Exercise 9.  Find the temperature function in the first quadrant  ,   

that satisfies the following boundary conditions (shown in Figure 11.29).  

                      

Solution 9.

See text and/or instructor's solution manual.

Answer.   .  

A Short Solution.   Map the quadrant onto the upper half-plane with the function   ,  then multiply the boundary values in Example 11.17 by  75

and consider  ,   then construct  .  

A Longer Solution.   The transformation    maps the first quadrant  ,   

onto the upper half plane     where the boundary values are

                      

Use the result in Example 11.17 where we found that the function   

                        has the boundary values  

                      

We can easily obtain by multiplying    by   75  and adding  25,

                    .     

Now construct  

                       

Therefore,

                    .     

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  ,    

                      

This solution is complements of the authors.

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