Exercise 10.  Find the temperature function in the infinite strip  ,  

that satisfies the following boundary conditions (shown in Figure 11.30).   

                      
        
Hint.  Use  .

Solution 10.

See text and/or instructor's solution manual.

Answer.   .  

A Short Solution.   Map the infinite strip onto the upper half-plane with the function   ,  then multiply the boundary values in Example 11.17 by

and consider  ,   then construct  .  

Solution.   The transformation    maps the infinite strip    onto the upper half plane     where the boundary values are

                      

Using the result of Example 11.17, and multiply the solution by to get

                    .  

Therefore the solution in the infinite strip    

                    .  

We are done.   

        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  ,   

                      

We are really really done.   

        We can explore the intermediate function   .  

                    A contour graph of the intermediate function  ,

                    where     for   .

                    A graph of the intermediate function  .  

                      

This solution is complements of the authors.

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