Electronic Engineering我 们 首 先 把 指 数 函 数 的 定 义 扩 充 到 整 个 复平面。要求复变数z=x+iy 的函数f(z)满足下列条件：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering由解析性，我们利用柯西-黎曼条件，有所以，因此，我们也重新得到欧拉公式：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering(2)指数函数的基本性质复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineeringyxz-平面uw-平面v复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering2.三角函数与双曲函数复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering 由于Euler 公式，对任何实数x，我们有：所以有因此，对任何复数z，定义余弦函数和正弦函数如下：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering三角函数的基本性质:则对任何复数z，Euler 公式也成立：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering关于复三角函数，有下面的基本性质：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Department of Electronic EngineeringDepartment of Electronic Engineering3、cosz 和sinz 是以为周期的周期函数:复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering证明：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering注解：由于负数可以开平方，所以由此不能得到例如z=2i 时，有复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering6、cosz 和sinz 在整个复平面解析，并且有：证明：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering7、cosz 和sinz 在 复 平 面 的 零 点：cosz 在 复 平 面 的零点是，sinz 在复平面的零点是8、同理可以定义其他三角函数：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering9、反正切函数：由函数所定义的函数w 称为z 的反正切函数，记作由于令，得到复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering从而所以反正切函数是多值解析函数，它的支点是无穷远点不是它的支点。复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering3.对数函数 和实变量一样，复变量的对数函数也定义为指数函数的反函数：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering注解、由于对数函数是指数函数的反函数，而指数函数是周期为2 的周期函数，所以对数函数必然是多值函数，事实上。复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering对数函数的主值：相应与幅角函数的主值，我们定义对数函数Ln z 的主值ln z 为：则这时，有复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering三种对数函数的联系与区别：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering对数函数的基本性质复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineeringuvw-平面xz-平面y复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering对数函数的单值化：相应与幅角函数的单值化，我们也可以将对数函数单值化：考虑复平面除去负实轴（包括0）而得的区域D。显然，在D 内，对数函数可以分解为无穷多个单值连续分支。复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering沿负实轴的割线的取值情况：上沿下沿复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering一般区域：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering对数函数的单值化：由于对数函数的每个单值连续分支都是解析的，所以我们也将它的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的无穷阶支点（对数支点）；复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering特点：1、当z 绕它们连续变化一周时，Ln z 连续变化到其它值；2、不论如何沿同一方向变化，永远不会回到同一个值。复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering例1复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering例2复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering例3复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering4.幂函数利用对数函数，可以定义幂函数：设a是任何复数，则定义z 的a次幂函数为复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering当a为正实数，且z=0时，还规定由于因此，对同一个的不同数值的个数等于不同数值的因子个数。复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering幂函数的基本性质：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering设在区域G 内，我们可以把Lnz 分成无穷个解析分支。对于Lnz 的一个解析分支，相应地有 一 个 单 值 连 续 分 支。根 据 复 合 函 数 求 导 法则，的 这 个 单 值 连 续 分 支 在G 内 解 析，并且其中应当理解为对它求导数的那个分支，lnz 应当理解为对数函数相应的分支。复变函数与积分变换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 内有无穷多个解析分支，是一个无穷值多值函数。复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering例如当n是大于1的整数时，称为根式函数，它是的反函数。当时，有这是一个n值函数。复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering在复平面上以负实轴（包括0）为割线而得的区域D 内，它有n个不同的解析分支：它们也可以记作这些分支在负实轴的上沿与下沿所取的值，与相应的连续分支在该处所取的值一致。复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering幂函数的映射性质:复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering关于幂函数当a为正实数时的映射性质，有下面的结论：设是一个实数，并且在z 平面上取正实数轴（包括原点）作为割线，得到一个区域D*。考虑D*内的角形，并取在D*内的一个解析分支复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering当z 描出A 内的一条射线时让从0增加到（不包括0及），那么射线l 扫过角形A，而相应的射线扫过角形（不包括0），w 在w 平面描出一条射线复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering因此把夹角为的角形双射成一个夹角为的 角 形，同 时，这 个 函 数 把A 中 以 原 点 为 心 的圆弧映射成中以原点为心的圆弧。复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering类似地，我们有，当n(1)是正整数时，的n个分支分别把区域D*双射成w 平面的n个角形复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering例1、作出一个含i 的区域，使得函数在这个区域内可以分解成解析分支；求一个分支在点i 个的值。解：我们知道可能的支点为0、1、2与无穷，具体分析见下图复变函数与积分变换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Department of Electronic EngineeringDepartment of Electronic Engineering可以用正实数轴作为割线，在所得区域上，函数可以分解成单值解析分支。同时，我们注意到因此也可以用0，1 与作割线。复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering我们求函数下述的解析分支在z=i 的值。在z=1 处，取在w 的两个解析分支为：复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering如下图，所以复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering例2、验证函数在区域D=C-0,1 内可以分解成解析分支；求出这个分支函数在(0,1)上沿取正实值的一个分支在z=-1 处的值及函数在（0，1）下沿的值。解：我们知道复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering复变函数与积分变换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Department of Electronic EngineeringDepartment of Electronic Engineering因此，在区域D=C-0,1 内函数可以分解成解析分支；若在（0，1）的上沿规定在w 的四个解析分支为：则对应的解析分支为k=0。在z=-1 处，有，复变函数与积分变换Department of Electronic EngineeringDepartment of Electronic Engineering所以对应分支在(0,1)下沿的取值为复变函数与积分变换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