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Poisson's Integral Formula for the Upper Half-Plane

 

11.3  Poisson's Integral Formula for the Upper Half-Plane

    The Dirichlet problem for the upper half-plane [Graphics:Images/PoissonIntegralMod_gr_1.gif]  is to find a function [Graphics:Images/PoissonIntegralMod_gr_2.gif] that is harmonic in the upper half plane and has the boundary values [Graphics:Images/PoissonIntegralMod_gr_3.gif],  where  [Graphics:Images/PoissonIntegralMod_gr_4.gif],  is a real-valued function of the real variable x.  An important method for solving this problem is our next result which is attributed to the French mathematician Siméon Poisson.

 

Theorem 11.3 (Poisson's Integral Formula).  Let U be a real-valued function that is piecewise continuous and bounded for all real t.  The function  

(11-12)        [Graphics:Images/PoissonIntegralMod_gr_5.gif]  

is  harmonic in the upper half plane [Graphics:Images/PoissonIntegralMod_gr_6.gif] and has the boundary values  

            [Graphics:Images/PoissonIntegralMod_gr_7.gif]  

wherever [Graphics:Images/PoissonIntegralMod_gr_8.gif] is continuous.

Proof.

Proof of Theorem 11.3 is in the book.
Complex Analysis for Mathematics and Engineering

 

Example 11.11.  Find the function [Graphics:Images/PoissonIntegralMod_gr_9.gif] that is harmonic in the upper half-plane [Graphics:Images/PoissonIntegralMod_gr_10.gif], which takes on the boundary values  

            [Graphics:Images/PoissonIntegralMod_gr_11.gif]  

[Graphics:Images/PoissonIntegralMod_gr_12.gif]

Solution. Using Equation (11-12), we obtain  

            [Graphics:Images/PoissonIntegralMod_gr_13.gif]

Using techniques from calculus we have the integration formula  [Graphics:Images/PoissonIntegralMod_gr_14.gif].  We obtain the solution as follows

            [Graphics:Images/PoissonIntegralMod_gr_15.gif]   

Explore Solution 11.11.

 

Extra Example 1.  Find the function [Graphics:Images/PoissonIntegralMod_gr_31.gif] that is harmonic in the upper half-plane [Graphics:Images/PoissonIntegralMod_gr_32.gif], which takes on the boundary values  

            [Graphics:Images/PoissonIntegralMod_gr_33.gif]  

[Graphics:Images/PoissonIntegralMod_gr_34.gif]

Explore Extra Solution 1.

 

Example 11.12.  Find the function [Graphics:Images/PoissonIntegralMod_gr_48.gif] that is harmonic in the upper half-plane [Graphics:Images/PoissonIntegralMod_gr_49.gif], which takes on the boundary values  

            [Graphics:Images/PoissonIntegralMod_gr_50.gif]  

[Graphics:Images/PoissonIntegralMod_gr_51.gif]

Solution.  Using Equation (11-12), we obtain

            [Graphics:Images/PoissonIntegralMod_gr_52.gif]  

Using techniques from calculus we have the integration formulas

            [Graphics:Images/PoissonIntegralMod_gr_53.gif],   and   [Graphics:Images/PoissonIntegralMod_gr_54.gif].  

We obtain the solution as follows

            [Graphics:Images/PoissonIntegralMod_gr_55.gif]  

The function [Graphics:Images/PoissonIntegralMod_gr_56.gif] is continuous in the upper half-plane, and on the boundary [Graphics:Images/PoissonIntegralMod_gr_57.gif], except at the discontinuities [Graphics:Images/PoissonIntegralMod_gr_58.gif] on the real axis.  The graph in Figure 11.14 shows this phenomenon.

Figure 11.14  The graph of [Graphics:Images/PoissonIntegralMod_gr_59.gif] with the boundary values [Graphics:Images/PoissonIntegralMod_gr_60.gif]

Explore Solution 11.12.

 

Example 11.13.  Use Poisson's Integral formula to find the harmonic function [Graphics:Images/PoissonIntegralMod_gr_77.gif] that is harmonic in the upper half-plane [Graphics:Images/PoissonIntegralMod_gr_78.gif], that takes on the boundary values  

            [Graphics:Images/PoissonIntegralMod_gr_79.gif]  

[Graphics:Images/PoissonIntegralMod_gr_80.gif]

Solution.  Using techniques from Section 11.2, we find that the function  

            [Graphics:Images/PoissonIntegralMod_gr_81.gif]  

is harmonic in the upper half-plane and has the boundary values  [Graphics:Images/PoissonIntegralMod_gr_82.gif]  [Graphics:Images/PoissonIntegralMod_gr_83.gif]  [Graphics:Images/PoissonIntegralMod_gr_84.gif]  This function can be added to the one in Example 11.12 to obtain the desired result:  

        [Graphics:Images/PoissonIntegralMod_gr_85.gif]  

Figure 11.15 shows the graph of [Graphics:Images/PoissonIntegralMod_gr_86.gif].  

Figure 11.15  The graph of [Graphics:Images/PoissonIntegralMod_gr_87.gif].

Explore Solution 11.13.

 

Extra Example 2.  Use Poisson's Integral formula to find the harmonic function [Graphics:Images/PoissonIntegralMod_gr_103.gif] that is harmonic in the upper half-plane [Graphics:Images/PoissonIntegralMod_gr_104.gif], that takes on the boundary values  

            [Graphics:Images/PoissonIntegralMod_gr_105.gif]  

[Graphics:Images/PoissonIntegralMod_gr_106.gif]

Explore Extra Solution 2.

 

Exercises for Section 11.3.  Poisson's Integral Formula for the Upper Half Plane

 

Library Research Experience for Undergraduates

Dirichlet Problem

Neumann Problem

Poisson Integral

 

 

 

 

A Companion Module is

Dirichlet Problem for the Disk

The Next Module is

Two-Dimensional Mathematical Models

 

 

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This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2012 John H. Mathews, Russell W. Howell