[2-2调和函数与解析函数.ppt

§2解析函数与调和函数的关系 2.2.1 调和函数的概念 2.2.4小结与思考 * 2.2.1 调和函数的定义 2.2.2 解析函数与调和函数的关系 2.2.3 由调和函数构造解析函数 2.2.4 小结与思考 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 定义2.3 如果二元实函数H(x,y)在区域D内有 二阶连续偏导数,且满足拉普拉斯方程，即： 则称H(x,y)为区域D内的调和函数。 注： 称为Laplace算子 例如： f(x,y)=x2-2xy2 不是一个调和函数 调和函数在流体力学和电磁场理论等实际问题中有很重要的应用. Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 设f(z)=u+iv在区域D内解析,则由C.-R.条件 得 同例,在D内有 即u及v都是D内的调和函数 2.2.2解析函数与调和函数的关系 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. v称为u在区域D内的共轭调和函数. 定理：设函数f(z)=u(x,y)+v(x,y)∈A(D) u(x,y),v(x,y)都是D内的调和函数 例如:设 f(z)=x-iy,则u(x,y),v(x,y)都是z平面上的调和函数,但f(z)=x-iy在z平面上处处不解析 原因: u(x,y),v(x,y)在D内不满足C-R条件 定义2.4 u(x,y),v(x,y)是D内的调和函数，且满足C.-R.条件： Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 定理2.4 若f(z)=u(x,y)+iv(x,y)在区域D内解析的充要条件是：在区域D内f(z)的虚部v(x,y)必为u(x,y)的共轭调和函数. 根据这个定理，便可利用一个调合函数和它的共轭调和函数作出一个解析函数。 由于共轭调和函数的这种关系,如果知道其中的一个,则可根据C-R条件求出另一个来。 Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0. Copyright 2004-2011 Aspose Pty Ltd. 例2.6 验证u(x,y)=x3—3xy2是z平面上的调和函数，并求以u(x,y)为实部的解析函数f(z),使合f(0)=i. 解： 要求f(z)，需先求v(x,y),一般可用以下方法求v(x,y) Evaluation only. Created with Aspos