Exercise 11.  Find the temperature function in the upper half-plane  ,  

that satisfies the following boundary conditions (shown in Figure 11.31).

                      

Solution 11.

See text and/or instructor's solution manual.

Answer.   Consider the mapping    and multiply the boundary values in Example 11.17 by  100  and construct

                    .  

A Short Solution.   Map the upper half-plane   onto the lower half-plane   with the function   .  

Notice that the function in Example 11.17 takes on the same boundary values in the lower half-plane, multiply this function by  100  

and consider  ,   then construct   .  

More details for the solution.   

For computational purposes we can use the formulas for the real and imaginary parts of  ,  that were derived in Section 10.4.

                    .  

In particular,

(10-26)         .  

If an explicit solution is required, then we can use Formula (10-26) in Section 10.4 and write  

                    ,    

where the function    has range values satisfying   .  

Now make the substitution      and  

                        and    .   

Thus,

                        

Therefore,

                    .  

We are really done.   

Aside.   We can graph the function   .   

                    A contour graph of the function   ,

                    where     for   .

                    A graph of the function   ,   

                      

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell